Weak-Hamiltonian dynamical systems

A big-isotropic structure E is an isotropic subbundle of TM T* M, endowed with the metric defined by pairing. The structure E is said to be integrable if the Courant bracket [X,Y] E, X,YE. Then, necessarily, one also has [X,Z] E⊥, Z E⊥ [Vaisman, I., "Isotropic subbundles of TM T* M, " Int. J. Geom. Methods Mod. Phys. 4, 487-516 (2007)]. A weak-Hamiltonian dynamical system is a vector field XH such that (XH, dH) E⊥ (H C∞ (M)). We obtain the explicit expression of XH 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) [Dalsmo, M. and van der Schaft, A. J., "On representations and integrability of mathematical structures in energy conserving physical systems," SIAM J. Control Optim. 37, 54-91 (1998)] may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems.