Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 23, 2006, 95–150

O. Krupková and P. Volný
DIFFERENTIAL EQUATIONS WITH CONSTRAINTS IN JET BUNDLES: LAGRANGIAN AND HAMILTONIAN SYSTEMS
(submitted by V.V. Lychagin)

Abstract. The paper is a survey of the theory of Lagrangian systems with non-holonomic constraints in jet bundles. The subject of the paper are systems of second-order ordinary and partial differential equations that arise as extremals of variational functionals in ﬁbered manifolds. A geometric setting for Euler-Lagrange and Hamilton equations, based on the concept of Lepage class is presented. A constraint is modeled in the underlying ﬁbered manifold as a ﬁbered submanifold endowed with a distribution (the canonical distribution). A constrained system is deﬁned by means of a Lepage class on the constraint submanifold. Constrained Euler-Lagrange equations and constrained Hamilton equations, and properties of the corresponding exterior differential systems, such as regularity, canonical form, or existence of a constraint Legendre transformation, are presented. The case of mechanics (ODEs) and ﬁeld theory (PDEs) are investigated separately, however, stress is put on a uniﬁed exposition, so that a direct comparison of results and formulas is at hand.

________________

2000 Mathematical Subject Classiﬁcation. primary: 35A30, secondary: 37J60, 49Q99.

Key words and phrases. Jet bundles, non-holonomic constraints, semiholonomic constraints, holonomic constraints, constrained Lagrangian systems, constrained Euler-Lagrange equations, Hamilton–De Donder equations, regularity of constrained systems, momenta, Hamiltonian, Legendre transformation.

Research supported by grants 201/06/0922 of the Czech Science Foundation, and MSM 6198959214 of the Czech Ministry of Education, Youth and Sports. O. Krupková also would like to acknowledge the hospitality of the Department of Mathematics, La Trobe University, Melbourne.

1. Introduction

Since the 30’s of the last century when the pioneer paper by Chetaev was published [4], the study of non-holonomic constrained systems has been of growing interest in mechanics, control theory and geometry. Namely during the past 15 years much effort has been devoted to developments of geometric methods and studies of geometric structures of non-holonomic mechanics; among the many contributions to the subject, let us mention here at least [3, 6, 9, 15, 17, 20, 26, 27, 28, 30, 31, 33, 34, 38], and references therein. Recently, several authors have started to study a more general situation of partial differential equations (ﬁeld theories) with constraints given by systems of ﬁrst-order partial differential equations [2, 22, 25, 37]. Since the geometric origin of these constraints is the same as in mechanics, it is natural also in this generalized situation to call such constraints “non-holonomic”.

The papers investigating non-holonomic systems differ in approaches, methods, geometric setting, kind of constraints studied, and many other aspects. Usually (and this is in no case speciﬁc for constrained systems), tools, structures and methods used in mechanics (i.e. ordinary differential equations) and ﬁeld theory (partial differential equations) are essentially different. The aim of this paper is to present foundations of a general geometric theory of non-holonomic systems as a part of the calculus of variations on ﬁbered manifolds. It is based on the theory of Lepage equivalents of Lagrangians (Krupka [10, 12]) and of dynamical forms (Krupková [14, 15, 16, 19]), and on study of exterior differential systems associated with variational equations (Krupková [14, 16, 19, 21]). It is also important to note that a constraint is modeled in the underlying ﬁbered manifold as a ﬁbered submanifold endowed with a distribution (called canonical distribution) [15, 22]. This structure plays a key role in studying the geometry of non-holonomic constrained systems, and represents a correct mathematical realization of the physical d’Alembert’s principle (that is ambiguous in case of velocity dependent constraints in mechanics, and completely unclear in ﬁeld theory). The setting of [15] and [22] brings a uniﬁed approach to mechanics and ﬁeld theory, both unconstrained and with constraints, and can be directly transferred to higher-order situation [18].

There are basically two different approaches to systems with constraints:

$∙$ constrained system is modeled as a modiﬁed unconstrained system, deﬁned on the same manifold as the unconstrained system (in mechanics this concerns so called “constraint forces” and dynamics governed by equations with Lagrange multipliers),

$∙$ constrained system is modeled as a system deﬁned on the constraint submanifold (dynamics are modeled by the so called “reduced equations”, without Lagrange multipliers).

As shown in [15] and [22], both these approaches are equivalent. In this paper, however, we prefer the latter one, since it is more geometrical, and enables us to study constrained systems by the same tools as unconstrained systems. We focus on variational systems, i.e. such that their dynamics are given by differential equations that arise as equations for extremals of Lagrangians (Euler–Lagrange equations). First, we recall basic facts on unconstrained Lagrangian systems and their associated Hamiltonian systems in jet bundles. Then we turn to the concept of non-holonomic constraint structure. Finally we study Lagrangian systems subjected to non-holonomic constraints, namely constrained Euler–Lagrange equations and constrained Hamilton equations, where we devote our attention to such problems as regularity of constrained systems, or existence of an appropriate “constraint Legendre transformation”. The cases of mechanics (ordinary differential equations) and ﬁeld theory (partial differential equations) are investigated separately, however, stress is put on a uniﬁed exposition, so that common features on one hand and differences on the other hand are transparent and their geometric origin becomes clear. We also tried to provide analogous results and formulas in such a way that the reader could compare them directly.

This work is basically a review paper, however, it contains also new, yet unpublished results (this concerns Sec. 4.5 and 4.6 on constrained Hamilton–De Donder equations and constraint Legendre transformation for general non-holonomic constraints in ﬁeld theory).

2. Calculus in jet bundles

We start with a brief introduction of notations, basic structures and the corresponding calculus to be used. For more details we refer to the books by Saunders [32] and Krupková [16], and the papers by Krupka [10, 12].

We consider a ﬁbered manifold $π : Y \to X$ with $dim X = n \geq 1$ , $dim Y = m + n$ , and its jet prolongations $π_{1} : J^{1} Y \to X$ and $π_{2} : J^{2} Y \to X$ . All manifolds and mappings are smooth, and the summation convention on repeated indices applies throughout.

A mapping $γ : U \to Y$ , where $U \subset X$ is an open set, is called a section of $π$ if $π \circ γ = {i d}_{U}$ . We denote by $J^{1} γ$ and $J^{2} γ$ the ﬁrst and the second jet prolongation of $γ$ , respectively. Note that $J^{1} γ$ (resp. $J^{2} γ$ ) is a section of $π_{1}$ (resp. $π_{2}$ ). A section $δ$ of $π_{1}$ is called holonomic if $δ = J^{1} γ$ for a section $γ$ of $π$ .

A vector ﬁeld $ξ$ on $Y$ is called $π$ -vertical if $T π \cdot ξ = 0$ , and $π$ -projectable if $T π \cdot ξ = ξ_{0} \circ π$ for a vector ﬁeld $ξ_{0}$ on $X$ . Considering the projections $π_{1} : J^{1} Y \to X$ , $π_{2} : J^{2} Y \to X$ , $π_{1, 0} : J^{1} Y \to Y$ , $π_{2, 1} : J^{2} Y \to J^{1} Y$ and $π_{2, 0} : J^{2} Y \to Y$ the concepts of the corresponding verticality and projectability are obtained quite similarly. For the module of vector ﬁelds (resp. $π_{r}$ -vertical vector ﬁelds) on $J^{r} Y$ , $r = 1, 2$ , we shall use the notation $X (J^{r} Y)$ (resp. $V (J^{r} Y)$ ).

Denote by $Ω^{q} (J^{1} Y)$ the module of $q$ -forms on $J^{1} Y$ . A form $η \in Ω^{q} (J^{1} Y)$ is called $π_{1}$ -horizontal (resp. $π_{1, 0}$ -horizontal) if $i_{ξ} η = 0$ for every $π_{1}$ -vertical (resp. $π_{1, 0}$ -vertical) vector ﬁeld $ξ$ on $J^{1} Y$ ; $η \in Ω^{q} (J^{1} Y)$ is called contact if $J^{1} γ^{*} η = 0$ for every section $γ$ of $π$ [10]. A contact form $η \in Ω^{q} (J^{1} Y)$ is called $1$ -contact if for every $π_{1}$ -vertical vector ﬁeld $ξ$ the form $i_{ξ} η$ is $π_{1}$ -horizontal; it is called $k$ -contact, where $2 \leq k \leq q$ , if for every $π_{1}$ -vertical vector ﬁeld $ξ$ the form $i_{ξ} η$ is $(k - 1)$ -contact [12]. We denote

$∙$ $Ω_{X}^{q} (J^{1} Y)$ the module of $π_{1}$ -horizontal $q$ -forms on $J^{1} Y$ ,

$∙$ $Ω^{q - i, i} (J^{1} Y)$ the module of $i$ -contact $q$ -forms on $J^{1} Y$ ,

$∙$ $Ω_{Y}^{q - i, i} (J^{1} Y)$ the submodule of $Ω^{q - i, i} (J^{1} Y)$ consisting of $π_{1, 0}$ -horizontal forms.

It is important to mention that every form $η \in Ω^{q} (J^{1} Y)$ has a unique decomposition into contact components as follows (Krupka [12]):

π_{2, 1}^{*} η = h η + p_{1} η + \dots + p_{q - 1} η + p_{q} η,

(2.1)

where $h$ and $p_{k}$ ( $k \geq 1$ ) denotes the horizontalization and $k$ -contactization operators, respectively, assigning to $η$ its horizontal (resp. $k$ -contact, $1 \leq k \leq q$ ) component.

Therefore, we shall also use the following notations:

$∙$ $Ω^{q (\geq k)} (J^{1} Y) = Ω^{q - k, k} (J^{1} Y) \oplus Ω^{q - k - 1, k + 1} (J^{1} Y) \oplus \dots \oplus Ω^{0, q} (J^{1} Y)$ , i.e. the module of $q$ -forms on $J^{1} Y$ that are at least $k$ -contact,

$∙$ $Ω_{Y}^{q (\geq k)} (J^{1} Y) = Ω_{Y}^{q - k, k} (J^{1} Y) \oplus Ω_{Y}^{q - k - 1, k + 1} (J^{1} Y) \oplus \dots \oplus Ω_{Y}^{0, q} (J^{1} Y)$ , i.e. $π_{1, 0}$ -horizontal $q$ -forms on $J^{1} Y$ that are at least $k$ -contact.

We denote by $(x^{i}, y^{σ})$ , where $1 \leq i \leq n$ , $1 \leq σ \leq m$ , local ﬁbered coordinates on $Y$ , and by $(x^{i}, y^{σ}, y_{j}^{σ})$ and $(x^{i}, y^{σ}, y_{j}^{σ}, y_{j k}^{σ})$ , where $1 \leq j \leq k \leq n$ , associated coordinates on $J^{1} Y$ and $J^{2} Y$ , respectively. We put

ω_{0} = d x^{1} \land \dots \land d x^{n}, ω_{j} = i_{\partial ∕ \partial x^{j}} ω_{0} .

(2.2)

In case that $dim X = 1$ , we write $(t, q^{σ})$ , where $1 \leq σ \leq m$ , to denote local ﬁbered coordinates on $Y$ , and $(t, q^{σ}, {\dot{q}}^{σ})$ (resp. $(t, q^{σ}, {\dot{q}}^{σ}, {\ddot{q}}^{σ})$ ) for associated coordinates on $J^{1} Y$ (resp. $J^{2} Y$ ).

In calculations we use either a canonical basis of one forms, i.e. $(d x^{i}, d y^{σ}, d y_{j}^{σ})$ on $J^{1} Y$ and $(d x^{i}, d y^{σ}, d y_{j}^{σ}, d y_{j l}^{σ})$ on $J^{2} Y$ (alternatively, if $dim X$ $= 1$ , $(d t, d q^{σ}, d {\dot{q}}^{σ})$ and $(d t, d q^{σ}, d {\dot{q}}^{σ}, d {\ddot{q}}^{σ})$ ), or better a basis adapted to the contact structure, i.e. $(d x^{i}, ω^{σ}, d y_{j}^{σ})$ on $J^{1} Y$ and $(d x^{i}, ω^{σ}, ω_{j}^{σ}, d y_{j l}^{σ})$ , on $J^{2} Y$ , where

ω^{σ} = d y^{σ} - y_{i}^{σ} d x^{i}, ω_{j}^{σ} = d y_{j}^{σ} - y_{j i}^{σ} d x^{i}

(2.3)

are local canonical contact $1$ -forms. Alternatively, if $dim X = 1$ , adapted bases take the form $(d t, ω^{σ}, d {\dot{q}}^{σ})$ and $(d t, ω^{σ}, {\dot{ω}}^{σ}, d {\ddot{q}}^{σ})$ , where

ω^{σ} = d q^{σ} - {\dot{q}}^{σ} d t, {\dot{ω}}^{σ} = d {\dot{q}}^{σ} - {\ddot{q}}^{σ} d t .

(2.4)

In an adapted basis to the contact structure every $k$ -contact component $p_{k} η$ of a $q$ -form $η$ (where $1 \leq k \leq q$ ) is expressed by means of a wedge product containing exactly $k$ of the canonical contact $1$ -forms above.

If $f$ is a function on $J^{1} Y$ , we have by (2.1) the exterior derivative $d f$ canonically splitted into the horizontal and contact component,

π_{2, 1}^{*} d f = h d f + p d f,

(2.5)

with

h d f = \frac{d f}{d x^{j}} d x^{j},

(2.6)

where $d ∕ d x^{j}$ , $1 \leq j \leq n$ , denotes the $j$ -th total derivative operator (also called $j$ -th formal derivative operator),

\frac{d}{d x^{j}} = \frac{\partial}{\partial x^{j}} + y_{j}^{σ} \frac{\partial}{\partial y^{σ}} + y_{i j}^{σ} \frac{\partial}{\partial y_{i}^{σ}} .

(2.7)

For convenience of notations we also use the ‘cut’ total derivative operators,

\frac{d^{'}}{d x^{j}} = \frac{\partial}{\partial x^{j}} + y_{j}^{σ} \frac{\partial}{\partial y^{σ}} = \frac{d}{d x^{j}} - y_{i j}^{σ} \frac{\partial}{\partial y_{i}^{σ}}, 1 \leq j \leq n .

(2.8)

If $dim X = 1$ , these formulas take the following form:

h d f = \frac{d f}{d t} d t,

(2.9)

\frac{d}{d t} = \frac{\partial}{\partial t} + {\dot{q}}^{σ} \frac{\partial}{\partial q^{σ}} + {\ddot{q}}^{σ} \frac{\partial}{\partial {\dot{q}}^{σ}},

(2.10)

\frac{d^{'}}{d t} = \frac{\partial}{\partial t} + {\dot{q}}^{σ} \frac{\partial}{\partial q^{σ}} = \frac{d}{d t} - {\ddot{q}}^{σ} \frac{\partial}{\partial {\dot{q}}^{σ}} .

(2.11)

Deﬁnition 2.1. $1$ -contact $(n + 1)$ -forms on $J^{r} Y$ , horizontal with respect to the projection $π_{r, 0}$ , are called dynamical forms of order $r$ [16].

Horizontal $n$ -forms on $J^{r} Y$ are called Lagrangians of order $r$ [10]. By a local Lagrangian (of order $r$ ) we shall mean a Lagrangian deﬁned on an open subset of $J^{r} Y$ .

Deﬁnition 2.2. [10] Let $λ$ be a Lagrangian on $J^{1} Y$ . An $n$ -form $ρ$ is called Lepage equivalent of $λ$ if $h ρ = λ$ and $p_{1} d ρ$ is a dynamical form. The form $p_{1} d ρ$ is then called the Euler–Lagrange form of $λ$ and denoted by $E_{λ}$ .

As proved in [12], every Lagrangian has a Lepage equivalent. For a Lagrangian of order $r$ Lepage equivalents are of order $2 r - 1$ , and the Euler–Lagrange form is of order $2 r$ . It should be stressed that while Lepage equivalent of a Lagrangian need not be unique, the Euler–Lagrange form always is unique.

3. Mechanical systems with constraints

Throughout this section we consider a ﬁbered manifold $π : Y \to X$ , $dim X = 1$ , and we assume $dim Y = m + 1$ , where $m \geq 1$ . Main sources for our exposition are the following: [1, 8, 11, 12, 35, 36] for the inverse variational problem, [15, 16, 21] for a geometric approach to variational ordinary differential equations, [7, 10, 12, 14, 16] for (unconstrained) Lagrangian and Hamiltonian mechanics in jet bundles, [15, 28] for the model of the non-holonomic constraint structure, and [3, 4, 6, 9, 15, 17, 20, 27, 28, 29, 30, 31, 33, 38] for non-holonomic Lagrangian and Hamiltonian systems.

3.1. Dynamical forms. Let $E$ be a dynamical form on $J^{2} Y$ . A section $γ$ of $π$ is called a path of $E$ if

E \circ J^{2} γ = 0 .

(3.1)

In ﬁbered coordinates $E$ reads

E = E_{σ} ω^{σ} \land d t,

(3.2)

where $E_{σ}$ are functions of $(t, q^{ν}, {\dot{q}}^{ν}, {\ddot{q}}^{ν})$ , and the equation for paths of $E$ takes the form of a system of $m$ second-order ordinary differential equations for the components $(γ^{ν})$ of $γ$ as follows:

E_{σ} (t, γ^{ν} (t), \frac{d γ^{ν}}{d t}, \frac{d^{2} γ^{ν}}{d t^{2}}) = 0, 1 \leq σ \leq m .

(3.3)

We stress that these equations need not be “solvable with respect to the second derivatives”, meaning that they need not be expressible in a normal form,

\frac{d^{2} γ^{σ}}{d t^{2}} = F^{σ} (t, γ^{ν} (t), \frac{d γ^{ν}}{d t}) .

(3.4)

Equations for paths of dynamical forms can be represented by means of exterior differential systems locally generated by $1$ -forms [14].

Proposition 3.1. Let $E$ be a dynamical form on $J^{2} Y$ . A section $γ$ of $π$ is a path of $E$ if and only if

J^{2} γ^{*} i_{ξ} α = 0 \forall ξ \in V (J^{2} Y),

(3.5)

where $α$ is any $2$ -form such that $p_{1} α = E$ .

Proof. By a direct computation we immediately obtain that (3.1) is equivalent with the condition $J^{2} γ^{*} i_{ξ} E = 0$ $\forall ξ \in V (J^{2} Y)$ . Now, since contraction by vertical vector ﬁelds is compatible with the decomposition of forms to contact components (2.1), and prolongations of sections annihilate contact forms, we can see that adding to $E$ (which is $1$ -contact) any $2$ -contact form $F$ gives us

J^{2} γ^{*} i_{ξ} (E + F) = J^{2} γ^{*} i_{ξ} E + J^{2} γ^{*} i_{ξ} F = J^{2} γ^{*} i_{ξ} E \forall ξ \in V (J^{2} Y) .

(3.6)

□

Deﬁnition 3.1. [15] Let $E \in Ω_{Y}^{2} (J^{2} Y)$ be a dynamical form. The equivalence class of $2$ -forms (on an open subset $U \subset J^{2} Y$ ) deﬁned by

α_{1} \sim α_{2} iff p_{1} α_{1} = p_{1} α_{2} = E ∣_{U}

(3.7)

is called Lepage class of $E$ on $U$ . The family of all local Lepage classes of $E$ will be referred to as Lepage class of $E$ and will be denoted by ${[α]}_{E}$ , or simply $[α]$ .

By the above proposition, the equation for paths of $E$ (on $U$ ) coincides with equations for holonomic integral sections of the distribution

Δ_{α} = a n n i h {i_{ξ} α ∣ \forall ξ \in V (J^{2} Y)} = s p a n {ζ \in X (J^{2} Y) ∣ i_{ζ} α = 0},

(3.8)

where $α$ is any representative of the Lepage class of $E$ (on $U$ ).

Deﬁnition 3.2. [15] Let $[α]$ be a Lepage class of $E$ . Every representative $α \in [α]$ is called a Hamiltonian system associated with $E$ . The distribution $Δ_{α}$ is called a dynamical distribution of $E$ . Equations for (all) integral sections of $Δ_{α}$ are called Hamilton equations associated with $E$ .

In what follows we shall be interested in dynamical forms that can be represented by ﬁrst-order Lepage classes. This means that the dynamics are described by dynamical distributions deﬁned on (open subsets of) $J^{1} Y$ .

Proposition 3.2. Let $E$ be a dynamical form on $J^{2} Y$ . The following conditions are equivalent:

Around each point in $J^{1} Y$ there exists a Lepage class of $E$ .
In every ﬁbered chart, $E$ takes the form $(3.2)$ , where the functions $E_{σ}$ are affine in the second derivatives, i.e.,
$E_{σ} = A_{σ} (t, q^{ν}, {\dot{q}}^{ν}) + B_{σ ρ} (t, q^{ν}, {\dot{q}}^{ν}) {\ddot{q}}^{ρ} .$ (3.9)

Proof. We have

\begin{matrix} α = E + F = E_{σ} ω^{σ} \land d t + F_{σ ν} ω^{σ} \land ω^{ν} + F_{σ ν}^{01} ω^{σ} \land {\dot{ω}}^{ν} + F_{σ ν}^{11} {\dot{ω}}^{σ} \land {\dot{ω}}^{ν} \\ = (E_{σ} - 2 F_{σ ν} {\dot{q}}^{ν} - F_{σ ν}^{01} {\ddot{q}}^{ν}) d q^{σ} \land d t + (F_{ν σ}^{01} {\dot{q}}^{σ} - 2 F_{σ ν}^{11} {\ddot{q}}^{ν}) d {\dot{q}}^{σ} \land d t \\ + F_{σ ν} d q^{σ} \land d q^{ν} + F_{σ ν}^{01} d q^{σ} \land d {\dot{q}}^{ν} + F_{σ ν}^{11} d {\dot{q}}^{σ} \land d {\dot{q}}^{ν} . \end{matrix}

(3.10)

Hence, $α$ is projectable onto an open subset of $J^{1} Y$ iff $F_{σ ν}$ and $F_{σ ν}^{01}$ do not depend on ${\ddot{q}}^{ρ}$ , $F_{σ ν}^{11} = 0$ , and

\frac{\partial E_{σ}}{\partial {\ddot{q}}^{ρ}} = F_{σ ρ}^{01};

(3.11)

consequently,

\frac{\partial^{2} E_{σ}}{\partial {\ddot{q}}^{ρ} \partial {\ddot{q}}^{ν}} = 0 .

(3.12)

The ﬁrst-order Lepage class is represented by $2$ -forms

α = A_{σ} ω^{σ} \land d t + B_{σ ν} ω^{σ} \land d {\dot{q}}^{ν} + F_{σ ν} ω^{σ} \land ω^{ν},

(3.13)

where $F_{σ ν}$ are arbitrary functions of $(t, q^{ρ}, {\dot{q}}^{ρ})$ . □

Deﬁnition 3.3. A dynamical form on $J^{2} Y$ that has a Lepage class around each point of $J^{1} Y$ is called $J^{1} Y$ -pertinent, or, a ﬁrst-order mechanical system.

Finally, we recall the concept of a regular dynamical form.

Deﬁnition 3.4. [14, 15] A ﬁrst-order mechanical system (respectively, a $J^{1} Y$ -pertinent dynamical form) $E$ is called regular if around each point of $J^{1} Y$ there exists a dynamical distribution $Δ_{α}$ , $α \in {[α]}_{E}$ such that $r a n k Δ_{α} = 1$ .

Proposition 3.3. [14, 15] The following conditions are equivalent:

A ﬁrst-order mechanical system $E$ is regular.
The following condition holds:
$det (\frac{\partial E_{σ}}{\partial {\ddot{q}}^{ν}}) = det (B_{σ ν}) \neq 0 .$ (3.14)
Equations for paths of $E$ have an equivalent normal form $(3.4)$ where $F^{σ} = - B^{σ ν} A_{ν}$ .

3.2. Variational ODE’s and related Hamiltonian systems. A dynamical form $E \in Ω_{Y}^{2} (J^{2} Y)$ is called (globally) variational if there exists a Lagrangian $λ$ such that (possibly up to a projection), $E = E_{λ}$ . $E$ is called locally variational if it is variational in a neighborhood of every point in $J^{2} Y$ [11, 12]. In ﬁbered coordinates this means that the components $E_{σ}$ of $E$ take the form of Euler–Lagrange expressions of $λ = L d t$ , i.e.

E_{σ} = \frac{\partial L}{\partial q^{σ}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}^{σ}}

(3.15)

if $λ$ is a ﬁrst order Lagrangian.

It is known that a locally variational form need not be globally variational [35]: obstructions come from the topology of the manifold $Y$ . Every (globally) variational form on $J^{2} Y$ possesses a global second-order Lagrangian. This Lagrangian is locally equivalent with ﬁrst-order Lagrangians (we say that it can be locally reduced to ﬁrst-order Lagrangians).

A dynamical form $E \in Ω_{Y}^{2} (J^{2} Y)$ is locally variational if and only if its components $E_{σ}$ , $1 \leq σ \leq m$ , satisfy the Helmholtz conditions [8]

\begin{matrix} \frac{\partial E_{σ}}{\partial {\ddot{q}}^{ν}} - \frac{\partial E_{ν}}{\partial {\ddot{q}}^{σ}} & = & 0, \\ \frac{\partial E_{σ}}{\partial {\dot{q}}^{ν}} + \frac{\partial E_{ν}}{\partial {\dot{q}}^{σ}} - 2 \frac{d}{d t} \frac{\partial E_{ν}}{\partial {\ddot{q}}^{σ}} & = & 0, \\ \frac{\partial E_{σ}}{\partial q^{ν}} - \frac{\partial E_{ν}}{\partial q^{σ}} + \frac{d}{d t} \frac{\partial E_{ν}}{\partial {\dot{q}}^{σ}} - \frac{d^{2}}{d t^{2}} \frac{\partial E_{ν}}{\partial {\ddot{q}}^{σ}} & = & 0 . \end{matrix}

(3.16)

Local (second-order) Lagrangians then can be constructed using the following formula [36]

L = q^{σ} \int_{0}^{1} E_{σ} (t, u q^{ν}, u {\dot{q}}^{ν}, u {\ddot{q}}^{ν}) d u .

(3.17)

Notice that from the Helmholtz conditions one easily gets that every second-order locally variational form is $J^{1} Y$ -pertinent, i.e., deﬁnes a ﬁrst-order mechanical system; it is called a ﬁrst-order Lagrangian system.

Recall that every representative $α$ of the Lepage class of $E$ is called a Hamiltonian system associated with $E$ .

As shown in [10], every ﬁrst-order Lagrangian $λ$ has a unique ﬁrst-order Lepage equivalent, the Cartan form, denoted by $Θ_{λ}$ . Consequently, local ﬁrst-order Lepage classes of $E$ are represented by $2$ -forms

α = d Θ_{λ} + F,

(3.18)

where $F$ is an arbitrary $2$ -contact form on the domain of deﬁnition of $d Θ_{λ}$ . Hence, if $λ$ is (any) Lagrangian for $E$ on $U \subset J^{1} Y$ , the Lepage class of $E$ on $U$ is given by $[α] ∣_{U} = [d Θ_{λ}]$ . To simplify notations, we write with an obvious inaccuracy,

[α] = [d Θ_{λ}] = d Θ_{λ} mod Ω_{Y}^{0, 2} (J^{1} Y) .

(3.19)

Moreover, we have the following stronger result:

Theorem 3.1. (Krupková [14, 15]). Every ﬁrst-order Lepage class of $E$ has a unique closed representative, deﬁned on $J^{1} Y$ .

The unique closed $2$ -form mentioned above is denoted by $α_{E}$ and called the Lepage equivalent of $E$ [14]. If $λ$ is a Lagrangian for $E$ (possibly local, of order $r \geq 1$ ) then (up to a projection) $α_{E} ∣_{U} = d Θ_{λ}$ ; here $U$ denotes the domain of deﬁnition of $d Θ_{λ}$ .

In ﬁbered coordinates, where $E = E_{σ} ω^{σ} \land d t$ and $λ = L d t$ , we have

α_{E} = E_{σ} ω^{σ} \land d t + \frac{1}{2} \frac{\partial E_{σ}}{\partial {\dot{q}}^{ν}} ω^{σ} \land ω^{ν} + \frac{\partial E_{σ}}{\partial {\ddot{q}}^{ν}} ω^{σ} \land {\dot{ω}}^{ν},

(3.20)

Θ_{λ} = L d t + \frac{\partial L}{\partial {\dot{q}}^{σ}} ω^{σ},

(3.21)

and, on the domain of deﬁnition of $λ$ ,

E = E_{λ} = p_{1} d Θ_{λ} = E_{σ} d q^{σ} \land d t where E_{σ} = \frac{\partial L}{\partial q^{σ}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}^{σ}} .

(3.22)

Since the functions $E_{σ}$ are affine in the $\ddot{q}$ ’s, we write

E_{σ} = A_{σ} + B_{σ ν} {\ddot{q}}^{ν},

(3.23)

where $A_{σ}$ and $B_{σ ν}$ are functions of $(t, q^{σ}, {\dot{q}}^{σ})$ ,

B_{σ ν} = - \frac{\partial^{2} L}{\partial {\dot{q}}^{σ} \partial {\dot{q}}^{ν}}, A_{σ} = \frac{\partial L}{\partial q^{σ}} - \frac{\partial^{2} L}{\partial t \partial {\dot{q}}^{σ}} - \frac{\partial^{2} L}{\partial q^{ν} \partial {\dot{q}}^{σ}} {\dot{q}}^{ν} .

(3.24)

Remark 3.1. In what follows we shall always assume that $E$ is deﬁned on $J^{2} Y$ and is everywhere nontrivially of order $2$ . This means that $(B_{σ ν})$ in (3.23) is everywhere a non-zero matrix, or, equivalently, for every Lagrangian $λ$ the Cartan $2$ -form $d Θ_{λ}$ is everywhere nontrivially of order one.

Deﬁnition 3.5. Paths of a locally variational form are called extremals. Equations for paths of a locally variational form (respectively, equations for holonomic integral sections of associated dynamical distributions) are called Euler–Lagrange equations. Equations for integral sections of the dynamical distributions are called Hamilton equations, their integral sections are then called Hamilton extremals. The dynamical distribution $Δ_{α_{E}}$ is called the Euler–Lagrange distribution.

Note that locally for every Lagrangian $λ$ of $E$ , $Δ_{α_{E}} = Δ_{d Θ_{λ}}$ .

A principal question in the theory of Lagrangian systems on ﬁbered manifolds is the relationship between Hamilton equations on one side and Euler–Lagrange equations on the other side [13]. It is clear that every extremal prolonged to $J^{1} Y$ is a Hamilton extremal. The converse, however need not hold: a Hamilton extremal need not be a solution of the Euler–Lagrange equations.

The problem of equivalence between the set of extremals and Hamilton extremals is solved by the following theorem.

Theorem 3.2. (Krupková [15]). If $E$ is regular then the dynamical distributions $Δ_{α}$ , where $α$ belongs to the ﬁrst-order Lepage class of $E$ , coincide on the common domain of deﬁnition (and their rank equals to $1$ ). Consequently, if $E$ is regular then for every $α \in {[α]}_{E}$ the Hamilton equations are equivalent with the Euler–Lagrange equations.

Notice that if $E$ is regular then every dynamical distribution is locally spanned by the following semispray:

ζ = \frac{\partial}{\partial t} + {\dot{q}}^{σ} \frac{\partial}{\partial q^{σ}} - B^{σ ρ} A_{ρ} \frac{\partial}{\partial {\dot{q}}^{σ}},

(3.25)

where $(B^{σ ρ})$ is the inverse matrix to $(B_{ρ ν})$ .

Deﬁnition 3.6. A Lagrangian $λ$ is called regular if its Euler–Lagrange form $E_{λ}$ is regular [14].

Proposition 3.3 and Theorem 3.1 easily imply that we have the following equivalent characterizations of a regular Lagrangian:

$r a n k Δ_{d Θ_{λ}} = c o r a n k d Θ_{λ} = 1$ .
If $λ$ is a ﬁrst-order Lagrangian,
$det (\frac{\partial^{2} L}{\partial {\dot{q}}^{σ} \partial {\dot{q}}^{ν}}) \neq 0 .$ (3.26)

In view of Theorem 3.1 and a theorem on a canonical form of the Lepage equivalent of $E$ ([14]) one obtains the following result:

Proposition 3.4. Let $E$ be regular. Then in a neighborhood of every point in $J^{1} Y$ there is a local coordinate transformation $(t, q^{σ}, {\dot{q}}^{σ}) \to (t, q^{σ}, p_{σ})$ such that every $α$ belonging to the ﬁrst-order Lepage class of $E$ takes the canonical form

α = - d H \land d t + d p_{σ} \land d q^{σ} + F,

(3.27)

where $F$ is a $2$ -contact $π_{1, 0}$ -horizontal $2$ -form.

This transformation is called Legendre transformation.

Functions $H$ and $p_{σ}$ above can be expressed in terms of a ﬁrst-order Lagrangian $λ$ for $E$ ; it holds

p_{σ} = \frac{\partial L}{\partial {\dot{q}}^{σ}}, H = - L + p_{σ} {\dot{q}}^{σ} .

(3.28)

In Legendre coordinates Hamilton equations take the “canonical form”

\frac{d (p_{σ} \circ δ)}{d t} = - \frac{\partial H}{\partial q^{σ}}, \frac{d (q^{σ} \circ δ)}{d t} = \frac{\partial H}{\partial p_{σ}} .

(3.29)

Summarizing, for a regular Lagrangian system (represented by a regular locally variational form $E$ on $J^{2} Y$ ), all related Hamiltonian systems are (locally) equivalent and Hamilton equations are equivalent with Euler–Lagrange equations. Hamilton extremals coincide with prolongations of extremals, and are solutions of the canonical equations (3.29).

3.3. Nonholonomic constraints. Let us introduce the non-holonomic constraint structure in $J^{1} Y$ , as deﬁned in [15].

Deﬁnition 3.7. By a constraint submanifold or a non-holonomic constraint in $J^{1} Y$ we shall understand a submanifold $Q \subset J^{1} Y$ , ﬁbered over $Y$ , precisely speaking, a surjective submersion $π_{1, 0} ∣_{Q} : Q \to Y$ .

We denote by $k$ the codimension of $Q$ and assume that $1 \leq k \leq m - 1$ .

A nonholonomic constraint $Q$ in $J^{1} Y$ of codimension $k$ can be locally expressed by equations

f^{a} (t, q^{σ}, {\dot{q}}^{σ}) = 0, 1 \leq a \leq k,

(3.30)

where

r a n k (\frac{\partial f^{a}}{\partial {\dot{q}}^{σ}}) = k,

(3.31)

or, equivalently, by equations in a normal form,

{\dot{q}}^{m - k + a} - g^{a} (t, q^{σ}, {\dot{q}}^{1}, \dots, {\dot{q}}^{m - k}) = 0, 1 \leq a \leq k .

(3.32)

A section $γ$ of $π$ deﬁned on an open set $W \subset X$ is called a holonomic path in $Q$ if for every $x \in W$

J^{1} γ (x) \in Q .

(3.33)

Given a constraint submanifold $Q$ in $J^{1} Y$ there naturally arise the following local distributions, deﬁned on the domain $U$ of deﬁnition of the functions $f^{a}$ :

(1) $D_{U} = a n n i h {d f^{a}, 1 \leq a \leq k}$ ; $r a n k D_{U}$ is constant on $U$ due to (3.31) and equal to $2 m + 1 - k$ .

(2) ${\tilde{C}}_{U} = a n n i h {φ^{a}, 1 \leq a \leq k}$ , where

φ^{a} = f^{a} d t + \frac{\partial f^{a}}{\partial {\dot{q}}^{σ}} ω^{σ}, 1 \leq a \leq k .

(3.34)

(3) $C_{U} = a n n i h {φ^{a}, d f^{a}, 1 \leq a \leq k}$ .

${\tilde{C}}_{U}$ is called extended constraint distribution; it has a constant rank equal to $2 m + 1 - k$ . $C_{U}$ is called constraint distribution related to the constraint submanifold $Q$ on $U$ , and its rank equals to $2 m + 1 - 2 k$ .

The following assertions hold ([15]):

Proposition 3.5. $Q \cap U$ is an integral submanifold of $D_{U}$ . For every point $x \in Q$ , the forms $d f^{a} (x), 1 \leq a \leq k$ , annihilate the tangent space $T_{x} Q$ to the manifold $Q$ at $x$ , i.e. along $Q$ , $D = a n n i h {d f^{a}, 1 \leq a \leq k} = T Q$ .

Corollary 3.1. Let $Q$ be a non-holonomic constraint of codimension $k$ in $J^{1} Y$ , and let $f^{a} = 0$ and ${\bar{f}}^{a} = 0$ , where $1 \leq a \leq k$ , be two sets of equations of $Q$ on an open set $U \subset J^{1} Y$ . Then there are functions $γ_{j}^{a}$ on $U$ such that at each point of $U$ , $(γ_{j}^{a})$ is a regular matrix, and $d {\bar{f}}^{a} = γ_{j}^{a} d f^{j}$ . In particular, at each point $x \in Q \cap U$ ,

\frac{\partial {\bar{f}}^{a}}{\partial {\dot{q}}^{σ}} = γ_{j}^{a} \frac{\partial f^{j}}{\partial {\dot{q}}^{σ}} .

(3.35)

Proposition 3.6. $C_{U}$ is a subdistribution of both ${\tilde{C}}_{U}$ and $D_{U}$ . At the points of $Q \cap U$ , the distributions $C_{U}$ and ${\tilde{C}}_{U} \cap D$ coincide, and deﬁne a distribution of corank $k$ on $Q \cap U$ .

Theorem 3.3. Let $Q$ be a non-holonomic constraint in $J^{1} Y$ , let $ι : Q \to J^{1} Y$ be the canonical embedding of the submanifold $Q$ into $J^{1} Y$ . Put

{\bar{ϕ}}^{a} = ι^{*} φ^{a}, 1 \leq a \leq k .

(3.36)

Then

C = a n n i h {{\bar{ϕ}}^{a}, 1 \leq a \leq k}

(3.37)

is a distribution of corank $k$ on $Q$ .

Note that in ﬁbered coordinates

{\bar{ϕ}}^{a} = (\frac{\partial f^{a}}{\partial {\dot{q}}^{σ}} \circ ι) {\bar{ω}}^{σ} = {\bar{ω}}^{m - k + a} - \frac{\partial g^{a}}{\partial {\dot{q}}^{l}} {\bar{ω}}^{l}

(3.38)

where we have denoted

{\bar{ω}}^{σ} = ι^{*} ω^{σ} = d q^{σ} - ({\dot{q}}^{σ} \circ ι) d t,

(3.39)

i.e.,

\begin{matrix} {\bar{ω}}^{l} = d q^{l} - {\dot{q}}^{l} d t, 1 \leq l \leq m - k, \\ {\bar{ω}}^{m - k + a} = d q^{m - k + a} - g^{a} d t, 1 \leq a \leq k . \end{matrix}

(3.40)

Deﬁnition 3.8. [15] The distribution $C$ (3.37) on $Q$ is called canonical distribution. $1$ -forms belonging to the annihilator $C^{0}$ of $C$ , are called canonical constraint $1$ -forms. The ideal in the exterior algebra of differential forms on $Q$ generated by $C^{0}$ is called canonical constraint ideal, and denoted by $ℐ (C^{0})$ ; its homogeneous component of degree $p$ is denoted by $ℐ^{p} (C^{0})$ . Elements of the ideal $ℐ (C^{0})$ are called canonical constraint forms.

One can show by a direct computation that the canonical distribution can be equivalently locally spanned by the following system of vector ﬁelds:

\begin{matrix} \frac{\partial_{c}}{\partial t} \equiv \frac{\partial}{\partial t} + \sum_{a = 1}^{k} (g^{a} - \frac{\partial g^{a}}{\partial {\dot{q}}^{l}} {\dot{q}}^{l}) \frac{\partial}{\partial q^{m - k + a}}, \\ \frac{\partial_{c}}{\partial q^{s}} \equiv \frac{\partial}{\partial q^{s}} + \sum_{a = 1}^{k} \frac{\partial g^{a}}{\partial {\dot{q}}^{s}} \frac{\partial}{\partial q^{m - k + a}}, 1 \leq s \leq m - k, \\ \frac{\partial}{\partial {\dot{q}}^{s}}, 1 \leq s \leq m - k . \end{matrix}

(3.41)

For the sake of simplicity we shall also use the following notations:

\begin{matrix} \frac{d_{c}}{d t} = \frac{\partial}{\partial t} + {\dot{q}}^{l} \frac{\partial}{\partial q^{l}} + g^{a} \frac{\partial}{\partial q^{m - k + a}} + {\ddot{q}}^{l} \frac{\partial}{\partial {\dot{q}}^{l}} = \frac{\partial_{c}}{\partial t} + {\dot{q}}^{l} \frac{\partial_{c}}{\partial q^{l}} + {\ddot{q}}^{l} \frac{\partial}{\partial {\dot{q}}^{l}}, \\ \frac{d_{c}^{'}}{d t} = \frac{\partial}{\partial t} + {\dot{q}}^{l} \frac{\partial}{\partial q^{l}} + g^{a} \frac{\partial}{\partial q^{m - k + a}} = \frac{\partial_{c}}{\partial t} + {\dot{q}}^{l} \frac{\partial_{c}}{\partial q^{l}} \\ = \frac{d_{c}}{d t} - {\ddot{q}}^{l} \frac{\partial}{\partial {\dot{q}}^{l}} = \frac{d^{'}}{d t} \circ ι . \end{matrix}

(3.42)

In general, the canonical distribution is not completely integrable. There are two interesting particular cases of non-holonomic constraints as follows:

Deﬁnition 3.9. [15] A non-holonomic constraint $Q$ is called

(1) simple if the canonical distribution $C$ is projectable onto a distribution on $Y$ ,

(2) semiholonomic if the canonical distribution $C$ is completely integrable.

It can be proved [15] that every semiholonomic constraint is simple. Consequently it can be equivalently modeled as either

$∙$ a ﬁbered submanifold $Q \subset J^{1} Y$ with the canonical distribution $C$ completely integrable (i.e., the canonical constraint ideal $ℐ (C^{0})$ closed), or

$∙$ a completely integrable, nowhere vertical distribution on the total space $Y$ .

Another result shows [15] that a non-holonomic constraint is simple if and only if it is locally deﬁned by equations affine in velocities. Consequently, a simple non-holonomic constraint can be equivalently modeled as either

$∙$ a ﬁbered submanifold $Q \subset J^{1} Y$ with the canonical distribution $C$ projectable onto a distribution on $Y$ , or

$∙$ a nowhere vertical distribution on $Y$ , which need not be completely integrable.

3.4. Constrained Lagrangian systems. Let us consider a Lagrangian system on $J^{1} Y$ . Recall that it is deﬁned by a locally variational form $E$ on $J^{2} Y$ , as the ﬁrst-order Lepage class $[α]$ of $E$ (see (3.19))

[α] = [d Θ_{λ}] = d Θ_{λ} mod Ω_{Y}^{0, 2} (J^{1} Y) .

(3.43)

If $ι : Q \to J^{1} Y$ is a non-holonomic constraint and $ℐ (C^{0})$ the corresponding canonical constraint ideal, we have another equivalence, denoted by $\approx$ , on $2$ -forms on $Q$ (with the same domain of deﬁnition):

η_{1} \approx η_{2} iff η_{1} - η_{2} = \bar{F} + ϕ,

(3.44)

where $\bar{F}$ is a (local) $2$ -contact $2$ -form on $Q$ , and $ϕ$ is a constraint $2$ -form. We denote by $[[η]]$ the class of $η$ . If $[α]$ is a Lepage class on $J^{1} Y$ associated with a locally variational form $E \in Ω^{2} (J^{2} Y)$ then for any of its two elements deﬁned on the same subset of $J^{1} Y$ ,

α_{1} \sim α_{2} \Rightarrow ι^{*} α_{1} \approx ι^{*} α_{2} .

(3.45)

Deﬁnition 3.10. [15, 20] Let $[α] = [d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ . By the associated constrained Lagrangian system we mean the class $[[ι^{*} α]] = [[ι^{*} d Θ_{λ}]]$ . Each form $ι^{*} d Θ_{λ} + ϕ$ , where $ϕ \in ℐ^{2} (C^{0})$ , is called constrained Cartan $2$ -form of $λ$ .

Note that every element of $[[ι^{*} d Θ_{λ}]]$ is of the form

\bar{α} = ι^{*} d Θ_{λ} + \bar{F} + ϕ,

(3.46)

where $\bar{F} \in Ω_{Y}^{0, 2} (Q)$ and $ϕ \in ℐ^{2} (C^{0})$ .

In ﬁbered coordinates, where $Q$ is given by (3.32) and the Euler–Lagrange form of $λ$ is represented by (3.23), (3.24), we have [15, 20]

\bar{α} = {\bar{A}}_{l} {\bar{ω}}^{l} \land d t + {\bar{B}}_{l s} {\bar{ω}}^{l} \land d {\dot{q}}^{s} + {\bar{F}}_{l s} {\bar{ω}}^{l} \land {\bar{ω}}^{s} + ϕ,

(3.47)

where ${\bar{ω}}^{l} = ι^{*} ω^{l}$ , $ϕ \in ℐ^{2} (C^{0})$ , ${\bar{F}}_{l s}$ are arbitrary, and

\begin{matrix} {\bar{A}}_{l} = (A_{l} + A_{m - k + i} \frac{\partial g^{i}}{\partial {\dot{q}}^{l}} + (B_{l, m - k + i} + B_{m - k + j, m - k + i} \frac{\partial g^{j}}{\partial {\dot{q}}^{l}}) \frac{d^{'} g^{i}}{d t}) \circ ι, \\ {\bar{B}}_{l s} = (B_{l s} + B_{l, m - k + i} \frac{\partial g^{i}}{\partial {\dot{q}}^{s}} + B_{s, m - k + i} \frac{\partial g^{i}}{\partial {\dot{q}}^{l}} + B_{m - k + i, m - k + j} \frac{\partial g^{i}}{\partial {\dot{q}}^{l}} \frac{\partial g^{j}}{\partial {\dot{q}}^{s}}) \circ ι, \end{matrix}

(3.48)

and summations run over $l, s = 1, 2, \dots, m - k$ and $i, j = 1, 2, \dots, k$ . Since $(B_{σ ν})$ is a symmetric matrix, the above formula gives us that the matrix $({\bar{B}}_{l s})$ is symmetric.

Deﬁnition 3.11. [15] The constraint dynamical distribution related with a $2$ -form $\bar{α}$ , denoted by $Δ_{\bar{α}}$ , is deﬁned to be the subdistribution of the canonical distribution $C$ , annihilated by the $1$ -forms $i_{ξ} \bar{α}$ , where $ξ$ runs over all $π_{1}$ -vertical vector ﬁelds on $Q$ belonging to $C$ . This means that

Δ_{\bar{α}} = a n n i h {{\bar{ϕ}}^{a}, 1 \leq a \leq k, i_{ξ} \bar{α}, \forall ξ \in V (J^{1} Y) \cap C} .

(3.49)

In particular, the constraint dynamical distribution related with a constrained Cartan 2-form is called constraint Euler–Lagrange distribution.

Deﬁnition 3.12. [15, 20] Let $[[ι^{*} d Θ_{λ}]]$ be a constrained Lagrangian system. Then for any representative $\bar{α}$ of the class $[[ι^{*} d Θ_{λ}]]$ , equations for holonomic integral sections of the constraint dynamical distribution $Δ_{\bar{α}}$ , i.e., the equations

J^{1} γ^{*} i_{ξ} \bar{α} = 0 for every π_{1} -vertical vector ﬁeld ξ \in C,

(3.50)

where $γ : W \to Y$ , $J^{1} γ (W) \subset Q$ , are called constrained Euler–Lagrange equations. Solutions of constrained Euler–Lagrange equations are called constrained extremals.

We note that (locally) constrained Euler–Lagrange equations do not depend upon the choice of a representative $\bar{α}$ of the class $[[ι^{*} d Θ_{λ}]]$ . This means that with help of a (local, possibly higher-order) Lagrangian $λ$ for $E$ we can write the constrained Euler–Lagrange equations in the form

J^{1} γ^{*} i_{ξ} (ι^{*} d Θ_{λ}) = 0 for every π_{1} -vertical vector ﬁeld ξ \in C,

(3.51)

where $γ : W \to Y$ , $J^{1} γ (W) \subset Q$ .

For $λ = L d t$ denote

\bar{L} = L \circ ι, {\bar{L}}_{a} = \frac{\partial L}{\partial {\dot{q}}^{m - k + a}} \circ ι, 1 \leq a \leq k,

(3.52)

and

Θ_{ι^{*} λ} = \bar{L} d t + \sum_{s = 1}^{m - k} \frac{\partial \bar{L}}{\partial {\dot{q}}^{s}} {\bar{ω}}^{s} .

(3.53)

We get the following relation between the forms $ι^{*} Θ_{λ}$ and $Θ_{ι^{*} λ}$ :

Proposition 3.7. [20]

ι^{*} Θ_{λ} = Θ_{ι^{*} λ} + {\bar{L}}_{a} {\bar{ϕ}}^{a} .

(3.54)

For convenience we shall use the notation

μ_{s} = \frac{\partial_{c}}{\partial q^{s}} - \frac{d_{c}}{d t} \frac{\partial}{\partial {\dot{q}}^{s}}, μ_{s}^{'} = \frac{\partial_{c}}{\partial q^{s}} - \frac{d_{c}^{'}}{d t} \frac{\partial}{\partial {\dot{q}}^{s}}

(3.55)

for the so called $C$ -modiﬁed Euler–Lagrange operator and cut $C$ -modiﬁed Euler–Lagrange operator, respectively.

In ﬁbered coordinates constrained Euler–Lagrange equations take the form a mixed system of $m - k$ second-order and $k$ ﬁrst-order ODE’s for sections $γ$ of $π$ as follows:

Theorem 3.4. [15, 20] Let $[d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ , $Q \subset J^{1} Y$ a non-holonomic constraint. A section $γ : W \to Y$ of $π$ is a constrained extremal if and only if it satisﬁes $J^{1} γ (W) \subset Q$ , i.e.

f^{a} \circ J^{1} γ = 0, 1 \leq a \leq k,

(3.56)

and the constrained Euler–Lagrange equations (3.51) . The latter take one of the following equivalent coordinate forms:

By means of $L$ ,
$({\bar{A}}_{l} + {\bar{B}}_{l s} {\ddot{q}}^{s}) \circ J^{2} γ = 0, 1 \leq l \leq m - k,$ (3.57)

where ${\bar{A}}_{l}$ , ${\bar{B}}_{l s}$ are given by (3.24), (3.48).
By means of $\bar{L}$ and ${\bar{L}}_{a}$ ,
$(μ_{l} (\bar{L}) - {\bar{L}}_{a} μ_{l} (g^{a})) \circ J^{2} γ = 0, 1 \leq l \leq m - k .$ (3.58)

Consequently, the functions ${\bar{A}}_{l}$ , ${\bar{B}}_{l s}$ are equivalently expressed as follows:

{\bar{A}}_{l} = μ_{l}^{'} (\bar{L}) - {\bar{L}}_{a} μ_{l}^{'} (g^{a}), {\bar{B}}_{l s} = - \frac{\partial^{2} \bar{L}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}} + {\bar{L}}_{a} \frac{\partial^{2} g^{a}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}} .

(3.59)

Deﬁnition 3.13. The operator

ℰ_{s}^{C} (\bar{L}, {\bar{L}}_{a}) = μ_{s} (\bar{L}) - {\bar{L}}_{a} μ_{s} (g^{a}), 1 \leq s \leq m - k

(3.60)

is called the constraint Euler–Lagrange operator.

The deﬁnition of a regular constrained system is quite similar to the unconstrained case.

Deﬁnition 3.14. [15] A constrained Lagrangian system $[[ι^{*} d Θ_{λ}]]$ is called regular if around each point of $Q$ there exists a constraint dynamical distribution $Δ_{\bar{α}}$ such that $r a n k Δ_{\bar{α}} = 1$ .

Regular constrained systems are characterized as follows (see [15, 38]):

Theorem 3.5. Let $[[ι^{*} d Θ_{λ}]]$ be a constrained Lagrangian system. The following conditions are equivalent:

$[[ι^{*} d Θ_{λ}]]$ is regular.
The $(m - k) \times (m - k)$ -matrix $({\bar{B}}_{s l})$ is regular, i.e.,
$det ({\bar{B}}_{s l}) \neq 0 .$ (3.61)

Every ﬁrst-order Lagrangian

λ = L d t

satisﬁes the regularity condition

\begin{matrix} det ((\frac{\partial^{2} L}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}} & + & \frac{\partial^{2} L}{\partial {\dot{q}}^{m - k + a} \partial {\dot{q}}^{s}} \frac{\partial g^{a}}{\partial {\dot{q}}^{l}} + \frac{\partial^{2} L}{\partial {\dot{q}}^{m - k + a} \partial {\dot{q}}^{l}} \frac{\partial g^{a}}{\partial {\dot{q}}^{s}} \\ + & \frac{\partial^{2} L}{\partial {\dot{q}}^{m - k + a} \partial {\dot{q}}^{m - k + b}} \frac{\partial g^{a}}{\partial {\dot{q}}^{l}} \frac{\partial g^{b}}{\partial {\dot{q}}^{s}}) \circ ι) \neq 0 . \end{matrix}

(3.62)

Every ﬁrst-order Lagrangian $λ = L d t$ satisﬁes the regularity condition
$det (\frac{\partial^{2} \bar{L}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}} - (\frac{\partial L}{\partial {\dot{q}}^{m - k + a}} \circ ι) \frac{\partial^{2} g^{a}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}}) \neq 0,$ (3.63)

where $\bar{L} = L \circ ι$ .
Every constraint dynamical distribution is locally spanned by the following constraint semispray:
$ζ = \frac{\partial}{\partial t} + \sum_{l = 1}^{m - k} {\dot{q}}^{l} \frac{\partial}{\partial q^{l}} + \sum_{a = 1}^{k} g^{a} \frac{\partial}{\partial q^{m - k + a}} - \sum_{l, s = 1}^{m - k} {\bar{B}}^{l s} {\bar{A}}_{s} \frac{\partial}{\partial {\dot{q}}^{l}},$ (3.64)

where $({\bar{B}}^{l s})$ is the inverse matrix to $({\bar{B}}_{l s})$ .
The constrained Euler–Lagrange equations have an equivalent form
$\begin{matrix} {\dot{q}}^{m - k + a} & = & g^{a} (t, q^{σ}, {\dot{q}}^{1}, \dots, {\dot{q}}^{m - k}), 1 \leq a \leq k, \\ {\ddot{q}}^{l} & = & - {\bar{B}}^{l s} {\bar{A}}_{s}, 1 \leq l \leq m - k . \end{matrix}$ (3.65)

We stress that, as one can see from any of the above equivalent regularity conditions, a constrained system arising from a regular Lagrangian system need not be regular.

Corollary 3.2. If $Q \subset J^{1} Y$ is a simple non-holonomic constraint then the regularity condition reads

det (\frac{\partial^{2} \bar{L}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}}) \neq 0,

(3.66)

where $\bar{L} = L \circ ι$ .

3.5. Constrained Hamilton equations. Constrained Hamiltonian systems were studied in detail in [3, 9, 38].

Let $[α] = [d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ , $ι : Q \to J^{1} Y$ a non-holonomic constraint, $[[ι^{*} α]] = [[ι^{*} d Θ_{λ}]]$ the corresponding constrained system. For every $\bar{α} = ι^{*} α \in [[ι^{*} d Θ_{λ}]]$ we have the constraint dynamical distribution $Δ_{\bar{α}}$ deﬁned on the domain of deﬁnition of $\bar{α}$ , say $U \subset Q$ .

Directly from the deﬁnition of constraint dynamical distribution we can see that if ${\bar{α}}_{1}$ and ${\bar{α}}_{2}$ differ by a constraint form, their constraint dynamical distributions $Δ_{{\bar{α}}_{1}}$ and $Δ_{{\bar{α}}_{2}}$ coincide.

Deﬁnition 3.15. [38] Let $[d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ . For every $α \in [d Θ_{λ}]$ the equivalence class

{\bar{α}}_{ℐ} = ι^{*} α mod ℐ^{2} (C^{0})

(3.67)

is called constrained Hamiltonian system related with $α$ and the constraint $Q$ .

Equations for integral sections of the corresponding constraint dynamical distribution $Δ_{\bar{α}}$ , i.e.

δ^{*} {\bar{ϕ}}^{a} = 0, 1 \leq a \leq k, δ^{*} i_{ξ} \bar{α} = 0 \forall ξ \in V (Q) \cap C,

(3.68)

where $\bar{α} \in {\bar{α}}_{ℐ}$ and $δ$ is a section of $π_{1} ∣_{Q} : Q \to X$ , are called constrained Hamilton equations.

Note that for every $\bar{α} \in [[ι^{*} d Θ_{λ}]]$ on $U$ , holonomic integral sections of $Δ_{\bar{α}}$ coincide with prolongations of constrained extremals in $U$ .

Theorem 3.6. [38] Let $[[ι^{*} d Θ_{λ}]]$ be a regular constrained Lagrangian system. Then for any two its Hamiltonian systems ${\bar{α}}_{1 ℐ}$ , ${\bar{α}}_{2 ℐ}$ on an open subset $U$ their constrained dynamical distributions coincide, i.e. $Δ_{{\bar{α}}_{1}} = Δ_{{\bar{α}}_{2}}$ .

Corollary 3.3. If the regularity condition (3.61) is satisﬁed then constrained Euler–Lagrange equations are equivalent with (any) constrained Hamilton equations.

For regular constrained systems we can introduce a constraint Legendre transformation.

Theorem 3.7. [38] Let $ι : Q \to J^{1} Y$ be a non-holonomic constraint, $[[ι^{*} d Θ_{λ}]]$ a constrained Lagrangian system. Let $x \in Q$ be a point. Suppose that in a neighborhood of $x$ ,

\frac{\partial {\bar{B}}_{l s}}{\partial {\dot{q}}^{r}} = \frac{\partial {\bar{B}}_{l r}}{\partial {\dot{q}}^{s}}, 1 \leq l, r, s \leq m - k .

(3.69)

Then there exists a neighborhood $U \subset Q$ of $x$ , and, on $U$ , functions $P_{l}$ , $1 \leq l \leq m - k$ , and a $1$ -form $η$ , such that the class $[[ι^{*} d Θ_{λ}]]$ has a representative of the form

{\bar{α}}^{'} = η \land d t + d P_{l} \land d q^{l} .

(3.70)

If, moreover, the constrained system $[[ι^{*} d Θ_{λ}]]$ is regular, then $(t, q^{σ}, {\dot{q}}^{l}) \to (t, q^{σ}, P_{l})$ is a coordinate transformation on $U$ .

Proof. In a neighborhood of $x$ , let us consider the elements of the equivalence class $[[ι^{*} d Θ_{λ}]]$ in the form (3.47). By assumption, from the Poincaré Lemma we get a neighborhood $U \subset Q$ of $x$ and functions $P_{l}$ , $1 \leq l \leq m - k$ , on $U$ such that

{\bar{B}}_{l s} = - \frac{\partial P_{l}}{\partial {\dot{q}}^{s}} .

(3.71)

Hence, in the class $[[ι^{*} d Θ_{λ}]]$ there is a local representative of the form

\begin{matrix} \bar{α} = {\bar{A}}_{l} {\bar{ω}}^{l} \land d t + \frac{\partial P_{l}}{\partial {\dot{q}}^{s}} d {\dot{q}}^{s} \land {\bar{ω}}^{l} \\ = {\bar{A}}_{l} {\bar{ω}}^{l} \land d t + d P_{l} \land {\bar{ω}}^{l} - \frac{\partial P_{l}}{\partial t} d t \land {\bar{ω}}^{l} - \frac{\partial P_{l}}{\partial q^{s}} d q^{s} \land {\bar{ω}}^{l} \\ - \frac{\partial P_{l}}{\partial q^{m - k + a}} d q^{m - k + a} \land {\bar{ω}}^{l} \\ = ({\bar{A}}_{l} + \frac{\partial P_{l}}{\partial t}) d q^{l} \land d t + d P_{l} \land d q^{l} - {\dot{q}}^{l} d P_{l} \land d t \\ - \frac{\partial P_{l}}{\partial q^{s}} ({\bar{ω}}^{s} + {\dot{q}}^{s} d t) \land {\bar{ω}}^{l} - \frac{\partial P_{l}}{\partial q^{m - k + a}} ({\bar{ω}}^{m - k + a} + g^{a} d t) \land {\bar{ω}}^{l} \\ = ({\bar{A}}_{l} + \frac{\partial P_{l}}{\partial t} + \frac{\partial P_{l}}{\partial q^{s}} {\dot{q}}^{s} + \frac{\partial P_{l}}{\partial q^{m - k + a}} g^{a}) d q^{l} \land d t - {\dot{q}}^{l} d P_{l} \land d t \\ + d P_{l} \land d q^{l} + \frac{\partial P_{l}}{\partial q^{s}} {\bar{ω}}^{l} \land {\bar{ω}}^{s} - \frac{\partial P_{l}}{\partial q^{m - k + a}} {\bar{ω}}^{m - k + a} \land {\bar{ω}}^{l} . \end{matrix}

(3.72)

This means that we also have a representative

\begin{matrix} {\bar{α}}^{'} = ({\bar{A}}_{l} + \frac{\partial P_{l}}{\partial t} + \frac{\partial P_{l}}{\partial q^{s}} {\dot{q}}^{s} + \frac{\partial P_{l}}{\partial q^{m - k + a}} g^{a}) d q^{l} \land d t - {\dot{q}}^{s} d P_{s} \land d t \\ + d P_{l} \land d q^{l} \\ = ({\bar{A}}_{l} + \frac{\partial P_{l}}{\partial t} + (\frac{\partial P_{l}}{\partial q^{s}} - \frac{\partial P_{s}}{\partial q^{l}}) {\dot{q}}^{s} + \frac{\partial P_{l}}{\partial q^{m - k + a}} g^{a}) d q^{l} \land d t \\ - \frac{\partial P_{s}}{\partial q^{m - k + a}} {\dot{q}}^{s} d q^{m - k + a} \land d t - \frac{\partial P_{s}}{\partial {\dot{q}}^{l}} {\dot{q}}^{s} d {\dot{q}}^{l} \land d t + d P_{l} \land d q^{l} . \end{matrix}

(3.73)

We can write it in the form ${\bar{α}}^{'} = η \land d t + d P_{l} \land d q^{l}$ with

η = {\bar{η}}_{0} d t + {\bar{η}}_{l} d q^{l} + {\bar{η}}_{m - k + a} d q^{m - k + a} + {\tilde{η}}_{l} d {\dot{q}}^{l},

(3.74)

where ${\bar{η}}_{0}$ is an arbitrary function on $U$ , and

\begin{matrix} {\bar{η}}_{l} & = & {\bar{A}}_{l} + \frac{\partial P_{l}}{\partial t} + (\frac{\partial P_{l}}{\partial q^{s}} - \frac{\partial P_{s}}{\partial q^{l}}) {\dot{q}}^{s} + \frac{\partial P_{l}}{\partial q^{m - k + a}} g^{a}, \\ {\tilde{η}}_{l} & = & - \frac{\partial P_{s}}{\partial {\dot{q}}^{l}} {\dot{q}}^{s}, {\bar{η}}_{m - k + a} = - \frac{\partial P_{l}}{\partial q^{m - k + a}} {\dot{q}}^{l} . \end{matrix}

(3.75)

Finally, the regularity condition for the transformation $(t, q^{σ}, {\dot{q}}^{l}) \to (t, q^{σ}, P_{l})$ coincides with (3.61). □

Remark 3.2. Condition (3.69) rewritten in terms of a ﬁrst-order Lagrangian reads

(\frac{\partial}{\partial {\dot{q}}^{r}} (\frac{\partial L}{\partial {\dot{q}}^{m - k + a}} \circ ι)) \frac{\partial^{2} g^{a}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}} = (\frac{\partial}{\partial {\dot{q}}^{s}} (\frac{\partial L}{\partial {\dot{q}}^{m - k + a}} \circ ι)) \frac{\partial^{2} g^{a}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{r}} .

(3.76)

The integrability condition for the ${\bar{B}}_{s l}$ ’s ((3.69), (3.76)) ensures that one can express functions $P_{l}$ explicitly. To this purpose we consider a mapping $χ : [0, 1] \times W \to W$ deﬁned by $(u, t, q^{σ}, {\dot{q}}^{l}) \to (t, q^{σ}, u {\dot{q}}^{l})$ , where $W \subset Q$ is an appropriate open set. Then Poincaré Lemma gives us a solution [38]

\begin{matrix} P_{l} = - {\dot{q}}^{s} \int_{0}^{1} ({\bar{B}}_{l s} \circ χ) d u \\ = \frac{\partial \bar{L}}{\partial {\dot{q}}^{l}} - {\dot{q}}^{s} \int_{0}^{1} ((\frac{\partial L}{\partial {\dot{q}}^{m - k + a}} \circ ι) \frac{\partial^{2} g^{a}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}}) \circ χ d u . \end{matrix}

(3.77)

Deﬁnition 3.16. [38] We call the above functions $P_{l}$ , $1 \leq l \leq k$ , constraint momenta, and the corresponding coordinate transformation constraint Legendre transformation. The $1$ -form $η$ in (3.70) is called a constraint energy $1$ -form.

The $1$ -form $η$ is determined up to a constraint $1$ -form, and need not be closed. In constraint Legendre coordinates we can write

η_{ℐ} = η_{0} d t + η_{l} d q^{l} + η^{l} d P_{l} mod ℐ^{1} (C^{0}) .

(3.78)

Corollary 3.4. If the Lagrangian system $[[ι^{*} d Θ_{λ}]]$ is regular, then the constraint Euler–Lagrange equations are equivalent with constraint Hamilton equations. In constraint Legendre coordinates Hamilton equations take the following canonical form

\frac{d}{d t} (P_{l} \circ δ) = η_{l}, \frac{d}{d t} (q^{l} \circ δ) = - η^{l}, \frac{d}{d t} (q^{m - k + a} \circ δ) = g^{a},

(3.79)

where $1 \leq l \leq m - k$ , $1 \leq a \leq k$ . Constraint Hamilton equations depend upon the choice of a representative $α \in [d Θ_{λ}]$ , rather than on a particular Lagrangian $λ$ .

For simple non-holonomic constraints, which, as we have seen in Sec. 3.3, can be modeled by a distribution on $Y$ , and are given by (3.30) (resp. (3.32)) where the functions $f^{a}$ (resp. $g^{a}$ ) are affine in the velocities) the situation essentially simpliﬁes:

Theorem 3.8. Assume that $Q$ is a simple non-holonomic constraint. Then (3.69) is fulﬁlled identically and the constraint momenta are deﬁned by

P_{l} = \frac{\partial \bar{L}}{\partial {\dot{q}}^{l}}, 1 \leq l \leq m - k .

(3.80)

Regularity condition takes the form

det (\frac{\partial^{2} \bar{L}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}}) \neq 0 .

(3.81)

Moreover, if the constraint $Q$ is semiholonomic then the family of energy $1$ -forms (3.78) contains a closed $1$ -form equal to $- d \bar{H}$ , where

\bar{H} = - \bar{L} + P_{l} {\dot{q}}^{l} .

(3.82)

3.6. Holonomic Lagrangian systems. The case of Lagrangian systems subjected to holonomic constraints can be considered as a special case of the nonholonomic theory (see [15]).

Deﬁnition 3.17. Let $π : Y \to X$ be a ﬁbered manifold, $dim X = 1$ , $dim Y = m + 1$ , $m > 1$ . By a holonomic constraint in $Y$ we mean a ﬁbered submanifold $π_{0} : Q_{0} \to X$ of $π$ .

If $X = R$ and $Y = R \times M$ where $M$ is a manifold of dimension $m$ , we also speak about a rheonomic constraint. If a rheonomic constraint is of the form $Q_{0} = R \times N$ where $N$ is a submanifold of $M$ , it is called skleronomic.

We denote by $ι_{0} : Q_{0} \to Y$ the canonical inclusion of $Q_{0}$ into $Y$ , and assume $c o d i m Q_{0} = k$ , where $1 \leq k < m$ . The constraint $Q_{0}$ is locally deﬁned by a system of algebraic equations

u^{a} (t, q^{σ}) = 0, 1 \leq a \leq k,

(3.83)

where the functions $u^{a}$ satisfy the rank condition

r a n k (\frac{\partial u^{a}}{\partial q^{σ}}) = k .

(3.84)

Hence, around every point $x \in Q_{0}$ there is a ﬁbered chart $(U, χ)$ on $Y$ , adapted to the submanifold $Q_{0}$ , i.e. $χ = (t, q^{1}, \dots, q^{m - k}, u^{1}, \dots, u^{k})$ .

The ﬁbered submanifold $Q_{0}$ can be prolonged to $J^{1} Q_{0} \subset J^{1} Y$ , $c o d i m$ $J^{1} Q_{0} = 2 k$ , locally deﬁned by the equations

u^{a} = 0, \frac{d u^{a}}{d t} = 0, 1 \leq a \leq k .

(3.85)

$J^{1} Q_{0}$ is a submanifold of the manifold $Q \subset J^{1} Y$ , deﬁned by the equations

f^{a} \equiv \frac{d u^{a}}{d t} = 0, 1 \leq a \leq k .

(3.86)

We can see that $Q$ is a semiholonomic constraint in $J^{1} Y$ , $c o d i m Q = k$ .

The only admissible holonomic paths in $Q$ are sections $γ$ of the ﬁbered manifold $Q_{0} \to X$ , i.e. such that $J^{1} γ \in J^{1} Q_{0}$ . This means that for a Lagrangian system $[d Θ_{λ}]$ on $J^{1} Y$ , the corresponding constrained system $[[ι^{*} d Θ_{λ}]]$ can be restricted to $J^{1} Q_{0}$ .

Proposition 3.8. Let $Q_{0} \subset Y$ be a holonomic constraint, $Q$ the semiholonomic constraint related with $Q_{0}$ . Let $C$ be the canonical distribution on $Q$ . Then for every $x \in J^{1} Q_{0}$

C (x) = T_{x} J^{1} Q_{0} .

(3.87)

Equivalently, the annihilator $C^{0}$ of the canonical distribution on $J^{1} Q_{0}$ is trivial, $C^{0} (x) = {0}$ .

Corollary 3.5. Let $Q_{0}$ be a holonomic constraint in $Y$ , $Q \subset J^{1} Y$ the associated semiholonomic constraint. Then the canonical distribution $C$ on $Q$ is completely integrable and projects onto a (completely integrable) distribution on $Y$ . Along $J^{1} Q_{0}$ the canonical distribution coincides with the tangent bundle to $J^{1} Q_{0}$ and projects onto the tangent bundle $T Q_{0} \to Q_{0}$ .

Proposition 3.9. Let $ι_{0} : Q_{0} \to Y$ be a holonomic constraint, $[d Θ_{λ}]$ a Lagrangian system on $J^{1} Y$ . Then for every $α \in [d Θ_{λ}]$

{(J^{1} ι_{0}^{*} α)}_{ℐ} = J^{1} ι_{0}^{*} α .

(3.88)

Moreover, for every ﬁrst-order Lagrangian $λ$

J^{1} ι_{0}^{*} Θ_{λ} = Θ_{J^{1} ι_{0}^{*} λ} .

(3.89)

This means that the corresponding constrained system on $J^{1} Q_{0}$ satisﬁes

\begin{matrix} [[J^{1} ι_{0}^{*} d Θ_{λ}]] = [J^{1} ι_{0}^{*} d Θ_{λ}] = J^{1} ι_{0}^{*} [d Θ_{λ}] = [d Θ_{J^{1} ι_{0}^{*} λ}] \\ \equiv d Θ_{J^{1} ι_{0}^{*} λ} mod Ω_{Y}^{0, 2} (J^{1} Q_{0}) . \end{matrix}

(3.90)

For simplicity of notations we write

\bar{λ} = J^{1} ι_{0}^{*} λ, \bar{L} = L \circ J^{1} ι_{0}

(3.91)

for the restricted Lagrangian.

By the above propositions, contrary to the non-holonomic case, holonomic constraints represent no constraints in the tangent bundle to the constraint submanifold. Consequently, holonomic constrained systems are treated in the same way as unconstrained systems on ﬁbered manifolds. Simply, instead of a Lagrangian $λ$ and a constraint $Q_{0}$ in $Y$ one can consider the restricted Lagrangian $\bar{λ}$ on $J^{1} Q_{0}$ .

Let us summarize some of the main properties of holonomic systems:

Corollary 3.6. The holonomic constraint Euler–Lagrange form satisﬁes

E_{λ}^{C} = E_{\bar{λ}},

(3.92)

and constraint Euler–Lagrange equations become simply equations for sections $γ$ of the ﬁbered manifold $π_{0} : Q_{0} \to X$ as follows:

J^{1} γ^{*} i_{ξ} d Θ_{\bar{λ}} = 0 \forall ξ \in V (J^{1} Q_{0}),

(3.93)

or, in adapted ﬁbered coordinates,

\frac{\partial \bar{L}}{\partial q^{l}} - \frac{d}{d t} \frac{\partial \bar{L}}{\partial {\dot{q}}^{l}} = 0, 1 \leq l \leq m - k .

(3.94)

Hamilton equations are then equations for sections of the prolonged manifold ${(π_{0})}_{1} : J^{1} Q_{0} \to X$ ,

δ^{*} i_{ξ} d Θ_{\bar{λ}} = 0 \forall ξ \in V (J^{1} Q_{0}) .

(3.95)

The regularity condition reads

det (\frac{\partial^{2} \bar{L}}{\partial {\dot{q}}^{l} \partial {\dot{q}}^{s}}) \neq 0 .

The class (3.90) has the canonical form

- d \bar{H} \land d t + d P_{l} \land d q^{l} mod Ω_{Y}^{0, 2} (J^{1} Q_{0}),

(3.96)

where the Hamiltonian and momenta take the form

\bar{H} = - \bar{L} + P_{l} {\dot{q}}^{l}, P_{l} = \frac{\partial \bar{L}}{\partial {\dot{q}}^{l}}, 1 \leq l \leq m - k .

(3.97)

The holonomic Hamilton equations then take the canonical form

\frac{d}{d t} (P_{l} \circ δ) = - \frac{\partial \bar{H}}{\partial q^{l}}, \frac{d}{d t} (q^{l} \circ δ) = \frac{\partial \bar{H}}{\partial P_{l}} .

(3.98)

3.7. Example: A sleigh on an inclined plane. Let us consider an example of a nonholonomic motion. The situation is presented on the following picture:

There is an object on the inclined plane and a cutting knife. The center of mass of the object lies on the straight line along the knife edge at a distance $a$ from the point $(x, y)$ of contact of the knife and the plane and $β$ measures the angle between the straight line along the knife edge and $x$ axis, see [29].

This mechanical system is modeled on the ﬁbered manifold $π : ℝ \times S^{1} \times ℝ^{2} \to ℝ$ , $(t, β, x, y)$ are coordinates on $ℝ \times S^{1} \times ℝ^{2}$ .

We introduce “generalized coordinates”

q^{1} = β, q^{2} = x, q^{3} = y,

(3.99)

$β \in (- \frac{π}{2}, \frac{π}{2})$ .

The Lagrange function $L$ of the system consists of three parts, the ﬁrst one represents the energy of rotation, the second one characterizes the kinetic energy of translation of the mechanical system and the third one is the potential energy of the system.

Because the center of mass $C$ does not coincide with the point of contact, the energy of rotation has to be modiﬁed. At ﬁrst we write down the formulas for angular velocities associated with Euler angles which represent rotational motion of this system,

\begin{matrix} ω_{x}^{'} & = & 0, \\ ω_{y}^{'} & = & 0, \\ ω_{z}^{'} & = & \dot{β}, \end{matrix}

(3.100)

for $\bar{β} = β$ , $\bar{Θ} = 0$ , $\bar{ψ} = 0$ . Then

T^{r o t} = \frac{1}{2} J {\dot{β}}^{2} + m v_{t r} (ω^{'} \times R^{'}),

(3.101)

where $ω^{'}$ is the angular velocity vector, $R^{'} = (x^{'}, y^{'}, z^{'})$ is the position vector between point of contact and the center of mass and $v_{t r}$ is the velocity vector of a translation. The following identities hold

\begin{matrix} ω^{'} \times R^{'} & = & (ω_{x}^{'}, ω_{y}^{'}, ω_{z}^{'}) \times (x^{'}, y^{'}, z^{'}) \\ = & (ω_{y}^{'} z^{'} - ω_{z}^{'} y^{'}, - ω_{x}^{'} z^{'} + ω_{z}^{'} x^{'}, ω_{x}^{'} y^{'} - ω_{y}^{'} x^{'}) \\ = & (- \dot{β} y^{'}, \dot{β} x^{'}, 0) = (- \dot{β} a sin β, \dot{β} a cos β, 0) . \end{matrix}

(3.102)

The second term of (3.101) now can be expressed as follows

m v_{t r} (ω^{'} \times R^{'}) = m (ẋ, ẏ, ż) (ω^{'} \times R^{'}) = m a \dot{β} (ẏ cos β - ẋ sin β) .

(3.103)

The kinetic energy of translation and the potential energy take the form

T^{t r} = \frac{1}{2} m (ẋ^{2} + ẏ^{2}), V = - m g x sin α,

(3.104)

where angle $α$ represents an inclination of the plane.

Finally the Lagrange function $L$ of this mechanical system is expressed by

L = \frac{1}{2} m (ẋ^{2} + ẏ^{2}) + \frac{1}{2} J {\dot{β}}^{2} + m a \dot{β} (ẏ cos β - ẋ sin β) - m g x sin α,

(3.105)

$g$ is the gravitational acceleration. For the variations of $d x$ and $d y$ we obtain

d y = tan β d x .

(3.106)

This is equivalent to

ẏ = ẋ tan β .

(3.107)

Then constraint function $f_{1}$ is expressed by

f_{1} = ẏ - g_{1} = ẏ - ẋ tan β .

(3.108)

With respect to (3.24) we have

\begin{matrix} A_{1} & = & 0, \\ A_{2} & = & - m g sin α + m a {\dot{β}}^{2} cos β, \\ A_{3} & = & m a {\dot{β}}^{2} sin β, \end{matrix}

(3.109)

and

B_{σ ν} = (\begin{matrix} - J & m a sin β & - m a cos β \\ m a sin β & - m & 0 \\ - m a cos β & 0 & - m \end{matrix}) .

(3.110)

The matrix $B_{σ ν}$ is regular, $det (B_{σ ν}) \neq 0$ . The Euler–Lagrange equations are then

\begin{matrix} \ddot{β} = - \frac{a m g sin α sin β}{J - m a^{2}}, \\ \ddot{x} = - \frac{J g sin α - J a {\dot{β}}^{2} cos β - m a^{2} g sin α {cos}^{2} β + m a^{3} {\dot{β}}^{2} cos β}{J - m a^{2}}, \\ \ddot{y} = \frac{a sin β (m a g sin α cos β + J {\dot{β}}^{2} - m a^{2} {\dot{β}}^{2})}{J - m a^{2}} . \end{matrix}

(3.111)

Let us return to the constrained case. Now $a = 1$ , $k = 1$ . The rank condition (3.31) is satisﬁed. Indeed,

r a n k (\frac{\partial f^{1}}{\partial {\dot{q}}^{σ}}) = r a n k (\begin{matrix} 0, & - tan β, & 1 \end{matrix}) = 1 .

(3.112)

So, constraint (3.108) generates a constraint submanifold $Q \subset J^{1} Y$ by

Q = \{(t, β, x, y, \dot{β}, ẋ, ẏ); ẏ = ẋ tan β\} .

(3.113)

The constraint distribution on $Q$ is annihilated by one $1$ -form ${\bar{ϕ}}^{1} = ι^{*} φ^{1}$ , where

φ^{1} = d y - tan β d x .

(3.114)

The constraint distribution in not completely integrable, i.e. there does not exist any $1$ -form $μ$ such that

d {\bar{ϕ}}^{1} = μ \land {\bar{ϕ}}^{1} .

(3.115)

Indeed,

d {\bar{ϕ}}^{1} = d (d y - tan β d x) = - \frac{1}{{cos}^{2} β} d β \land d x .

(3.116)

The constraint is affine in the velocities, i.e. the constraint is simple and it can be equivalently represented by a (non-integrable) distribution on $Y$ , generated by $φ^{1}$ .

From (3.48) we get

\begin{matrix} {\bar{A}}_{1} & = & - \frac{m a \dot{β} ẋ}{cos β}, \\ {\bar{A}}_{2} & = & - \frac{m (g sin α {cos}^{3} β - a {\dot{β}}^{2} {cos}^{2} β + \dot{β} ẋ sin β)}{{cos}^{3} β}, \end{matrix}

(3.117)

and

\bar{B} = (\begin{matrix} - J & 0 \\ 0 & - m - m {tan}^{2} β \end{matrix}) .

(3.118)

The determinant of $\bar{B}$ is non-zero, so the constrained system is regular.

Constrained Euler–Lagrange equations (3.65) consist of two equations of the second order

\begin{matrix} \ddot{β} & = & - \frac{m a \dot{β} ẋ}{J cos β}, \\ \ddot{x} & = & - g sin α {cos}^{2} β + a {\dot{β}}^{2} cos β - \dot{β} ẋ tan β, \end{matrix}

(3.119)

and of one equation of the ﬁrst order

ẏ = ẋ tan β .

(3.120)

The “constraint Lagrangian” $\bar{L} = L \circ ι$ is expressed by

\bar{L} = \frac{1}{2} \frac{m ẋ^{2}}{{cos}^{2} β} + \frac{1}{2} J {\dot{β}}^{2} - m g x sin α,

(3.121)

and we can check by a direct computation that the same equations are obtained from the functions $\bar{L}$ and

{\bar{L}}_{1} = m ẋ tan β + m a \dot{β} cos β

(3.122)

using Euler–Lagrange equations in the form (3.58).

The constraint Legendre transformation is given by

(t, β, x, y, \dot{β}, ẋ) \to (t, β, x, y, P_{β}, P_{x}),

(3.123)

where constraint momenta (3.77) take the form

\begin{matrix} P_{β} & = & J \dot{β}, \\ P_{x} & = & \frac{m ẋ}{{cos}^{2} β} . \end{matrix}

(3.124)

The class of constraint energy $1$ -forms is then expressed in constraint Legendre coordinates by

\begin{matrix} η_{ℐ} = {\bar{η}}_{0} d t \\ - \frac{a P_{β} P_{x} cos β}{J} d β + (- m g sin α + \frac{m a P_{β}^{2}}{J^{2} cos β} + \frac{P_{β} P_{x} sin β}{J cos β}) d x \\ - \frac{P_{β}}{J} d P_{β} - \frac{P_{x} {cos}^{2} β}{m} d P_{x} mod ℐ^{1} (C^{0}) . \end{matrix}

(3.125)

Hence, constrained Hamilton equations consist of four ﬁrst-order equations which, in simpliﬁed notation, can be written as follows:

\begin{matrix} Ṗ_{β} = - \frac{a P_{β} P_{x} cos β}{J}, \\ Ṗ_{x} = - m g sin α + \frac{m a P_{β}^{2}}{J^{2} cos β} + \frac{P_{β} P_{x} sin β}{J cos β}, \\ \dot{β} = \frac{P_{β}}{J}, \\ ẋ = \frac{P_{x} {cos}^{2} ϕ}{m}, \end{matrix}

(3.126)

and of the equation of the constraint $ẏ = ẋ tan β$ .

4. Fields with differential constraints

In the sequel we consider a ﬁbered manifold $π : Y \to X$ where $dim X = n$ , and its jet prolongations. As above, $m$ denotes the ﬁber dimension (i.e. $dim Y = m + n$ ), and we assume $m > 1$ .

First, we summarize main concepts from the theory of unconstrained Lagrangian and Hamiltonian systems, then we turn to the constraint structure in $J^{1} Y$ , and ﬁnally we are interested in Lagrangian and Hamiltonian constrained ﬁeld equations. Main sources for this section are [7, 10, 12, 13, 19, 21] for the unconstrained theory, [22] for the non-holonomic constraint structure, and [2, 22, 37, 38], for the nonholonomic constrained systems. In this section we also present new results concerning constrained Hamilton–De Donder systems. We study constrained Hamilton–De Donder equations, regularity, and existence of constraint Legendre transformation.

4.1. Dynamical forms and locally variational forms. Let $E$ be a dynamical form on $J^{2} Y$ . In ﬁbered coordinates,

E = E_{σ} ω^{σ} \land ω_{0},

(4.1)

where $E_{σ}$ are functions of $(x^{i}, y^{ν}, y_{i}^{ν}, y_{i j}^{ν})$ , and $ω_{0}$ denotes the local volume element (2.2). The coordinate form of the equation for paths of $E$ (3.1) is a system of $m$ second-order partial differential equations for the components $(γ^{ν})$ of sections $γ$ of $π$ as follows:

E_{σ} (x^{i}, γ^{ν} (x^{i}), \frac{\partial γ^{ν}}{\partial x^{i}}, \frac{\partial^{2} γ^{ν}}{\partial x^{i} \partial x^{j}}) = 0, 1 \leq σ \leq m .

(4.2)

Again, equations for paths of dynamical forms can be represented by means of exterior differential systems; now, however, locally generated by $n$ -forms.

Proposition 4.1. [19] Let $E$ be a dynamical form on $J^{2} Y$ . A section $γ$ of $π$ is a path of $E$ if and only if

J^{2} γ^{*} i_{ξ} α = 0 \forall ξ \in V (J^{2} Y),

(4.3)

where $α$ is any $(n + 1)$ -form such that $p_{1} α = E$ .

Proof of this statement is the same as that of Proposition 3.1.

In the “PDE situation” we can proceed in full analogy with the case of mechanics, and consider Lepage classes and corresponding Hamiltonian systems:

Deﬁnition 4.1. [22] Let $E \in Ω_{Y}^{n + 1} (J^{2} Y)$ be a dynamical form. The equivalence class of $(n + 1)$ -forms (on an open subset $U \subset J^{2} Y$ ) deﬁned by

α_{1} \sim α_{2} iff p_{1} α_{1} = p_{1} α_{2} = E ∣_{U}

(4.4)

is called Lepage class of $E$ on $U$ . The family of all local Lepage classes of $E$ is referred to as Lepage class of $E$ and is denoted by ${[α]}_{E}$ , or simply $[α]$ .

By the above proposition, the equation for paths of $E$ (on $U$ ) coincides with equations for holonomic integral sections of the exterior differential system $ℋ_{α}$ , generated by the following system of $n$ -forms

i_{ξ} α \forall ξ \in V (J^{2} Y),

(4.5)

where $α$ is any representative of the Lepage class of $E$ (on $U$ ).

Deﬁnition 4.2. [19, 22] Let $[α]$ be a Lepage class of $E$ . Every representative $α \in [α]$ is called a Hamiltonian system associated with $E$ . The exterior differential system $ℋ_{α}$ is called a Hamiltonian EDS related to $E$ . Equations for (all) integral sections of $ℋ_{α}$ are called Hamilton equations associated with $E$ .

Let us turn to locally variational dynamical forms. By deﬁnition this means that around each point $E = E_{λ}$ , for a local Lagrangian $λ$ (recall that a Lagrangian of order $r$ is deﬁned to be a horizontal $n$ -form on $J^{r} Y$ ). In ﬁbered coordinates, the components $E_{σ}$ of $E$ take the form of Euler–Lagrange expressions of $λ = L ω_{0}$ . Necessary and sufficient conditions for a (second-order) dynamical form to be locally variational read [1, 11]

\begin{matrix} \frac{\partial E_{σ}}{\partial y_{i j}^{ν}} - \frac{\partial E_{ν}}{\partial y_{i j}^{σ}} & = & 0, \\ \frac{\partial E_{σ}}{\partial y_{i}^{ν}} + \frac{\partial E_{ν}}{\partial y_{i}^{σ}} - 2 \frac{d}{d x^{j}} \frac{\partial E_{ν}}{\partial y_{i j}^{σ}} & = & 0, \\ \frac{\partial E_{σ}}{\partial y^{ν}} - \frac{\partial E_{ν}}{\partial y^{σ}} + \frac{d}{d x^{i}} \frac{\partial E_{ν}}{\partial y_{i}^{σ}} - \frac{d}{d x^{i}} \frac{d}{d x^{j}} \frac{\partial E_{ν}}{\partial y_{i j}^{σ}} & = & 0, \end{matrix}

(4.6)

and the corresponding Tonti Lagrangian for $E$ is

L = y^{σ} \int_{0}^{1} E_{σ} (x^{i}, u y^{ν}, u y_{i}^{ν}, u y_{i j}^{ν}) d u .

(4.7)

Notice that, contrary to the case of ordinary differential equations, variationality conditions (4.6) do not imply that the locally variational form $E$ should be affine in the second derivatives. Moreover, second-order locally variational forms need not come from Lagrangians of the ﬁrst order. This means that Tonti Lagrangian need not be reducible to a ﬁrst-order Lagrangian. (From the formula for Euler–Lagrange expressions we can see immediately that a necessary condition for reducibility is that $E_{σ}$ should be affine in the second derivatives $y_{i j}^{ν}$ ).

Let us summarize basic deﬁnitions:

Deﬁnition 4.3. The Lepage class $[α]$ of a locally variational form $E$ is called Lagrangian system. Every element $α \in [α]$ is called a Hamiltonian system associated with $E$ . Paths of a locally variational form $E$ are called extremals. Equations for paths of a locally variational form (respectively, equations for holonomic integral sections of associated Hamiltonian EDS) are called Euler–Lagrange equations. Equations for integral sections of the Hamiltonian EDS are called Hamilton equations, their integral sections are then called Hamilton extremals.

4.2. Euler–Lagrange and Hamilton–De Donder equations in ﬁrst-order ﬁeld theory. In what follows we shall be concerned merely with locally variational forms that arise from local ﬁrst-order Lagrangians. Such dynamical forms, among others, have the following properties:

$∙$ Around each point in $J^{2} Y$ it holds $E = E_{λ}$ , where $λ$ is a Lagrangian deﬁned on an open subset of $J^{1} Y$ . In ﬁbered coordinates where $λ = L ω_{0}$ ,

E_{λ} = E_{σ} ω^{σ} \land ω_{0}, E_{σ} = \frac{\partial L}{\partial y^{σ}} - \frac{d}{d x^{j}} \frac{\partial L}{\partial y_{j}^{σ}} .

(4.8)

This means that the Euler–Lagrange equations in ﬁbered coordinates take the familiar form

(\frac{\partial L}{\partial y^{σ}} - \frac{d}{d x^{j}} \frac{\partial L}{\partial y_{j}^{σ}}) \circ J^{2} γ = 0, 1 \leq σ \leq m .

(4.9)

$∙$ In every ﬁbered chart, components $E_{σ}$ of $E$ are affine in the second derivatives, i.e.

E_{σ} = A_{σ} + B_{σ ν}^{i j} y_{i j}^{ν} = A_{σ} + B_{σ ν}^{(i j)} y_{i j}^{ν},

(4.10)

where $A_{σ}$ and $B_{σ ν}^{i j}$ are functions of $(x^{i}, y^{ρ}, y_{l}^{ρ})$ that in terms of a ﬁrst-order Lagrangian for $E$ take the form

\begin{matrix} B_{σ ν}^{i j} = - \frac{\partial^{2} L}{\partial y_{i}^{ν} \partial y_{j}^{σ}}, \\ A_{σ} = \frac{\partial L}{\partial y^{σ}} - \frac{d^{'}}{d x^{j}} \frac{\partial L}{\partial y_{j}^{σ}} = \frac{\partial L}{\partial y^{σ}} - \frac{\partial^{2} L}{\partial x^{j} \partial y_{j}^{σ}} - \frac{\partial^{2} L}{\partial y^{ν} \partial y_{j}^{σ}} y_{j}^{ν} . \end{matrix}

(4.11)

Note that the $B_{σ ν}^{i j}$ need not be symmetric in the upper indices. $B_{σ ν}^{(i j)}$ denotes the symmetric part in the $i, j$ .

$∙$ Every ﬁrst-order Lagrangian for $E$ has a ﬁrst-order Lepage equivalent that is not unique. Lepage equivalents of $λ$ take the form

ρ = Θ_{λ} + d ν + μ,

(4.12)

where

Θ_{λ} = L ω_{0} + \frac{\partial L}{\partial y_{j}^{σ}} ω^{σ} \land ω_{j},

(4.13)

$μ$ is an arbitrary at least $2$ -contact $n$ -form, and $ν$ is an arbitrary contact $(n - 1)$ -form. $Θ_{λ}$ is called the Poincaré–Cartan form associated with $λ$ . It is the unique at most $1$ -contact $n$ -form such that $h Θ_{λ} = λ$ and $p_{1} d Θ_{λ}$ is $π_{1, 0}$ -horizontal. This means that the Euler–Lagrange form $E_{λ}$ of $λ$ is also unique, since by (4.12) it does not depend upon the choice of a Lepage equivalent $ρ$ of $λ$ :

E_{λ} = p_{1} d ρ = p_{1} d Θ_{λ} .

(4.14)

Note that contrary to mechanics, one generally has $d Θ_{λ_{1}} \neq d Θ_{λ_{2}}$ for equivalent Lagrangians $λ_{1}$ , $λ_{2}$ .

$∙$ $E$ can be locally represented by a ﬁrst-order Lepage class that is called Lagrangian system associated with the locally variational form $E$ . With a similar inaccuracy as in Sec. 3, in order to simplify notations, we write

[α] = [d Θ_{λ}] = d Θ_{λ} mod Ω^{n + 1, (\geq 2)} (J^{1} Y) .

(4.15)

Consequently, extremals and Hamilton extremals are described by Hamiltonian exterior differential systems deﬁned on (open subsets of) $J^{1} Y$ .

Remark 4.1. In the sequel we shall again assume that $E$ , deﬁned on $J^{2} Y$ , is everywhere nontrivially of order $2$ . This means that $(B_{σ ν}^{i j})$ in (4.10) is everywhere a non-zero matrix, or, equivalently, for every ﬁrst-order Lagrangian $λ$ the Poincaré–Cartan $(n + 1)$ -form $d Θ_{λ}$ is everywhere nontrivially of order one.

As we can see, the Euler–Lagrange and Hamilton equations in ﬁrst-order ﬁeld theory now have the following EDS formulation:

Proposition 4.2. Let $[α]$ be a Lagrangian system on $J^{1} Y$ , $E$ the corresponding locally variational form.

A section $γ$ of $π$ is an extremal of $E$ (on an open set $W \subset X$ in $d o m γ$ ) if and only if

J^{1} γ^{*} i_{ξ} α = 0 \forall ξ \in V (J^{1} Y),

(4.16)

where $α$ is any $(n + 1)$ -form belonging to the class $[α]$ (deﬁned on $π_{1}^{- 1} (W)$ ).

A section $δ$ of $π_{1}$ is a Hamilton extremal of $E$ , related with the Hamiltonian system $α \in [α]$ (deﬁned in $π_{1}^{- 1} (d o m δ)$ ) if and only if

δ^{*} i_{ξ} α = 0 \forall ξ \in V (J^{1} Y) .

(4.17)

The concept of regularity of a Lagrangian system is, similarly as in mechanics, related with the properties of the associated Hamiltonian exterior differential systems. The situation in ﬁeld theory is, however, much more rich and interesting than that in mechanics: the reason is the non-uniqueness of the Poincaré–Cartan form $Θ_{λ}$ of a Lagrangian $λ$ . For more details we refer to [19, 23, 24]. In this paper we shall study the most simple case related just to the properties of the Hamiltonian differential systems $ℋ_{d Θ_{λ}}$ related with the forms $d Θ_{λ}$ (Hamilton–De Donder equations [5, 7]).

To this end, let us recall the following deﬁnition [25].

Deﬁnition 4.4. Let $[α]$ (on $J^{1} Y$ ) be a Lagrangian system related with a locally variational form $E$ . An element $α$ of the class $[α]$ is called Hamilton–De Donder system related with $E$ if

α = d Θ_{λ} + F,

(4.18)

where $λ$ is a Lagrangian for $E$ and $F \in Ω_{Y}^{n + 1, (\geq 2)} (J^{1} Y)$ . The corresponding Hamilton equations, i.e.

δ^{*} i_{ξ} (d Θ_{λ} + F) = 0 \forall ξ \in V (J^{1} Y)

(4.19)

are called Hamilton–De Donder equations.

It is easy to see that Hamilton–De Donder systems can be locally expressed in the so-called canonical form as follows:

Proposition 4.3. Let $α = d Θ_{λ} + F$ be a Hamilton–De Donder system on an open set $U \subset J^{1} Y$ ; we may assume that $U$ is endowed with ﬁbered coordinates $(x^{i}, y^{σ}, y_{j}^{σ})$ . Then there exist functions $H$ and $p_{σ}^{i}$ , $1 \leq σ \leq m$ , $1 \leq i \leq n$ , such that

α = - d H \land ω_{0} + d p_{σ}^{i} \land d y^{σ} \land ω_{i} + F .

(4.20)

$H$ and $p_{σ}^{i}$ are deﬁned by

p_{σ}^{i} = \frac{\partial L}{\partial y_{i}^{σ}}, H = - L + p_{ν}^{j} y_{j}^{ν},

(4.21)

where $L ω_{0}$ is a ﬁrst-order Lagrangian whose Poincaré–Cartan $(n + 1)$ -form coincides with $d Θ_{λ}$ .

Deﬁnition 4.5. Functions $H$ and $p_{σ}^{i}$ , $1 \leq σ \leq m$ , $1 \leq i \leq n$ , deﬁned by formula (4.20) are called a Hamiltonian and momenta of the Hamilton–De Donder system $α$ .

Note that the family of a Hamiltonian and momenta (4.21) of a Hamilton–De Donder system is non-unique and depends upon the choice of a Lagrangian for the form $d Θ_{λ}$ . On the other hand, in the following subclass of the equivalence class (4.15),

{[α]}_{Y} = {[d Θ_{λ}]}_{Y} \equiv d Θ_{λ} mod Ω_{Y}^{n + 1, (\geq 2)} (U),

(4.22)

all elements posses the same families of momenta $&$ Hamiltonian. This is due to the fact that if $α_{1}, α_{2} \in [α] ∣_{U}$ are such that $α_{1} = d Θ_{λ_{1}} + F_{1}$ and $α_{2} = d Θ_{λ_{2}} + F_{2}$ where $d Θ_{λ_{1}} \neq d Θ_{λ_{2}}$ and $F_{1}, F_{2} \in Ω_{Y}^{n + 1, (\geq 2)} (U)$ , then $α_{1}$ and $α_{2}$ are not equivalent in the sense of (4.22), and vice versa (see [25], Proposition 3.1 and its proof).

Let us turn to the concept of regularity of a Hamilton–De Donder system ([19]).

Deﬁnition 4.6. A Hamilton–De Donder system $α$ is called regular if $ℋ_{α}$ contains all the canonical contact $n$ -forms

ω^{σ} \land ω_{i}, 1 \leq σ \leq m, 1 \leq i \leq n .

(4.23)

A Lagrangian system $[α]$ is called De Donder regular if around each point in $J^{1} Y$ there exists a related regular Hamilton–De Donder system.

Proposition 4.4. Let $α$ be a regular Hamilton–De Donder system. Then every integral section of $ℋ_{α}$ is holonomic. Consequently, Hamilton–De Donder equations of $α$ are equivalent with the Euler–Lagrange equations of $[α]$ .

Proof. If a Hamilton–De Donder system $α$ is regular then for every integral section $δ$ of $ℋ_{α}$ ,

\begin{matrix} δ^{*} (ω^{σ} \land ω_{i}) = (\frac{\partial (y^{σ} \circ δ)}{\partial x^{i}} - (y_{i}^{σ} \circ δ)) ω_{0} = 0, \\ 1 \leq σ \leq m, 1 \leq i \leq n, \end{matrix}

(4.24)

meaning that $δ = J^{1} γ$ for a section $γ$ of $π$ . □

Theorem 4.1. Let $α = d Θ_{λ} + F$ be a Hamilton–De Donder system. The following conditions are equivalent:

The Hamilton–De Donder system $α$ is regular.
A system of generators of $ℋ_{α}$ has maximal rank (i.e. equal to $m + m n$ ).
Every Lagrangian $λ$ for $d Θ_{λ}$ satisﬁes the regularity condition
$det (\frac{\partial^{2} L}{\partial y_{i}^{σ} \partial y_{j}^{ν}}) \neq 0 .$ (4.25)

Proof. Computing explicitly generators of $ℋ_{α}$ we obtain the following system of $m + m n$ differential $n$ -forms:

\begin{matrix} A_{σ} ω_{0} + (2 F_{σ ν}^{j} + \frac{\partial^{2} L}{\partial y^{σ} \partial y_{j}^{ν}} - \frac{\partial^{2} L}{\partial y^{ν} \partial y_{j}^{σ}}) ω^{ν} \land ω_{j} + B_{σ ν}^{i j} d y_{j}^{ν} \land ω_{i} + μ_{σ}, \\ B_{σ ν}^{i j} ω^{ν} \land ω_{j}, \end{matrix}

(4.26)

where $1 \leq σ \leq m$ , $1 \leq i \leq n$ , $A_{σ}$ and $B_{σ ν}^{i j}$ are given by (4.11), and $μ_{σ}$ are at least $2$ -contact (precisely, $μ_{σ}$ is the at least $2$ -contact part of $i_{\partial ∕ \partial y^{σ}} F$ ). This means that the matrix of generators of $ℋ_{α}$ is the following matrix with $m + m n$ rows (and $1 + m n + m n^{2} + \dots$ columns):

(\begin{matrix} A_{σ} & 2 F_{σ ν}^{j} + \frac{\partial^{2} L}{\partial y^{σ} \partial y_{j}^{ν}} - \frac{\partial^{2} L}{\partial y^{ν} \partial y_{j}^{σ}} & B_{σ ν}^{i j} & \dots \\ 0 & B_{σ ν}^{i j} & 0 & 0 \end{matrix}) .

(4.27)

First, we prove the equivalence of (1) and (2).

If $α$ is regular then all the generators $B_{σ ν}^{i j} ω^{ν} \land ω_{j}$ are independent, meaning that the matrix $(B_{σ ν}^{i j})$ is regular. Consequently, all rows of $(B_{σ ν}^{i j})$ (labelled by $(σ, i)$ ) are linearly independent, for every ﬁxed $i$ . Then, however, the matrix $(B_{σ ν}^{i j})$ with $m$ rows labelled by $σ$ , and $m n$ columns labelled by $(ν, j)$ , has the maximal rank, $m$ , and the rank of the matrix (4.27) is equal to $m + m n$ , as desired.

Conversely, if the rank of the matrix (4.27) is maximal then its square submatrix $(B_{σ ν}^{i j})$ (with rows labelled by $(σ, i)$ ) is regular. This means that all the forms $ω^{σ} \land ω_{i}$ are independent. Hence $α$ is regular.

The equivalence of (3) and (2) is now clear: by the above, $ℋ_{α}$ has maximal rank iff the matrix $(B_{σ ν}^{i j})$ is regular, However, in terms of the Lagrangian $λ$ for $α$ ,

B_{σ ν}^{i j} = - \frac{\partial^{2} L}{\partial y_{i}^{σ} \partial y_{j}^{ν}} .

(4.28)

This completes the proof. □

Theorem 4.2. If $d Θ_{λ} + F$ is a regular Hamilton–De Donder system then every form $α \in {[d Θ_{λ}]}_{Y}$ is regular. Consequently,

For every $α \in {[d Θ_{λ}]}_{Y}$ , all Hamilton extremals are holonomic.
For every $α \in {[d Θ_{λ}]}_{Y}$ , the Hamilton equations are equivalent with the Euler–Lagrange equations of $λ$ .
For every $α \in {[d Θ_{λ}]}_{Y}$ , every Hamilton extremal of $α$ is a prolongation of an extremal of $λ$ .
Hamilton equations of all elements in the class ${[d Θ_{λ}]}_{Y}$ are equivalent.

Proof. Looking at the generators (4.26) we can see immediately that regularity does not depend upon the choice of functions $F_{σ ν}^{j}$ , i.e., upon the choice of $α \in {[d Θ_{λ}]}_{Y}$ .

The rest of the proof is elementary. □

Corollary 4.1. Let $α = d Θ_{λ} + F$ be a regular Hamilton–De Donder system. Then momenta $p_{σ}^{j}$ , $1 \leq σ \leq m$ , $1 \leq j \leq n$ , of $α$ are independent, and $(x^{i}, y^{σ}, p_{σ}^{j})$ are local coordinates on $J^{1} Y$ , called Legendre coordinates. The Hamiltonian differential system $ℋ_{α}$ has generators that in Legendre coordinates take the form (since $ω^{ν} \land ω_{i} \in ℋ_{α}, \forall ν, i$ ),

\frac{\partial H}{\partial y^{σ}} ω_{0} + d p_{σ}^{j} \land ω_{j} + μ_{σ}, \frac{\partial H}{\partial p_{σ}^{j}} ω_{0} - d y^{σ} \land ω_{j},

(4.29)

where $μ_{σ} = i_{\partial ∕ \partial y^{σ} F} \in Ω_{Y}^{n (\geq 2)} (J^{1} Y)$ . Hamilton equations of $α$ in Legendre coordinates then read

\frac{\partial p_{σ}^{j}}{\partial x^{j}} = - \frac{\partial H}{\partial y^{σ}}, \frac{\partial y^{σ}}{\partial x^{j}} = \frac{\partial H}{\partial p_{σ}^{j}},

(4.30)

where the appearing functions are considered along sections $δ$ of $π_{1}$ .

4.3. Non-holonomic constraints in ﬁeld theory. The aim of the section is to present the concept of the non-holonomic constraint structure [22].

Non-holonomic constraints in the case $dim X > 1$ are deﬁned in the same way as for one independent variable:

Deﬁnition 4.7. By a constraint submanifold or a non-holonomic constraint in $J^{1} Y$ we shall understand a submanifold $Q \subset J^{1} Y$ , ﬁbered over $Y$ , precisely speaking, a surjective submersion $π_{1, 0} ∣_{Q} : Q \to Y$ .

Put $c o d i m Q = κ$ and assume $1 \leq κ \leq m n - 1$ . Locally $Q$ can be expressed by a system of ﬁrst-order partial differential equations

f^{α} (x^{i}, y^{σ}, y_{j}^{σ}) = 0, 1 \leq α \leq κ,

(4.31)

such that

r a n k (\frac{\partial f^{α}}{\partial y_{j}^{σ}}) = κ, where α labels rows and σ, j columns.

(4.32)

Deﬁnition 4.8. Let $Q$ be a non-holonomic constraint in $J^{1} Y$ , $c o d i m Q = κ$ , $1 \leq κ \leq m n - 1$ . If

r a n k (\frac{\partial f^{α}}{\partial y_{j}^{σ}}) = k, where α, j label rows and σ columns,

(4.33)

for some $k$ , $1 \leq k \leq m - 1$ , we say that $Q$ is a regular non-holonomic constraint of corank $(κ, k)$ .

It can be shown that the above deﬁnition is correct (coordinate independent) [22].

Given a regular non-holonomic constraint $Q$ in $J^{1} Y$ there naturally arise the following local distributions, deﬁned on the domain $U$ of deﬁnition of the functions $f^{α}$ :

(1) $D_{U} = a n n i h {d f^{α}, 1 \leq α \leq κ}$ ; $r a n k D_{U}$ is constant on $U$ due to (4.32) and equal to $m + n + m n - κ$ .

(2) ${\tilde{C}}_{U} = a n n i h {φ^{α j}}$ , where

φ^{α j} = f^{α} d x^{j} + \frac{1}{n} \frac{\partial f^{α}}{\partial y_{j}^{σ}} ω^{σ}, 1 \leq α \leq κ, 1 \leq j \leq n .

(4.34)

The forms (4.34) are not linearly independent, however, due to rank condition (4.33), there exist functions $c_{α j}^{a}$ , $1 \leq a \leq k$ , $1 \leq α \leq κ$ , $1 \leq j \leq n$ , on $U$ , such that the $(k \times m)$ -matrix

M = (M_{σ}^{a}), where M_{σ}^{a} = \frac{1}{n} c_{α j}^{a} \frac{\partial f^{α}}{\partial y_{j}^{σ}},

(4.35)

has maximal rank equal to $k$ . Thus,

\begin{matrix} φ^{a} = c_{α j}^{a} φ^{α j} = c_{α j}^{a} f^{α} d x^{j} + \frac{1}{n} c_{α j}^{a} \frac{\partial f^{α}}{\partial y_{j}^{σ}} ω^{σ} = c_{α j}^{a} f^{α} d x^{j} + M_{σ}^{a} ω^{σ}, \\ 1 \leq a \leq k, \end{matrix}

(4.36)

are linearly independent at each point in $U$ . Hence, the distribution ${\tilde{C}}_{U}$ has constant corank equal to $k$ , i.e. $r a n k {\tilde{C}}_{U} = m + n + m n - k$ .

(3) $C_{U} = a n n i h {φ^{a}, d f^{α}, 1 \leq a \leq k, 1 \leq α \leq κ}$ .

The following results have been obtained in [22]:

Proposition 4.5. $Q \cap U$ is an integral submanifold of $D_{U}$ . Hence, for every $x \in Q$ , the forms $d f^{α} (x)$ , $1 \leq α \leq κ$ , annihilate the tangent space $T_{x} Q$ to the manifold $Q$ at $x$ , i.e., along $Q$ , $D = a n n i h {d f^{α}, 1 \leq α \leq κ} = T Q$ .

Corollary 4.2. Let $Q$ be a constraint of codimension $κ$ in $J^{1} Y$ , and let $f^{α} = 0$ and ${f^{'}}^{α} = 0$ , where $1 \leq α \leq κ$ , be two sets of equations of $Q$ on an open set $U \subset V_{1} \subset J^{1} Y$ . Then there are functions $γ_{β}^{α}$ on $U$ such that at each point of $U$ , $(γ_{β}^{α})$ is a regular matrix, and $d {f^{'}}^{α} = γ_{β}^{α} d f^{β}$ . In particular, at each point $x \in Q \cap U$ ,

\frac{\partial {f^{'}}^{α}}{\partial y_{j}^{σ}} = γ_{β}^{α} \frac{\partial f^{β}}{\partial y_{j}^{σ}} .

(4.37)

Proposition 4.6. $C_{U}$ is a subdistribution of both ${\tilde{C}}_{U}$ and $D_{U}$ . At the points of $Q \cap U$ , the distributions $C_{U}$ and ${\tilde{C}}_{U} \cap D$ coincide, and deﬁne a distribution of corank $k$ on $Q \cap U$ .

The local distributions on $Q$ mentioned above unite into a (global) distribution on $Q$ :

Theorem 4.3. Let $Q$ be a regular non-holonomic constraint in $J^{1} Y$ of corank $(κ, k)$ , let $ι : Q \to J^{1} Y$ be the canonical embedding of the submanifold $Q$ into $J^{1} Y$ . If $φ^{a}$ , $1 \leq a \leq k$ , are independent $1$ -forms $(4.36)$ , put

ϕ^{a} = ι^{*} φ^{a} = (M_{σ}^{a} \circ ι) ι^{*} ω^{σ}, 1 \leq a \leq k .

(4.38)

Then

C = a n n i h {ϕ^{a}, 1 \leq a \leq k}

(4.39)

is a distribution of corank $k$ on $Q$ .

The proof of the theorem can be found in [22].

Deﬁnition 4.9. The distribution $C$ (4.39) on $Q$ is called canonical distribution. $1$ -forms belonging to the annihilator, $C^{0}$ , of $C$ , are called canonical constraint $1$ -forms. The ideal in the exterior algebra of differential forms on $Q$ generated by $C^{0}$ is called canonical constraint ideal, and denoted by $ℐ (C^{0})$ ; its homogeneous component of degree $p$ is denoted by $ℐ^{p} (C^{0})$ . Elements of the ideal $ℐ (C^{0})$ are called canonical constraint forms.

Theorem 4.4. The canonical distribution $C$ on $Q$ is locally spanned by the following vector ﬁelds:

\begin{matrix} \frac{\partial_{c}}{\partial x^{i}} \equiv \frac{\partial}{\partial x^{i}} + \sum_{a = 1}^{k} (F_{i}^{a} \circ ι) \frac{\partial}{\partial y^{m - k + a}}, 1 \leq i \leq n, \\ \frac{\partial_{c}}{\partial y^{s}} \equiv \frac{\partial}{\partial y^{s}} + \sum_{a = 1}^{k} (G_{s}^{a} \circ ι) \frac{\partial}{\partial y^{m - k + a}}, 1 \leq s \leq m - k, \\ \frac{\partial}{\partial z^{J}}, 1 \leq J \leq n m - κ, \end{matrix}

(4.40)

where $(x^{i}, y^{σ}, z^{J}, f^{α})$ , $1 \leq i \leq n$ , $1 \leq σ \leq m$ , $1 \leq J \leq n m - κ$ , $1 \leq α \leq κ$ , denote ﬁbered coordinates adapted to the submanifold $ι : Q \to J^{1} Y$ , the functions $G_{s}^{a}$ represent (at each point) a fundamental system of solutions of the system of independent homogeneous algebraic equations for $m$ unknowns $Ξ^{σ}$ , $1 \leq σ \leq m$ ,

M_{σ}^{a} Ξ^{σ} = 0, 1 \leq a \leq k,

(4.41)

and, for every $i = 1, 2, \dots, n$ , the $F_{i}^{a}$ are solutions of the equations

M_{σ}^{a} F_{i}^{σ} = M_{σ}^{a} y_{i}^{σ} - f^{α} c_{α i}^{a}, 1 \leq a \leq k,

(4.42)

(where $y_{j}^{σ}$ are considered as functions of $z^{J}, f^{β}$ ) corresponding to the choice of all the parameters equal to zero.

A section $γ$ of $π$ deﬁned on an open set $W \subset X$ is called a holonomic path in $Q$ if for every $x \in W$

J^{1} γ (x) \in Q .

(4.43)

Remark 4.2. We shall use the following notations and objects, adapted to the constraint structure, introduced in [22].

(i) Conventions concerning notation of indices:

\begin{matrix} 1 \leq i, j, l \leq n, 1 \leq α, β, γ \leq κ, 1 \leq J \leq n m - κ, \\ 1 \leq σ, ν, ρ \leq m, 1 \leq a, b, c \leq k, 1 \leq p, r, s \leq m - k . \end{matrix}

(4.44)

(ii) Taking into account that the matrix (4.35) in (4.36) has maximal rank, $k$ , one can express $k$ of the contact $1$ -forms $ω^{σ}$ by means of the constraint forms $φ^{a}$ , $1 \leq a \leq m$ , and the remaining $ω^{ν}$ ’s. Without loss of generality we may suppose that this concerns the forms $ω^{m - k + a}$ , where $1 \leq a \leq k$ . In an adapted basis $(x^{i}, y^{σ}, z^{J}, f^{α})$ , and in the notations of the above theorem it holds

\begin{matrix} ω^{m - k + a} = μ_{b}^{a} (φ^{b} - M_{s}^{b} ω^{s} - c_{α j}^{b} f^{α} d x^{j}) \\ = μ_{b}^{a} φ^{b} + G_{s}^{a} ω^{s} + (F_{j}^{a} + G_{s}^{a} y_{j}^{s} - y_{j}^{m - k + a}) d x^{j}, \end{matrix}

(4.45)

where $(μ_{b}^{a})$ is an appropriate regular matrix. Here and in what follows, $y_{j}^{σ}$ are considered as functions of the coordinates $(x^{i}, y^{σ}, z^{J}, f^{α})$ . Similarly, the rank condition (4.32) guarantees that one can express the forms $d z^{α}$ by means of $(d f^{β}, d x^{i}, d y^{σ}, d z^{J})$ . Thus, we have on $J^{1} Y$ the following bases of $1$ -forms, adapted to the constraint structure:

(d x^{i}, d y^{s}, φ^{a}, d z^{J}, d f^{α}), or (d x^{i}, ω^{s}, φ^{a}, d z^{J}, d f^{α});

(4.46)

Consequently, with obvious notations we may write

{\bar{ω}}^{m - k + a} \equiv ι^{*} ω^{m - k + a} = {\bar{ϕ}}^{a} + {\bar{G}}_{s}^{a} {\bar{ω}}^{s},

(4.47)

where ${\bar{ω}}^{s} = ι^{*} ω^{s}$ , and ${\bar{ϕ}}^{a} = ι^{*} (μ_{b}^{a} φ^{b}) = (μ_{b}^{a} \circ ι) ϕ^{b}$ . We can see that, on $Q$ , instead of a canonical basis $(d x^{i}, d y^{σ}, d z^{J})$ , or a basis $(d x^{i}, {\bar{ω}}^{σ}, d z^{J})$ adapted to the induced contact structure, it is worth to work with bases adapted to the constraint structure, where the canonical constraint $1$ -forms appear:

(d x^{i}, d y^{s}, {\bar{ϕ}}^{a}, d z^{J}), (d x^{i}, {\bar{ω}}^{s}, {\bar{ϕ}}^{a}, d z^{J}) .

(4.48)

(iii) Keeping the above notations we can express the functions $G_{s}^{a}$ and $F_{j}^{a}$ appearing in (4.40) as follows:

G_{s}^{a} = μ_{b}^{a} M_{s}^{b}, F_{j}^{a} = y_{j}^{m - k + a} - G_{s}^{a} y_{j}^{s} - μ_{b}^{a} c_{α j}^{b} f^{α} .

(4.49)

We also put

y_{j}^{σ} \circ ι = g_{j}^{σ} .

(4.50)

With this notation,

{\bar{G}}_{s}^{a} = (μ_{b}^{a} M_{s}^{b}) \circ ι, {\bar{F}}_{j}^{a} = g_{j}^{m - k + a} - {\bar{G}}_{s}^{a} g_{j}^{s},

(4.51)

i.e.

g_{j}^{m - k + a} = {\bar{F}}_{j}^{a} + {\bar{G}}_{s}^{a} g_{j}^{s} .

(4.52)

(iv) The vector ﬁelds $\partial_{c} ∕ \partial x^{i}$ and $\partial_{c} ∕ \partial y^{s}$ on $Q$ deﬁned by (4.40) are called constraint partial derivative operators. We put

\begin{matrix} \frac{d_{c}^{'}}{d x^{i}} = \frac{\partial_{c}}{\partial x^{i}} + g_{i}^{s} \frac{\partial_{c}}{\partial y^{s}}, \\ \frac{d_{c}}{d x^{i}} = \frac{\partial_{c}}{\partial x^{i}} + g_{i}^{s} \frac{\partial_{c}}{\partial y^{s}} + z_{i}^{J} \frac{\partial}{\partial z^{J}} = \frac{d_{c}^{'}}{d x^{i}} + z_{i}^{J} \frac{\partial}{\partial z^{J}}, \end{matrix}

(4.53)

and call the above operators the $i$ -th cut constraint total derivative operator and $i$ -th constraint total derivative operator, respectively.

(v) The exterior derivative of a function $f$ on $Q$ is expressed as follows:

d f = \frac{d_{c}^{'} f}{d x^{j}} d x^{j} + \frac{\partial_{c} f}{\partial y^{s}} {\bar{ω}}^{s} + \frac{\partial f}{\partial y^{m - k + a}} {\bar{ϕ}}^{a} + \frac{\partial f}{\partial z^{J}} d z^{J} .

(4.54)

(vi) Next, denote

C_{J j}^{a} = {\bar{G}}_{s}^{a} \frac{\partial g_{j}^{s}}{\partial z^{J}} - \frac{\partial g_{j}^{m - k + a}}{\partial z^{J}} = - \frac{\partial {\bar{F}}_{j}^{a}}{\partial z^{J}} - \frac{\partial {\bar{G}}_{s}^{a}}{\partial z^{J}} g_{j}^{s}, C_{j s}^{a i} = C_{J j}^{a} \frac{\partial z^{J}}{\partial y_{i}^{s}} .

(4.55)

(vii) For $d {\bar{ϕ}}^{a}$ we have

\begin{matrix} d {\bar{ϕ}}^{a} = ({\bar{G}}_{s}^{a} \frac{d_{c}^{'} g_{j}^{s}}{d x^{i}} - \frac{d_{c}^{'} g_{j}^{m - k + a}}{d x^{i}}) d x^{i} \land d x^{j} \\ + ({\bar{G}}_{s}^{a} \frac{\partial_{c} g_{j}^{s}}{\partial y^{r}} + \frac{d_{c}^{'} {\bar{G}}_{r}^{a}}{d x^{j}} - \frac{\partial_{c} g_{j}^{m - k + a}}{\partial y^{r}}) {\bar{ω}}^{r} \land d x^{j} \\ + C_{J j}^{a} d z^{J} \land d x^{j} - \frac{\partial_{c} {\bar{G}}_{s}^{a}}{\partial y^{r}} {\bar{ω}}^{r} \land {\bar{ω}}^{s} - \frac{\partial {\bar{G}}_{s}^{a}}{\partial z^{J}} d z^{J} \land {\bar{ω}}^{s} \\ + ({\bar{G}}_{s}^{a} \frac{\partial g_{j}^{s}}{\partial y^{m - k + b}} - \frac{\partial g_{j}^{m - k + a}}{\partial y^{m - k + b}}) {\bar{ϕ}}^{b} \land d x^{j} - \frac{\partial {\bar{G}}_{s}^{a}}{\partial y^{m - k + b}} {\bar{ϕ}}^{b} \land {\bar{ω}}^{s} . \end{matrix}

(4.56)

There are several interesting particular cases of regular non-holonomic constraints in ﬁeld theory. We wish to mention here very brieﬂy the following ones (precise deﬁnitions and further properties can be found in [22]:

$∙$ Constraints whose canonical distribution is projectable onto a distribution on $Y$ , i.e. constraints that can be modeled by a distribution or codistribution on $Y$ .

$∙$ Constraints whose canonical distribution is completely integrable; these constraints are called semiholonomic, and can be equivalently modeled by a completely integrable, nowhere vertical distribution on $Y$ .

$∙$ Lagrangian constraints: these are characterized by the property that the codistributions ${\tilde{C}}_{U}^{0}$ can be generated by a system of (independent) Lepage $1$ -forms; for Lagrangian constraints it holds

C_{J j}^{a} = 0, C_{j s}^{a i} = 0

(4.57)

for all values of indices. In this context it is interesting to note that for $dim X = 1$ (mechanics) all non-holonomic constraints are Lagrangian.

$∙$ $π$ -adapted constraints: can be locally represented by equations “in normal form”,

y_{j}^{m - k + a} = g_{j}^{a} (x^{i}, y^{σ}, y_{l}^{s}), 1 \leq j \leq n, 1 \leq a \leq k .

(4.58)

These constraints are Lagrangian.

Lagrangian and Hamiltonian systems subjected to $π$ -adapted constraints are studied in detail in [25].

$∙$ Holonomic constraints, deﬁned as ﬁbered submanifolds of $π$ , can again be easily treated in terms of the theory of regular non-holonomic constraints as a (very) particular case. The situation is completely analogous to that in mechanics (for details see [22]).

4.4. Constrained Lagrangian systems. The aim of the section is to introduce the concept of the constrained Lagrangian systems. For more details and the proofs of the assertions we refer to [22].

Let us consider a Lagrangian system on $J^{1} Y$ . Recall from Sec. 4.2 that it is deﬁned to be a ﬁrst-order Lepage class. We write it in the form

[α] = [d Θ_{λ}] = d Θ_{λ} mod Ω^{n + 1, (\geq 2)} (J^{1} Y) .

