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 fc-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.1991 Mathematics subject classification (Amer. Math. Soc): primary 53C15, 58Axx.

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 velocitydependent 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 invariance of the action functional, and that of invariance of dO (0 being the Cartan-form). A number of arguments are discussed in favor of this last version of a generalized Noether theorem. Throughout the analysis we pay attention to practical considerations, such as the complexity of the Killing-type partial differential equations in each approach, which must be solved in order to identify "Noether-transformations".

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.

In this paper we intend to unify different approaches to the construction of an 'almost-Poisson' bracket for mechanical systems with nonholonomic constraints. This almost-Poisson structure is subsequently used to describe the phase-space dynamics of a nonholonomic system. It is shown that when dealing with 'nonhomogeneous' constraints, the Hamiltonian equations of motion cannot be expressed in terms of the almost-Poisson bracket alone. This fact is illustrated in the case of mechanical systems with affine constraints. The problem of a rolling ball on a rotating table is treated as an example.

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