Nonholonomic Constraints: a New Viewpoint
The purpose of this paper is to show that, at least for Lagrangians of mechanical type, nonholonomic Euler-Lagrange equations for a nonholonomic linear constraint may be viewed as non-constrained Euler-Lagrange equations but on a new (generally not Lie) algebroid structure on . The proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of Neoether Theorem producing actual first integrals in case of symmetries.
Key words and phrases:General algebroids, Lie algebroids, double vector bundle, Nonholonomic mechanics, Lagrange-d’Alembert’s equations, nonholonomic bracket.
2000 Mathematics Subject Classification:70F25, 53D17, 70G45, 17B66, 70H03, 70H45
There are many approaches to geometric mechanics in the literature. We will work with a natural generalization of the framework for studying mechanical systems proposed by W. M. Tulczyjew [Tul1, Tul2] (see also [TU] and references therein). In the simplest form, the phase dynamics of the system is understand as the lagrangian submanifold of the symplectic manifold equipped with tangent lift of the canonical symplectic form of . Here, represents the configuration manifold of the system and is obtained from the lagrangian submanifold induced by the Lagrangian via the canonical isomorphism . In other words, the phase dynamics, as well as the Euler-Lagrange equations, are obtained in a simple way by means of the Tulczyjew differential . It is important to observe that both and are double vector bundles over and (see [GU2] and references therein). The resulting submanifold of is a particular case of modelling dynamical systems as implicit differential equations defined by differential inclusions (see [MMT1, MMT2]).
This framework admits an immediate generalization for more general morphisms of canonical double vector bundles associated with a vector bundle (see [GGU, GG]), inducing the Tulczyjew differentials . This generalization includes as a particular case a theory of mechanical systems based on Lie algebroids, as proposed by A. Weinstein [We] and developed by many authors, however in a different geometrical setting (see, for instance, [CoLeMaMaMa, CoLeMaMa, LMM, Li, mart, Medina]). The motivation for study systems on Lie algebroids is that they often appear naturally as results of some reduction procedures. This is is a situation similar to the one known in the theory of Hamiltonian systems: reductions may lead from a symplectic to a Poisson structure.
An additional challenge and one of the most fascinating topics in geometric mechanics is the study of constraints in this context. Of course, a general problem of putting constraints for the system in a variational setting involves constraints for velocities as well as constraints for virtual displacements, as was noticed already in [Tu2]. In some cases, however, one assumes that the constraints can be determined from a constraint subset of (or, of in the algebroid case) by certain well-described procedures. The best known approaches of this type refer to the so called vakonomic and nonholonomic constraints. In the simplest situation, for being a linear nonholonomic constraint, i.e. just a vector subbundle of (or, of in the algebroid context), this procedure describes the nonholonomic Euler-Lagrange equations by means of the d’Alembert principle, having analogs also in the algebroid case [CoLeMaMa, LMMdD, M4, GG]. We should stress that our nonholonomic constraints are linear in the broader sense, i.e. they are subbundles over submanifolds of the original base manifold. The nonholonomic Euler-Lagrange equations are commonly viewed as being not variational equations. In [GG] it has been pointed out that it is not exactly the case, if we extend slightly our understanding of Variational Calculus.
In this paper we continue studying nonholonomic constraints on algebroids and showing that, at least for Lagrangians of mechanical type, the nonholonomic Euler-Lagrange equations are just non-constrained Euler-Lagrange equations but for a special algebroid structure on the constraint subbundle (see also [LMMdD]). This shows that mechanical systems based on general (not necessary Lie) algebroids appear naturally in the presence of nonholonomic constraints and gives a powerful geometrical tool when dealing with constrained systems. In particular, we get a version of Noether Theorem with true first integrals for nonholonomic systems. We do not get all possible algebroids on applying our procedure. In particular, if the original structure was a Lie algebroid, then the new algebroid bracket is automatically skew- symmetric, so we deal with a quasi-Lie algebroid. One can the associate with the sequence of procedures, like reduction by symmetries and passing to a nonholonomic constraint, the sequence of the corresponding novel structures serving as appropriate geometrical tools in describing the systems:
All this is of course closely related to the discovery of the role of the nonholonomic quasi-Poisson brackets [Mr, SM, IbLeMaMa, CaLeMa], this time not in the Hamilton but in the Lagrange picture.
The paper is organized as follows. In the next section, we recall after [GGU] the basic ideas of developing mechanics on a general algebroid , in particular, the Euler-Lagrange equations. Then, we construct in this setting an analog of the Tulczyjew differential for linear nonholonomic constraints, together with the corresponding nonholonomic Euler-Lagrange equations. In Section 4 we study reductions of a general algebroid to an algebroid on a nonholonomic constraint (satisfying a natural admissibility condition) along a given projection. We discuss also the problem, what algebroids can be obtained in this way, if we start with Lie algebroids.
Section 5 is the most important part of our paper. For Lagrangian functions of mechanical type (‘kinetic energy - potential’), we show that the non-constrained Euler-Lagrange equations on , derived for the reduced Lagrangian and for the reduced algebroid structure on along the orthogonal projection associated with the kinetic energy, coincide with the nonholonomic Euler-Lagrange equations. Moreover, for this nonholonomic case, we can apply therefore the generalization of the Noether Theorem proved in [GGU] to obtain actual first integrals. Passing to the nonholonomic constraints does not requires therefore any change in our unified algebroid approach to mechanics: nonholonomic Euler-Lagrange equations are included in our framework.
We end up with two well-known examples of nonholonomic constraints, the Chaplygin sleigh and the snakebord, to show how simply the corresponding equations of motions can be derived by means of our method.
2. Geometric Mechanics on general algebroids
Let be a smooth manifold and , , be a coordinate system in . Denote by the tangent vector bundle, with induced coordinates , and by the contangent bundle, with induced coordinates .
Let be a vector bundle and its dual bundle. Taking a local basis of sections of , then we have the corresponding local coordinates on , where is the th-coordinate of in the given basis. We denote by the corresponding coordinates of the dual bundle . One can also say that is the fiber-wise linear local function on corresponding to the local section of . We have also adapted local coordinates:
It is well known (cf. [KU]) that the cotangent bundles and are examples of double vector bundles: