. SG ] 5 J an 2 00 7 Weak - Hamiltonian dynamical systems by Izu Vaisman

A big-isotropic structure E is an isotropic subbundle of T M ⊕ T * M , endowed with the metric defined by pairing, and E is said to be in-brackets) [7]. A weak-Hamiltonian dynamical system is a vector field X H such that (X H , dH) ∈ E ⊥ (H ∈ C ∞ (M)). We obtain the explicit expression of X H and of the integrability conditions of E under the regularity condition dim(pr T * M E) = const. We show that the port-controlled, Hamilto-nian systems (in particular, constrained mechanics) [1, 3] may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems. 1 Big-isotropic structures In this section we recall some basic facts concerning the big-isotropic structures that were studied in our paper [7]. All the manifolds and mappings are of class C ∞ and we use the standard notation of Differential Geometry, e.g., [4]. In particular, M is an m-dimensional manifold, χ k (M) is the space of k-vector fields, Ω k (M) is the space of differential k-forms, Γ indicates the space of global cross sections of a vector bundle, X, Y, .. are either contravariant vectors or vector fields, α, β, ... are either covariant vectors or 1-forms, d is the exterior differential and L is the Lie derivative.