# Variational calculus with constraints on general algebroids

###### Abstract

Variational calculus on a vector bundle equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for .

MSC 2000: 70H03, 70H25, 53D17, 17B66, 53D10.

Key words: Lie algebroids, variational calculus, Lagrangian functions, Euler-Lagrange equations, vakonomic constraints, nonholonomic constraints.

## 1 Introduction

The Classical Analytical Mechanics is an old and well-established part of both mathematics and physics. Nevertheless many people still look for the best mathematical tools in describing various aspects of mechanical systems. A use of Lie algebroids and Lie groupoids for describing some systems of the Classical Mechanics was proposed by P. Libermann [Li] and A. Weinstein [We] more than ten years ago. This turned out to be a very fruitful idea and since then much work has been done (e.g. [CLMMM, GGU3, LMM, IMMS, IMPS, Mar1, M1]) making use of Lie algebroids in various aspects of Classical Mechanics and Classical Field Theory. The need of extending the geometrical tools of the Lagrangian formalism from just tangent bundles to Lie algebroids is justified by the fact that reductions usually move us out of the environment of the tangent bundles (think on the rigid body). It is similar to the better-known situation of passing from the symplectic to the Poisson structures in the Hamiltonian formalism.

In the paper [GGU3] it was observed that, following some ideas of W. M. Tulczyjew and using general algebroids instead of just Lie algebroids, one can describe a larger class of systems in a simple and elegant way, both in the Lagrangian and in the Hamiltonian formulation. Moreover, the proposed geometric picture does not require considering prolongations of Lie algebroids we start with, as it was in the case of previous approaches known in the literature. A further paper [GGU4], was devoted, in turn, to the construction of Euler-Lagrange equations in the affine setting of so called special affgebroids which is particularly suitable for time-dependent systems.

In the present paper we concentrate on variational calculus and constraints in the algebroid setting. We work with a general algebroid, defined in [GU2] as a double vector bundle morphism

(1.1) |

covering the identity on . Here is a vector bundle playing the role of kinematic configurations. To some extent then, our paper can be understood as a natural generalization of [Mar3], where a variational calculus on Lie algebroids has been developed according to the original ideas of A. Weinstein [We], and of [CLMM, IMMS], where constraints on Lie algebroids have been considered. On the other hand, our approach is definitely different from the approaches known in the literature, even when the equations we obtain cover the corresponding Euler-Lagrange equations in the Lie algebroid case. This is mainly because we adapt the framework of the Tulczyjew triple [Tu1, Tu3, TU], working simply with the morphism (1.1) rather than following the Klein’s method [Kl] generalized to Lie algebroids, in which the bundles tangent to and are replaced by the prolongations of with respect to the vector bundle projections and . This, in our opinion, simplifies the whole formalism substantially.

To define a variational problem on an algebroid we have to specify a manifold of paths whose tangent space represents all possible variations and an action functional on . Then we have to choose a submanifold of admissible paths and a set (generalized distribution) of admissible variations of admissible paths. In [Mar3] admissible variations are constructed out of homotopies of admissible paths as defined in [CF]. For general algebroids we need different way of constructing admissible variations, since we have to accept the fact that they are not tangent to the submanifold of admissible paths in general. Therefore we construct admissible variations for an admissible path in out of vertical variations of in , i.e. out of vertical vector fields along . Note that the variations are defined in (which is in the standard variational calculus), not in . This is because the variational calculus on algebroids leads to first-order differential equations in rather than to second-order equations in . This is only the case of the canonical Lie algebroid when paths in are in one to one correspondence with admissible paths in , this time – just tangent prolongations of paths in , and admissible variations are tangent prolongations of variations of paths in . For a general algebroid the admissible variations are constructed from the vertical ones by means of the double vector bundle relation

It is clear from our variational picture that putting constraints must result in defining a subset of . In the case of a general algebroid our classification of the constraints is based on the way in which the constrained admissible variations are constructed. According to the tradition we call them: vakonomic, non-holonomic, and holonomic constraints. Starting from a subset of , classically understood as a geometric constraint for velocities, we have at least two natural possibilities of constructing a constraint in admissible variations: one is to consider only admissible variations which are tangent to (vakonomic constraint), the other – to consider only admissible variations coming from those vertical ones which are tangent to (nonholonomic constraint). Note that our approach allows to understand nonholonomic constraint as a constrained variational problem, contrary to the commonly accepted conviction. A nonholonomic constraint is called holonomic if the constrained admissible variations are tangent to (are vakonomic). Sometimes it is hard to decide without making an experiment which method should be used to describe the real behavior of the system.

