Weak-hamiltonian Dynamical Systems

A big-isotropic structure E is an isotropic subbundle of T M ⊕ T * M , endowed with the metric defined by pairing. The structure E is said to be 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, Hamiltonian systems (in particular, constrained mechanics) [1, 4] 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 [9]. All the manifolds and mappings are of class C ∞ and we use the standard notation of Differential Geometry, e.g., [5]. 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.