(4.59)

If $ι : Q \to J^{1} Y$ is a regular non-holonomic constraint and $ℐ (C^{0})$ the corresponding canonical constraint ideal, we have another equivalence, denoted by $\approx$ , on $(n + 1)$ -forms on $Q$ (with the same domain of deﬁnition):

η_{1} \approx η_{2} iff η_{1} - η_{2} = \bar{F} + ϕ,

(4.60)

where $\bar{F}$ is a (local) at least $2$ -contact $(n + 1)$ -form on $Q$ , and $ϕ$ is a constraint $(n + 1)$ -form. We denote by $[[η]]$ the class of $η$ . If $[α]$ is a Lepage class on $J^{1} Y$ then for any of its two elements deﬁned on the same subset of $J^{1} Y$ ,

α_{1} \sim α_{2} \Rightarrow ι^{*} α_{1} \approx ι^{*} α_{2} .

(4.61)

Deﬁnition 4.10. Let $[α] = [d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ and $ι : Q \to J^{1} Y$ a regular non-holonomic constraint. By the associated constrained Lagrangian system we mean the class $[[ι^{*} α]] = [[ι^{*} d Θ_{λ}]]$ . Each form $ι^{*} d Θ_{λ} + ϕ$ , where $ϕ \in ℐ^{n + 1} (C^{0})$ , is called constrained Poincaré-Cartan $(n + 1)$ -form of $λ$ .

Note that every element of $[[ι^{*} d Θ_{λ}]]$ is of the form

\bar{α} = ι^{*} d Θ_{λ} + \bar{F} + ϕ,

(4.62)

where $\bar{F} \in Ω^{n + 1, (\geq 2)} (Q)$ and $ϕ \in ℐ^{n + 1} (C^{0})$ .

Deﬁnition 4.11. Consider the following system of forms on $d o m \bar{α} \subset Q$ :

i_{ξ} \bar{α} \forall ξ \in V (Q) \cap C, ϕ \in C^{0} .

(4.63)

The exterior differential system generated by (4.63) is called constraint Hamiltonian EDS related with $\bar{α}$ , and is denoted by $ℋ_{\bar{α}}$ .

Deﬁnition 4.12. Let $[[ι^{*} d Θ_{λ}]]$ be a constrained Lagrangian system. Then for any representative $\bar{α}$ of the class $[[ι^{*} d Θ_{λ}]]$ , equations for holonomic integral sections of the constraint Hamiltonian exterior differential system $ℋ_{\bar{α}}$ , i.e., the equations

J^{1} γ^{*} i_{ξ} \bar{α} = 0 for every π_{1} -vertical vector ﬁeld ξ \in C,

(4.64)

where $γ : W \to Y$ , $J^{1} γ (W) \subset Q$ , are called constrained Euler–Lagrange equations. Solutions of constrained Euler–Lagrange equations are called constrained extremals.

We note that (on an open subset of $Q$ ) constrained Euler–Lagrange equations do not depend upon the choice of a representative $\bar{α}$ of the class $[[ι^{*} d Θ_{λ}]]$ . This means that with help of a local Lagrangian $λ$ for $E$ we can write the constrained Euler–Lagrange equations in the form

J^{1} γ^{*} i_{ξ} (ι^{*} d Θ_{λ}) = 0 for every π_{1} -vertical vector ﬁeld ξ \in C,

(4.65)

where $γ : W \to Y$ , $J^{1} γ (W) \subset Q$ .

For $λ = L ω_{0}$ denote

\bar{L} = L \circ ι, {\bar{L}}_{a}^{j} = \frac{\partial L}{\partial y_{j}^{m - k + a}} \circ ι,

(4.66)

considered as functions of adapted ﬁbered coordinates $(x^{i}, y^{σ}, z^{J})$ on $Q$ , and put

Θ_{ι^{*} λ} = \bar{L} ω_{0} + \frac{\partial \bar{L}}{\partial y_{j}^{s}} {\bar{ω}}^{s} \land ω_{j} = \bar{L} ω_{0} + \frac{\partial \bar{L}}{\partial z^{J}} \frac{\partial z^{J}}{\partial y_{j}^{s}} {\bar{ω}}^{s} \land ω_{j} .

(4.67)

In keeping with notations in Remark 4.2 we can easily ﬁnd the following relation:

Proposition 4.7.

ι^{*} Θ_{λ} = Θ_{ι^{*} λ} + {\bar{L}}_{a}^{i} C_{i s}^{a j} {\bar{ω}}^{s} \land ω_{j} + {\bar{L}}_{a}^{j} {\bar{ϕ}}^{a} \land ω_{j} .

(4.68)

For convenience of notations let us introduce the $C$ -modiﬁed Euler–Lagrange operator and cut $C$ -modiﬁed Euler–Lagrange operator, respectively:

\begin{matrix} μ_{s} & = \frac{\partial_{c}}{\partial y^{s}} - \frac{d_{c}}{d x^{i}} (\frac{\partial}{\partial y_{i}^{s}}) - \frac{\partial_{c} g_{j}^{r}}{\partial y^{s}} \frac{\partial}{\partial y_{j}^{r}} \\ = \frac{\partial_{c}}{\partial y^{s}} - \frac{d_{c}}{d x^{i}} (\frac{\partial z^{J}}{\partial y_{i}^{s}} \frac{\partial}{\partial z^{J}}) - \frac{\partial_{c} g_{j}^{r}}{\partial y^{s}} \frac{\partial z^{J}}{\partial y_{j}^{r}} \frac{\partial}{\partial z^{J}}, \\ μ_{s}^{'} & = \frac{\partial_{c}}{\partial y^{s}} - \frac{d_{c}^{'}}{d x^{i}} (\frac{\partial}{\partial y_{i}^{s}}) - \frac{\partial_{c} g_{j}^{r}}{\partial y^{s}} \frac{\partial}{\partial y_{j}^{r}} \\ = \frac{\partial_{c}}{\partial y^{s}} - \frac{d_{c}^{'}}{d x^{i}} (\frac{\partial z^{J}}{\partial y_{i}^{s}} \frac{\partial}{\partial z^{J}}) - \frac{\partial_{c} g_{j}^{r}}{\partial y^{s}} \frac{\partial z^{J}}{\partial y_{j}^{r}} \frac{\partial}{\partial z^{J}} . \end{matrix}

(4.69)

Theorem 4.5. Let $λ$ be a Lagrangian in $J^{1} Y$ , $Q \subset J^{1} Y$ a regular non-holonomic constraint. Denote by $γ$ local sections of the ﬁbered manifold $π : Y \to X$ such that $J^{1} γ (W) \subset Q$ . In adapted ﬁbered coordinates, the constrained Euler–Lagrange equations take one of the following equivalent forms:
$(1)$ By means of $L$ ,

(A_{s} + ℬ_{s J}^{i} z_{i}^{J}) \circ J^{2} γ = 0,

(4.70)

where $A_{s}$ , $ℬ_{s J}^{i}$ are given by

\begin{matrix} A_{s} = {\bar{A}}_{s} + {\bar{A}}_{m - k + a} {\bar{G}}_{s}^{a} + ({\bar{B}}_{s ν}^{j i} + {\bar{B}}_{m - k + a ν}^{j i} {\bar{G}}_{s}^{a}) \frac{d_{c}^{'} g_{j}^{ν}}{d x^{i}}, \\ ℬ_{s J}^{i} = ({\bar{B}}_{s ν}^{j i} + {\bar{B}}_{m - k + a ν}^{j i} {\bar{G}}_{s}^{a}) \frac{\partial g_{j}^{ν}}{\partial z^{J}}, \end{matrix}

(4.71)

where $(c f . (4.11))$

{\bar{A}}_{σ} = ɛ_{σ}^{'} (L) \circ ι, {\bar{B}}_{σ ν}^{i j} = - (\frac{\partial^{2} L}{\partial y_{i}^{σ} \partial y_{j}^{ν}}) \circ ι

(4.72)

$(2)$ By means of $\bar{L}$ and ${\bar{L}}_{a}^{j}$ ,

(μ_{s} (\bar{L}) - {\bar{L}}_{a}^{j} μ_{s} (g_{j}^{m - k + a}) - C_{j s}^{a i} \frac{d_{c} {\bar{L}}_{a}^{j}}{d x^{i}}) \circ J^{2} γ = 0,

(4.73)

meaning that the functions $A_{s}$ , $ℬ_{s J}^{i}$ are equivalently expressed as follows:

\begin{matrix} A_{s} = μ_{s}^{'} (\bar{L}) - {\bar{L}}_{a}^{j} μ_{s}^{'} (g_{j}^{m - k + a}) - C_{j s}^{a i} \frac{d_{c}^{'} {\bar{L}}_{a}^{j}}{d x^{i}}, \\ ℬ_{s J}^{i} = - \frac{\partial}{\partial z^{J}} (\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}}) + {\bar{L}}_{a}^{i} \frac{\partial}{\partial z^{J}} ({\bar{G}}_{s}^{a} δ_{i}^{j} - C_{i s}^{a j}) - C_{j s}^{a i} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} \\ = - \frac{\partial}{\partial z^{J}} (\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}}) + {\bar{L}}_{a}^{j} \frac{\partial}{\partial z^{J}} (\frac{\partial g_{j}^{m - k + a}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}}) - C_{j s}^{a i} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} . \end{matrix}

(4.74)

Deﬁnition 4.13. The operator deﬁned by (4.73), i.e.

ℰ_{s}^{C} (\bar{L}, {\bar{L}}_{a}^{j}) = μ_{s} (\bar{L}) - {\bar{L}}_{a}^{j} μ_{s} (g_{j}^{m - k + a}) - C_{j s}^{a i} \frac{d_{c} {\bar{L}}_{a}^{j}}{d x^{i}}

(4.75)

is called the constraint Euler–Lagrange operator.

Remark 4.3. Lagrangian and semiholonomic constraints. Recall that if the constraint $Q$ in $J^{1} Y$ is Lagrangian, then

C_{j s}^{a i} = 0, C_{J j}^{a} = 0

(4.76)

for all values of indices. Consequently, formulas become much simpler. In particular,

ι^{*} Θ_{λ} = Θ_{ι^{*} λ} + {\bar{L}}_{a}^{j} {\bar{ϕ}}^{a} \land ω_{j},

(4.77)

and the constraint Euler–Lagrange operator reads

ℰ_{s}^{C} (\bar{L}, {\bar{L}}_{a}^{j}) = μ_{s} (\bar{L}) - {\bar{L}}_{a}^{j} μ_{s} (g_{j}^{m - k + a}) .

(4.78)

If, moreover, $Q$ is a semiholonomic constraint, i.e., if $d {\bar{ϕ}}^{a} \in ℐ (C^{0})$ for all $a$ , we get

ι^{*} d Θ_{λ} = d Θ_{ι^{*} λ} + a constraint form .

(4.79)

This means that $d Θ_{ι^{*} λ} \approx ι^{*} d Θ_{λ}$ , and even that $d Θ_{ι^{*} λ}$ is a constrained Poincaré–Cartan $(n + 1)$ -form of $λ$ . Then, of course, the constrained Euler–Lagrange equations (4.65) have the equivalent form

J^{1} γ^{*} i_{ξ} d Θ_{ι^{*} λ} = 0 for every π_{1} -vertical vector ﬁeld ξ \in C .

(4.80)

Since in this case $κ = k n$ , we have on $Q$ local coordinates $(x^{i}, y^{σ}, y_{j}^{s})$ , and all formulas take a much simpler form (cf. e.g. [22, 25]).

4.5. Constrained Hamilton–De Donder equations. Let $[α] = [d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ , $ι : Q \to J^{1} Y$ a regular non-holonomic constraint of corank $(κ, k)$ , $[[ι^{*} α]] = [[ι^{*} d Θ_{λ}]]$ the corresponding constrained system on $Q$ . For every $\bar{α} = ι^{*} α \in [[ι^{*} d Θ_{λ}]]$ we have the constraint Hamiltonian exterior differential system $ℋ_{\bar{α}}$ deﬁned on the domain of deﬁnition of $\bar{α}$ , say $U \subset Q$ , and generated by the system of $n$ -forms and $1$ -forms (4.63).

Directly from the deﬁnition of constraint Hamiltonian EDS we can see that if ${\bar{α}}_{1}$ and ${\bar{α}}_{2}$ differ by a constraint form, then $ℋ_{{\bar{α}}_{1}} = ℋ_{{\bar{α}}_{2}}$ .

Deﬁnition 4.14. Let $[d Θ_{λ}]$ be a Lagrangian system on $J^{1} Y$ . For every $α \in [d Θ_{λ}]$ the equivalence class

{\bar{α}}_{ℐ} = ι^{*} α mod ℐ^{n + 1} (C^{0})

(4.81)

is called constrained Hamiltonian system related with $α$ and the constraint $Q$ .

Equations for integral sections of $ℋ_{\bar{α}}$ , i.e.

δ^{*} i_{ξ} \bar{α} = 0 \forall ξ \in V (Q) \cap C, δ^{*} {\bar{ϕ}}^{a} = 0, 1 \leq a \leq k,

(4.82)

where $\bar{α} \in {\bar{α}}_{ℐ}$ and $δ$ is a section of $π_{1} ∣_{Q} : Q \to X$ , are called constrained Hamilton equations.

Integral sections of $ℋ_{\bar{α}}$ are called constrained Hamilton extremals of $[[ι^{*} d Θ_{λ}]]$ .

Similarly as in the unconstrained case, in what follows, we will be interested in constrained Hamiltonian systems that can be completely characterized by constrained Poincaré–Cartan $(n + 1)$ -forms:

Deﬁnition 4.15. Let $[α]$ be a Lagrangian system on $J^{1} Y$ , $ι : Q \to J^{1} Y$ a regular non-holonomic constraint of corank $(κ, k)$ . A constrained Hamiltonian system ${\bar{α}}_{ℐ}$ deﬁned on $U \subset Q$ is called constrained Hamilton–De Donder system of $[α]$ if there exists a Lagrangian $λ$ for $[α]$ such that for every $\bar{α} \in {\bar{α}}_{ℐ}$ ,

\bar{α} = ι^{*} d Θ_{λ} + \bar{F}, where \bar{F} \in Ω_{Y}^{n + 1, (\geq 2)} (U) .

(4.83)

The corresponding constrained Hamilton equations, i.e.

δ^{*} i_{ξ} (ι^{*} d Θ_{λ} + \bar{F}) = 0 \forall ξ \in V (Q) \cap C, δ^{*} {\bar{ϕ}}^{a} = 0, 1 \leq a \leq k,

(4.84)

are called constrained Hamilton–De Donder equations.

Deﬁnition 4.16. A constrained Hamilton–De Donder system ${\bar{α}}_{ℐ}$ is called regular if $ℋ_{\bar{α}}$ contains all the canonical contact $n$ -forms

ι^{*} ω^{σ} \land ω_{i}, 1 \leq σ \leq m, 1 \leq i \leq n .

(4.85)

A constrained Lagrangian system is called De Donder regular if around each point in $Q$ there exists an associated regular constrained Hamilton–De Donder system.

Proposition 4.8. Let $[α]$ be a Lagrangian system on $J^{1} Y$ , $ι : Q \to J^{1} Y$ a regular non-holonomic constraint of corank $(κ, k)$ , ${\bar{α}}_{ℐ}$ an associated regular constrained Hamilton–De Donder system. Then (for all $\bar{α} \in {\bar{α}}_{ℐ}$ ) every integral section of $ℋ_{\bar{α}}$ is holonomic. Consequently, constrained Hamilton–De Donder equations of ${\bar{α}}_{ℐ}$ are equivalent with the constrained Euler–Lagrange equations.

Proof. Let $δ$ be an integral section of $ℋ_{\bar{α}}$ . If $\bar{α}$ is regular then, by deﬁnition,

δ^{*} ({\bar{ω}}^{σ} \land ω_{i}) = 0, 1 \leq σ \leq m, 1 \leq i \leq n .

(4.86)

This implies, however, that for all $σ$ , $δ^{*} {\bar{ω}}^{σ} = 0$ , meaning that $δ$ is a holonomic section in $Q$ . □

Theorem 4.6. Let ${\bar{α}}_{ℐ} = ι^{*} d Θ_{λ} + \bar{F} mod ℐ^{n + 1} (C^{0})$ , where $\bar{F} \in Ω_{Y}^{n + 1, (\geq 2)} (Q)$ , be a constrained Hamilton–De Donder system. The following conditions are equivalent:

For every $\bar{α} \in {\bar{α}}_{ℐ}$ , $\bar{α}$ is regular.
For every $\bar{α} \in {\bar{α}}_{ℐ}$ , a system of generators of $ℋ_{\bar{α}}$ has maximal rank (equal to $(n + 1) (m - k) + k$ ).
Every ﬁrst-order Lagrangian $λ$ for $d Θ_{λ}$ satisﬁes the constraint regularity condition
$r a n k (ℬ_{s J}^{i}) = n (m - k),$ (4.87)

where $ℬ_{s J}^{i}$ are given in terms of $λ = L ω_{0}$ by 4.71 or 4.74.

Proof. We can write

\bar{α} = A_{s} {\bar{ω}}^{s} \land ω_{0} + ℬ_{s J}^{i} {\bar{ω}}^{s} \land d z^{J} \land ω_{i} + ℱ + ϕ,

(4.88)

where $A_{s}$ and $ℬ_{s J}^{i}$ are given in terms of a ﬁrst-order Lagrangian $λ$ for $d Θ_{λ}$ by (4.71) or (4.74), $ℱ$ is the sum of $\bar{F}$ and the $2$ -contact part of $ι^{*} d Θ_{λ}$ , and $ϕ \in ℐ^{n + 1} (C^{0})$ . Denote

ℱ = ℱ_{l s}^{i} {\bar{ω}}^{l} \land {\bar{ω}}^{s} \land ω_{i} + ν, where ℱ_{l s}^{i} = - ℱ_{s l}^{i} .

(4.89)

Computing generators of $ℋ_{\bar{α}}$ , we obtain a mixed system of $k$ (linearly independent) $1$ -forms

{\bar{ϕ}}^{a} = {\bar{ω}}^{m - k + a} - \sum_{s = 1}^{m - k} {\bar{G}}_{s}^{a} {\bar{ω}}^{s}, 1 \leq a \leq k,

(4.90)

and $m - k + m n - κ$ $n$ -forms (that, in general, need not be independent) as follows:

\begin{matrix} A_{s} ω_{0} + 2 ℱ_{s l}^{i} {\bar{ω}}^{l} \land ω_{i} + ℬ_{s J}^{i} d z^{J} \land ω_{i} + μ_{s}, \\ ℬ_{s J}^{i} {\bar{ω}}^{s} \land ω_{i}, \end{matrix}

(4.91)

$1 \leq s \leq m - k$ , $1 \leq J \leq m n - κ$ , where $μ_{s} = i_{\partial ∕ \partial y^{s}} ν$ .

Suppose that ${\bar{α}}_{ℐ}$ is regular. Then $ℋ_{\bar{α}}$ is generated by the forms ${\bar{ϕ}}^{a}$ and

A_{s} ω_{0} + ℬ_{s J}^{i} d z^{J} \land ω_{i} + μ_{s}, {\bar{ω}}^{s} \land ω_{i},

(4.92)

which means that the matrix $ℬ_{1} = (ℬ_{s J}^{i})$ with $m n - κ$ rows labelled by $J$ and $n (m - k)$ columns labelled by $i, s$ , has rank $n (m - k)$ . Now, the rank of the system of generators of $ℋ_{\bar{α}}$ is $> n (m - k)$ . Let us compute the rank of the matrix $ℬ_{2} = (ℬ_{s J}^{i})$ with $m - k$ rows labelled by $s$ and $n (n m - κ)$ columns labelled by $i, J$ . By the above we can see that for every ﬁxed $i$ , the matrix $ℬ_{1}$ has $m - k$ linearly independent columns (labelled by $s$ ). Consequently, $ℬ_{2}$ has for every ﬁxed $i$ the submatrix $(ℬ_{s J}^{∙})$ with $m - k$ independent rows labelled by $s$ (equal to transposed submatrix of $ℬ_{2}$ with the corresponding values of indices), hence $r a n k ℬ_{2} = m - k$ . Summarizing, we have obtained that the forms (4.91) (resp. (4.92)) are independent, meaning that the rank of the system of generators of $ℋ_{\bar{α}}$ (for all $\bar{α} \in {\bar{α}}_{ℐ}$ ) is maximal and equal to $n (m - k) + m - k + k = (n + 1) (m - k) + k$ , as desired.

Next, suppose that the rank of $ℋ_{\bar{α}}$ is maximal. Then the forms (4.91) are independent, meaning that the constrained regularity condition (4.87) holds.

Finally, if (4.87) holds, then the forms $ℬ_{s J}^{i} {\bar{ω}}^{s} \land ω_{i}$ in (4.91) are independent, i.e. the forms ${\bar{ω}}^{s} \land ω_{i}$ , $1 \leq i \leq n$ , $1 \leq s \leq m - k$ , belong to $ℋ_{\bar{α}}$ . By deﬁnition of $ℋ_{\bar{α}}$ also all the $n$ -forms ${\bar{ϕ}}^{a} \land ω_{i} \in ℋ_{\bar{α}}$ . Hence, for all $a, i$ , we have $ω^{m - k + a} \land ω_{i} = {\bar{ϕ}}^{a} \land ω_{i} + {\bar{G}}_{s}^{a} {\bar{ω}}^{s} \land ω_{i} \in ℋ_{\bar{α}}$ , and we are done. □

Corollary 4.3. Let

{\bar{α}}_{ℐ} = ι^{*} d Θ_{λ} + \bar{F} mod ℐ^{n + 1} (C^{0}), where \bar{F} \in Ω_{Y}^{n + 1, (\geq 2)} (Q),

be a regular constrained Hamilton–De Donder system. Then Hamilton extremals of ${\bar{α}}_{ℐ}$ (i.e. integral sections of $(4.90)$ and $(4.92)$ ) do not depend upon the choice of $\bar{F}$ . This means that Hamilton–De Donder equations of all elements in the class

ι^{*} d Θ_{λ} mod Ω_{Y}^{n + 1, (\geq 2)} (Q) + ℐ^{n + 1} (C^{0})

(4.93)

are equivalent.

From the proof of Theorem 4.6 we can conclude that if ${\bar{α}}_{ℐ}$ is regular then the matrix $ℬ_{1}$ has $n m - κ$ rows and $n (m - k)$ columns where $n m - κ \geq n (m - k)$ . This means that $κ \leq n k$ , and we get the following result:

Corollary 4.4. Let $[α]$ be a Lagrangian system on $J^{1} Y$ , $ι : Q \to J^{1} Y$ a regular non-holonomic constraint of corank $(κ, k)$ . A necessary condition for the constrained Lagrangian system $[[ι^{*} α]]$ be De Donder regular is

κ \leq n k .

(4.94)

4.6. Constraint Legendre transformation.

Theorem 4.7. Let $ι : Q \to J^{1} Y$ be a regular non-holonomic constraint of corank $(κ, k)$ , $[[\bar{α}]]$ a constrained Lagrangian system. Let $x \in Q$ be a point. Suppose that in a neighborhood of $x$ ,

\frac{\partial ℬ_{s J}^{i}}{\partial z^{K}} = \frac{\partial ℬ_{s K}^{i}}{\partial z^{J}}, 1 \leq i \leq n, 1 \leq s \leq m - k, 1 \leq J, K \leq m n - κ .

(4.95)

Then there exists a neighborhood $U \subset Q$ of $x$ , and, on $U$ , functions $P_{s}^{i}$ , $1 \leq i \leq n$ , $1 \leq s \leq m - k$ , and a $n$ -form $η$ , such that the class $[[\bar{α}]]$ is represented by the $(n + 1)$ -form

\bar{α} = η \land ω_{0} + d P_{s}^{i} \land d y^{s} \land ω_{i} .

(4.96)

Proof. As we have seen in Sec. 4.4, around each point in $Q$ , the constrained Lagrangian system $[[\bar{α}]]$ has a representative

A_{s} {\bar{ω}}^{s} \land ω_{0} + ℬ_{s J}^{i} {\bar{ω}}^{s} \land d z^{J} \land ω_{i},

(4.97)

where $A_{s}$ and $ℬ_{s J}^{i}$ are deﬁned by $(4.71)$ or $(4.74)$ . Assume that the given Lagrangian system $[\bar{α}]$ has a Lagrangian $λ = L ω_{0}$ deﬁned around $x$ such that for the corresponding functions $ℬ_{s J}^{i}$ integrability conditions (4.95) are satisﬁed. Applying the Poincaré Lemma we get a neighborhood $U \subset Q$ of $x$ and functions $P_{s}^{i}$ , $1 \leq i \leq n$ , $1 \leq s \leq m - k$ , on $U$ that are given by

ℬ_{s J}^{i} = - \frac{\partial P_{s}^{i}}{\partial z^{J}} .

(4.98)

Hence, in the class $[[\bar{α}]]$ on $U$ we can ﬁnd the following representatives, equivalent with $ι^{*} d Θ_{λ}$ , where $λ$ is the above mentioned Lagrangian:

\begin{matrix} A_{s} {\bar{ω}}^{s} \land ω_{0} + \frac{\partial P_{s}^{i}}{\partial z^{J}} d z^{J} \land {\bar{ω}}^{s} \land ω_{i} \\ \approx A_{s} {\bar{ω}}^{s} \land ω_{0} + d P_{s}^{i} \land {\bar{ω}}^{s} \land ω_{i} - \frac{d_{c}^{'} P_{s}^{i}}{d x^{j}} d x^{j} \land {\bar{ω}}^{s} \land ω_{i} \\ = (A_{s} + \frac{d_{c}^{'} P_{s}^{i}}{d x^{i}}) d y^{s} \land ω_{0} - g_{i}^{s} d P_{s}^{i} \land ω_{0} + d P_{s}^{i} \land d y^{s} \land ω_{i} \\ \approx (A_{s} + \frac{d_{c}^{'} P_{s}^{i}}{d x^{i}} - g_{i}^{l} \frac{\partial_{c} P_{l}^{i}}{\partial y^{s}}) d y^{s} \land ω_{0} \\ - g_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{J}} d z^{J} \land ω_{0} + d P_{s}^{i} \land d y^{s} \land ω_{i} . \end{matrix}

(4.99)

In this way we have obtained a representative

\begin{matrix} \bar{α} = (A_{s} + \frac{d_{c}^{'} P_{s}^{i}}{d x^{i}} - g_{i}^{r} \frac{\partial_{c} P_{r}^{i}}{\partial y^{s}}) d y^{s} \land ω_{0} \\ - g_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{J}} d z^{J} \land ω_{0} + d P_{s}^{i} \land d y^{s} \land ω_{i} . \end{matrix}

(4.100)

Let us denote

\bar{α} = η \land ω_{0} + d P_{s}^{i} \land d y^{s} \land ω_{i},

(4.101)

with

η = {\tilde{η}}_{j} d x^{j} + {\bar{η}}_{s} d y^{s} + {\bar{η}}_{J} d z^{J},

(4.102)

where ${\tilde{η}}_{j}$ , $1 \leq j \leq n$ , are arbitrary functions on $U$ , and

\begin{matrix} {\bar{η}}_{s} & = & A_{s} + \frac{d_{c}^{'} P_{s}^{i}}{d x^{i}} - g_{i}^{r} \frac{\partial_{c} P_{r}^{i}}{\partial y^{s}}, 1 \leq r, s \leq m - k, \\ {\bar{η}}_{J} & = & - g_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{J}}, 1 \leq s \leq m - k, 1 \leq J \leq m n - κ . \end{matrix}

(4.103)

This completes the proof. $□$ □

Corollary 4.5. The class

{\bar{α}}_{ℐ} = η \land ω_{0} + d P_{s}^{i} \land d y^{s} \land ω_{i} mod ℐ^{n + 1} (C^{0})

(4.104)

constructed in the above Theorem is a constrained Hamilton–De Donder system, corresponding to the Lagrangian $λ = L ω_{0}$ .

Proof. It is sufficient to check the computations to see that that $\bar{α} - ι^{*} d Θ_{λ}$ is (up to a constraint form) a $2$ -contact form, horizontal with respect to the projection onto $Y$ . □

Note that

η

in (4.104) is determined up to a constraint form. In this way, for a constrained Hamilton–De Donder system we have to consider the class

η_{ℐ} = η mod ℐ^{n} (C^{0}) .

(4.105)

It should be stressed that in the class $η_{ℐ}$ there need not exist a closed representative.

Remark 4.4. Integrability condition (4.95) can be rewritten in terms of a Lagrangian $λ = L ω_{0}$ giving rise to the functions $ℬ_{s J}^{i}$ using $(4.71)$ or $(4.74)$ . For example, with help of $(4.74)$ it takes the form

\begin{matrix} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{K}} \frac{\partial}{\partial z^{J}} (\frac{\partial g_{j}^{m - k + a}}{\partial z^{I}} \frac{\partial z^{I}}{\partial y_{i}^{s}}) - \frac{\partial C_{j s}^{a i}}{\partial z^{K}} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} \\ = \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} \frac{\partial}{\partial z^{K}} (\frac{\partial g_{j}^{m - k + a}}{\partial z^{I}} \frac{\partial z^{I}}{\partial y_{i}^{s}}) - \frac{\partial C_{j s}^{a i}}{\partial z^{J}} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{K}} . \end{matrix}

(4.106)

Let us ﬁnd an explicit formula for the functions $P_{s}^{i}$ .

Proposition 4.9. Let $x \in U$ , and consider a mapping $χ : [0, 1] \times W \to W$ , deﬁned by

(u, x^{i}, y^{σ}, z^{J}) \to (x^{i}, y^{σ}, u z^{J}),

(4.107)

where $W \subset U \subset Q$ is an appropriate neighborhood of $x$ . Then for arbitrary functions $ψ_{s}^{i} (x^{j}, y^{ν})$ (resp. ${\tilde{ψ}}_{s}^{i} (x^{j}, y^{ν})$ ), $1 \leq s \leq m - k$ , $1 \leq i \leq n$ , the functions

\begin{matrix} P_{s}^{i} = - z^{J} \int_{0}^{1} (ℬ_{s J}^{i} \circ χ) d u + ψ_{s}^{i} (x^{j}, y^{ν}) \\ = \frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}} + z^{J} \int_{0}^{1} (C_{j s}^{a i} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} - {\bar{L}}_{a}^{j} \frac{\partial}{\partial z^{J}} (\frac{\partial g_{j}^{m - k + a}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}})) \circ χ d u \\ + {\tilde{ψ}}_{s}^{i} (x^{j}, y^{ν}), \end{matrix}

(4.108)

are solutions of $(4.98)$ .

Proof. Integrability condition (4.95) guarantees that in a neighborhood of any point of $U$ one can ﬁnd solutions of (4.98) using Poincaré Lemma. Put

P_{s}^{i} = - z^{J} \int_{0}^{1} (ℬ_{s J}^{i} \circ χ) d u + ψ_{s}^{i} (x^{j}, y^{ν}) .

(4.109)

Then with help of (4.95)

\begin{matrix} \frac{\partial P_{s}^{i}}{\partial z^{J}} & = - \int_{0}^{1} (ℬ_{s J}^{i} \circ χ) d u - z^{K} \int_{0}^{1} (\frac{\partial ℬ_{s K}^{i}}{\partial z^{J}} \circ χ) u d u \\ = - \int_{0}^{1} d (u (ℬ_{s J}^{i} \circ χ)) = - ℬ_{s J}^{i}, \end{matrix}

(4.110)

as desired.

Using formula (4.74) and (4.109) we get

\begin{matrix} P_{s}^{i} = z^{J} \int_{0}^{1} (\frac{\partial}{\partial z^{J}} (\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}}) + C_{j s}^{a i} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} \\ - {\bar{L}}_{a}^{j} \frac{\partial}{\partial z^{J}} (\frac{\partial g_{j}^{m - k + a}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}})) \circ χ d u + ψ_{s}^{i} (x^{j}, y^{ν}) \\ = \frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}} + z^{J} \int_{0}^{1} (C_{j s}^{a i} \frac{\partial {\bar{L}}_{a}^{j}}{\partial z^{J}} - {\bar{L}}_{a}^{j} \frac{\partial}{\partial z^{J}} (\frac{\partial g_{j}^{m - k + a}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}})) \circ χ d u \\ + {\tilde{ψ}}_{s}^{i} (x^{j}, y^{ν}), \end{matrix}

(4.111)

since

\begin{matrix} \int_{0}^{1} d (\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}} \circ χ) = {[\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}} \circ χ]}_{u = 0}^{u = 1} = \frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}} - h_{s}^{i} (x^{j}, y^{ν}) \\ = \int_{0}^{1} \frac{d}{d u} (\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}} \circ χ) d u = z^{J} \int_{0}^{1} (\frac{\partial}{\partial z^{J}} (\frac{\partial \bar{L}}{\partial z^{K}} \frac{\partial z^{K}}{\partial y_{i}^{s}})) \circ χ d u . \end{matrix}

This completes the proof. $□$ □

Keeping the above notations we can introduce the following concepts:

Deﬁnition 4.17. The local representative $\bar{α}$ (4.96) of the constrained Lagrangian system is called a representative in canonical form. Functions $P_{s}^{i}$ are called constraint momenta, and (any) $1$ -form $η$ in (4.104) is called energy $1$ -form associated with the corresponding Hamilton–De Donder system ${\bar{α}}_{ℐ}$ .

By the next theorem, constraint Legendre transformation, associated with a regular constrained Hamilton–De Donder system is deﬁned. It is a local coordinate transformation on the constraint submanifold $Q$ .

Theorem 4.8. Let $[[\bar{α}]]$ be a De Donder regular constrained Lagrangian system on $Q$ , let ${\bar{α}}_{ℐ}$ be a corresponding regular Hamilton–De Donder system on a coordinate neighborhood $U \subset Q$ , $P_{s}^{i}$ , $1 \leq i \leq n$ , $1 \leq s \leq m - k$ , its associated constraint momenta. Then the set of functions $(x^{i}, y^{σ}, P_{s}^{i})$ can be completed to coordinates on $U$ . In particular, the set of $n m - κ$ indices labelled by $J$ has a subset labelled by $B$ , $n (m - k) + 1 \leq B \leq n m - κ$ , such that

(x^{i}, y^{σ}, z^{J}) \to (x^{i}, y^{σ}, P_{s}^{i}, z^{B})

(4.112)

is a coordinate transformation on $U$ .

Proof. By assumption, the matrix $(ℬ_{s J}^{i}) = (\partial P_{s}^{i} ∕ \partial z^{J})$ has maximal rank (equal to $n (m - k)$ ). This means that it has a regular submatrix with $n (m - k)$ linearly independent rows: we can label them by $A$ , where $1 \leq A \leq n (m - k)$ . The remaining rows will be labelled by $B$ , i.e. $n (m - k) + 1 \leq B \leq n m - κ$ . Now, the map $(x^{i}, y^{σ}, z^{J}) \to (x^{i}, y^{σ}, P_{s}^{i}, z^{B})$ is regular, i.e. is a coordinate transformation on $U$ . $□$ □

Finally, let us ﬁnd the expression of constrained Hamilton–De Donder equations in constraint Legendre coordinates. First, we have up to a constraint form,

η = η_{j} d x^{j} + η_{s} d y^{s} + η_{i}^{s} d P_{s}^{i} + η_{B} d z^{B},

(4.113)

hence,

\begin{matrix} η = η_{i} d x^{i} + η_{s} d y^{s} + η_{i}^{s} (\frac{d_{c}^{'} P_{s}^{i}}{d x^{j}} d x^{j} + \frac{\partial_{c} P_{s}^{i}}{\partial y^{r}} {\bar{ω}}^{r} + \frac{\partial P_{s}^{i}}{\partial z^{J}} d z^{J}) + η_{B} d z^{B} \\ = (η_{j} + η_{i}^{s} \frac{d_{c}^{'} P_{s}^{i}}{d x^{j}} - η_{i}^{s} \frac{\partial_{c} P_{s}^{i}}{\partial y^{r}} g_{j}^{r}) d x^{j} + (η_{r} + η_{i}^{s} \frac{\partial_{c} P_{s}^{i}}{\partial y^{r}}) d y^{r} \\ + η_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{A}} d z^{A} + (η_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{B}} + η_{B}) d z^{B} . \end{matrix}

(4.114)

Comparing with (4.102), (4.103) we obtain

\begin{matrix} {\bar{η}}_{r} = η_{r} + η_{i}^{s} \frac{\partial_{c} P_{s}^{i}}{\partial y^{r}} = A_{r} + \frac{d_{c}^{'} P_{r}^{i}}{d x^{i}} - g_{i}^{s} \frac{\partial_{c} P_{s}^{i}}{\partial y^{r}}, \\ {\bar{η}}_{A} = η_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{A}} = - g_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{A}}, \\ {\bar{η}}_{B} = η_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{B}} + η_{B} = - g_{i}^{s} \frac{\partial P_{s}^{i}}{\partial z^{B}} . \end{matrix}

(4.115)

The second relation gives us

(η_{i}^{s} + g_{i}^{s}) \frac{\partial P_{s}^{i}}{\partial z^{A}} = 0, i . e . η_{i}^{s} = - g_{i}^{s} (x^{j}, y^{ν}, P_{l}^{j}, z^{B}),

(4.116)

since the matrix $(\frac{\partial P_{s}^{i}}{\partial z^{A}})$ is regular. The ﬁrst and third relation above then read

η_{r} = {\bar{η}}_{r} + g_{i}^{s} \frac{\partial_{c} P_{s}^{i}}{\partial y^{r}} = A_{r} + \frac{d_{c}^{'} P_{r}^{i}}{d x^{i}}, η_{B} = 0 .

(4.117)

Next, we can see that the vertical subbundle of the canonical distribution is in Legendre coordinates generated by the vector ﬁelds $\partial_{c} ∕ \partial y^{s}$ , $\partial ∕ \partial P_{s}^{i}$ , and $\partial ∕ \partial z^{B}$ . Computing contractions of a representative $\bar{α}$ of (4.104) by these vector ﬁelds, we get the constrained Hamilton–De Donder equations $δ^{*} i_{ξ} \bar{α} = 0$ , $δ^{*} {\bar{ϕ}}^{a} = 0$ in the following “canonical form”:

Theorem 4.9. Constrained Hamilton–De Donder equations $(4.84)$ in constraint Legendre coordinates take the form

\begin{matrix} \frac{\partial (P_{s}^{i} \circ δ)}{\partial x^{i}} = η_{s} \circ δ, \frac{\partial (y^{s} \circ δ)}{\partial x^{i}} = - η_{i}^{s} \circ δ, \\ 1 \leq s \leq m - k, 1 \leq i \leq n, \end{matrix}

(4.118)

\frac{\partial (y^{m - k + a} \circ δ)}{\partial x^{i}} = g_{i}^{m - k + a} \circ δ, 1 \leq a \leq k, 1 \leq i \leq n .

(4.119)

Remark 4.5. In view of Theorem 4.8 we can see that for a regular constrained Lagrangian system one has on $Q$ local adapted coordinates $(x^{i}, y^{σ}, y_{j}^{s}, z^{B})$ , where $1 \leq i \leq n$ , $1 \leq σ \leq m$ , $1 \leq s \leq k$ , $n (m - k) + 1 \leq B \leq n m - κ$ , and corresponding adapted bases of $1$ -forms $(d x^{i}, d y^{s}, {\bar{ϕ}}^{a}, d y_{j}^{s}, d z^{B})$ , respectively, $(d x^{i}, {\bar{ω}}^{s}, {\bar{ϕ}}^{a}, d y_{j}^{s}, d z^{B})$ . In these coordinates many formulas simplify, since $z^{A} = y_{j}^{s}$ (hence $\partial z^{B} ∕ \partial y_{j}^{s} = 0$ ).

The results on constrained Hamilton–De Donder systems can be easily reformulated for special cases of constraints. Let us recall the properties of semiholonomic constraints, that are quite similar to the unconstrained case [25].

Theorem 4.10. Let $Q$ be a semiholonomic constraint in $J^{1} Y$ (i.e. $κ = k n$ and $d ℐ (C^{0}) \subset ℐ (C^{0})$ ), ${\bar{α}}_{ℐ}$ a constrained Hamilton–de Donder system. Then the integrability condition $(4.95)$ is satisﬁed identically and the regularity condition reads

det (\frac{\partial^{2} \bar{L}}{\partial y_{i}^{l} \partial y_{j}^{s}}) \neq 0 .

(4.120)

Constrained momenta are given by the formula

P_{s}^{i} = \frac{\partial \bar{L}}{\partial y_{i}^{s}},

(4.121)

the class of energy $1$ -forms $η_{ℐ}$ contains a closed form $d \bar{H}$ , where

\bar{H} = - \bar{L} + P_{s}^{i} y_{i}^{s},

(4.122)

and constraint Legendre transformation is a local diffeomorphism

(x^{i}, y^{σ}, y_{j}^{s}) \to (x^{i}, y^{σ}, P_{s}^{j})

(4.123)

on the submanifold $Q \subset J^{1} Y$ .

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Department of Algebra and Geometry, Faculty of Science Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic; Department of Mathematics, La Trobe University, Melbourne, Victoria 3086, Australia

E-mail address: krupkova@inf.upol.cz

Petr Volný Department of Mathematics and Descriptive Geometry, VSB - Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic

E-mail address: Petr.Volny@vsb.cz

Received July 24, 2006