For all types of constraints we construct analogs of the Euler-Lagrange equation for systems that are subject to those three types of constraints in variational way. Note however that the corresponding equations describe ”regular” solutions rather than a general solution of the variational problem. Additionally, like for non-constrained cases in [GGU3], we derive the equations purely geometrically, without referring to the variational calculus.

The literature concerning constraints in Variational Calculus is so extensive that there it is impossible to cite it in a complete way. We decided to list among references only papers dealing actually with Lie algebroids or being direct inspiration for the framework we propose. Let us also make it clear that we see the meaning of the present paper not only as a generalization of formalisms of Classical Mechanics. Working with the case of a general algebroid forced us to propose a geometric approach which seems to be new and illustrative even when applied to very classical situations. The main observation is that an algebroid structure is a crucial geometric ingredient in constructing the dynamics of the system. It tells us not only the configurations, velocities and inner degrees of freedom, but it contains the information on how the admissible variations should be produced from a simple geometric model of variations of paths in a vector bundle – the vertical ones. This structure is encoded in a single map (1.1) respecting double vector bundle structures. The brackets and the Jacobi identity are therefore proven to play a minor role. The Jacobi identity for an algebroid bracket ensures some integrability conditions that allow us to integrate the Lie algebroid into an (at least local) Lie groupoid (see [CF]), but which is irrelevant for the possibility of constructing Euler-Lagrange equations. Fixing this geometric setting for our system, it is then the Lagrangian function which produces a concrete dynamics out of these data. However, we would like to stress that regularity of the Lagrangian is completely irrelevant for our picture. The general method of constructing dynamics out of the Lagrangian works for all Lagrangians, singular or not. The difficulty with singular Lagrangians is that the dynamics we obtain is really implicit and complicated. In other words, difficulty with singular Lagrangians lies in difficulty in solving equations, not in the geometric construction of the equations themselves.

Finally, if the variational calculus is concerned, only admissible paths come to the play. This is because we work on the bundle of kinematic configurations and considering only admissible paths corresponds, classically, to work with paths in the manifold of position configurations lifted canonically to the paths in . The geometrical model of (infinitesimal) variations of an admissible path is to consider vertical vector fields along . Now, the true (mechanical) admissible variations are vector fields along constructed from the vertical ones out of the algebroid structure . This is how the algebroid structure comes to the variational picture. Note that the role of the (Lie) algebroid structure in the classical setting is usually overlooked, since it is hidden behind structures of the tangent and cotangent bundles which are viewed as a natural part of the theory.

The paper is organized as follows. In Section 2 we set up the notation and we recall the notion of general algebroid as a double vector bundle morphism. Then we introduce the relation that is used for defining admissible variations. In Section 3 we discuss the Lagrange formalism without constraints on general algebroid. Then we pass in Section 4 to the variational calculus. We derive the variation of the Lagrangian and Euler-Lagrange equations. The final section is devoted to constraints. Geometric constraints as subsets give rise to variational constraints which are classified in pure geometrical terms as vakonomic, nonholonomic, or holonomic. We derive constrained equations using variational motivations and give them pure geometric interpretations.

## 2 Lie algebroids as double vector bundle morphisms

We start with introducing some notation.

Let be a smooth manifold and let , be a coordinate system in . We denote by the tangent vector bundle and by the cotangent vector bundle. We have the induced (adapted) coordinate systems in and in . Let be a vector bundle and let be the dual bundle. Let be a basis of local sections of and let be the dual basis of local sections of . We have the induced coordinate systems:

where the linear functions are given by the canonical pairing . Thus we have local coordinates

It is well known (cf. [KU, Ur]) that the cotangent bundles and are examples of double vector bundles: