Weak-Hamiltonian dynamical systems

Abstract

A big-isotropic structure $E$ is an isotropic subbundle of $TM\oplus T^{\ast }M$, 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 $dim(p{r}_{T^{\ast }M}E)=const.$ 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.

19 pages, minor improvements

Cited by in corpus (1)

• Geometric quantization of weak-Hamiltonian functions