  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.
Possible geometries induced on the spatial domain M are investigated. An
abstract representation space for sl(2,R) algebra with a concrete physical
realization by the Darboux-Halphen system is considered for demonstration. The
Poisson bi-vector on R x M is shown to possess two intrinsic infinitesimal
automorphisms one of which is known as the modular or curl vector field.
Anchored to these two, an infinite hierarchy of automorphisms can be generated.
Implications on the symmetry structure of Hamiltonian dynamical systems are
discussed. As a generalization of the isomorphism between contact flows and
their symplectifications, 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 in connection with
three-dimensional motion of an incompressible fluid.
