Weak-Hamiltonian dynamical systems

Abstract

A big-isotropic structure $E$ is an isotropic subbundle of $TM\oplus T^*M$, endowed with the metric defined by pairing. The structure $E$ is said to be integrable if the Courant bracket $[\mathcal{X},\mathcal{Y}]\in\Gamma E$, $\forall\mathcal{X},\mathcal{Y}\in\Gamma E$. Then, necessarily, one also has $[\mathcal{X},\mathcal{Z}]\in\Gamma E^\perp$, $\forall\mathcal{Z}\in\Gamma E^\perp$ \cite{V-iso}. A weak-Hamiltonian dynamical system is a vector field $X_H$ such that $(X_H,dH)\in E^\perp$ $(H\in C^\infty(M))$. 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, Hamiltonian systems (in particular, constrained mechanics) \cite{{BR},{DS}} may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems.