Frans Cantrijn

GENERALIZATIONS OF NOETHER'S THEOREM IN CLASSICAL MECHANICS*

In this paper, a review is presented of various approaches to the generalization of the version of Noether’s theorem, which is presented in most textbooks on classical mechanics. Its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity-dependent transformations. Our analysis is centered around the one factor common to all known treatments, namely the structure of the related first integral. We first discuss the most general framework, in which a function of the above-mentioned structure constitutes a first integral of a given Lagrangian system, and show that one cannot really talk about an “interrelationship” between symmetries and first integrals there. We then compare different proposed generalizations of Noether’s theorem, by describing the nature of the restrictions which characterize them, when they are situated within the broadest framework. We prove a seemingly new equivalence-result between the two main approaches: that of invaria...

On the geometry of multisymplectic manifolds.

A multisymplectic structure on a manifold is defined by a closed differential form with zero characteristic distribution. Starting from the linear case, some of the basic properties of multisymplectic structures are described. Various examples of multisymplectic manifolds are considered, and special attention is paid to the canonical multisymplectic structure living on a bundle of exterior k -forms on a manifold. For a class of multisymplectic manifolds admitting a ‘Lagrangian’ fibration, a general structure theorem is given which, in particular, leads to a classification of these manifolds in terms of a prescribed family of cohomology classes.

On the geometry of generalized Chaplygin systems

Some aspects of the geometry and the dynamics of generalized Chaplygin systems are investigated. First, two different but complementary approaches to the construction of the reduced dynamics are reviewed: a symplectic approach and an approach based on the theory of affine connections. Both are mutually compared and further completed. Next, a necessary and sufficient condition is derived for the existence of an invariant measure for the reduced dynamics of generalized Chaplygin systems of mechanical type. A simple example is then constructed of a generalized Chaplygin system which does not verify this condition, thereby answering in the negative a question raised by Koiller.

Reduction of nonholonomic mechanical systems with symmetries

Abstract A geometric reduction procedure is presented for Lagrangian systems subjected to nonlinear nonholonomic constraints in the presence of symmetries. Our approach is based on a geometrical method which enables one to deduce the constrained dynamics from the unconstrained one by projection.

Gradient vector fields on cosymplectic manifolds

The authors study some geometrical properties of gradient vector fields on cosymplectic manifolds, thereby emphasizing the close analogy with Hamiltonian systems on symplectic manifolds. It is shown that gradient vector fields and, more generally, local gradient vector fields can be characterized in terms of Lagrangian submanifolds of the tangent bundle with respect to an induced symplectic structure. In addition, the symmetry and reduction properties of gradient vector fields are investigated.

Derivations of forms along a map: the framework for time-dependent second-order equations

Abstract A comprehensive theory is presented concerning derivations of scalar and vector-valued forms along the projection π : R × TM → R × M . It is the continuation of previous work on derivations of forms along the tangent bundle projection and is prompted by the need for a scheme which is adapted to the study of time-dependent second-order equations. The overall structure of the theory closely follows the pattern of this preceding work, but there are many features which are certainly not trivial transcripts of the time-independent situation. As before, a crucial ingredient in the classification of derivations is a non-linear connection on the bundle π. In the presence of a given second-order system, such a connection is canonically defined and gives rise to two important operations: the dynamical covariant derivative, which is a derivation of degree 0, and the Jacobi endomorphism, which is a type (1, 1) tensor field along π. The theory is developed in such a way that all results readily apply to the more general situation of a bundle π : J 1 E → E , where E is fibred over R, but need not be the trivial fibration R × M → R .

Reduction of constrained systems with symmetries

A general model is proposed for constrained dynamical systems on a symplectic manifold which covers, among others, the description of Lagrangian and Hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular Lagrangian. The reduction properties of these systems in the presence of symmetry are investigated within this general framework.

Skinner-Rusk approach to time-dependent mechanics

The geometric approach to autonomous classical mechanical systems in terms of a canonical first-order system on the Whitney sum of the tangent and cotangent bundle, developed by Skinner and Rusk, is extended to the time-dependent framework.

Pseudo-symmetries, Noether's theorem and the adjoint equation

Pseudo-symmetries were introduced by Sarlet and Cantrijn (1984) for time-dependent non-conservative systems. They are reconsidered in the context of general autonomous second-order systems, relying on the new approach to such systems which was presented by Sarlet et al. They further introduce the notion of adjoint symmetries of a second-order system, as being associated to invariant 1-forms, and show how they may be related to first integrals or to Lagrangians under appropriate circumstances. Their results enable them to clarify a rather unusual account of Noethers theorem which was recently given by Gordon (1986).

Geometric aspects of nonholonomic field theories

A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.