A time-extended hamiltonian formalism

A Poisson structure on the time-extended space R X M is shown to be appropriate for a Hamiltonian formalism in which time is no more a privileged variable and no a priori geometry is assumed on the space M of motions. Any Poisson bi-vector on R X M with a volume is shown to possess two intrinsic infinitesimal automorphisms one of which is known as its modular vector field. An abstract representation space for sl(2,R) algebra with a physical realization by the Darboux-Halphen system is considered for the case in which these are not independent. For the generic case, it is shown that an infinite hierarchy of automorphisms can be generated. The relation between Hamiltonian flows on R X M and infinitesimal motions on M preserving a geometric structure therein is demonstrated for volume preserving diffeomorphisms. (C) 1999 Elsevier Science B.V. All rights reserved.