A time-extended Hamiltonian formalism

Abstract

A Poisson structure on the time-extended space I × 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 I ×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 I × 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.

Poisson structures for the Aristotelian model of three-body motion

“…A Poisson structure on a manifold N is defined by a skew symmetric contravariant bilinear form subjected to the Jacobi identity expressed as the vanishing of the Schouten bracket of the Poisson tensor with itself [14][15][16][17]. Following [4], for a Hamiltonian formalism on a time-extended space N = I × M, I ⊂ R or C and M ⊂ R 3 for the present context, we take the bi-vector field…”

“…This transformation which complexifies the real time variable is similar to the one that appeared in [1]. Using techniques of [4], we will present, in section 3, a formal Hamiltonian structure with this time-dependent Hamiltonian function.…”

“…They were considered to be a prototype for a large class of models exhibiting transitions from simple to complicated motions that can be explained as travel on Riemann surfaces (see also extensive list of references in [1] and the forthcoming article [3]). In this work, following [4] and [5], we will present Poisson structures of an equivalent system of differential equations obtained after the elimination of the linear onebody forces in equation (1).…”

Helicity invariants in 3D: kinematical aspects

“…Even though the Casimirs of the bracket (24) gives zero functional on the orbits, the geometric structures arising from this case, that is, from the nilpotency of W 1 is non-trivial and results in Godbillon-Vey type invariants [9], [33]- [35]. We refer to Ref.…”

“…We refer to Ref. [35] for an investigation of this case which requires a separate treatment, its relation with the symplectic structure Ω as well as physically relevant applications. To this end, we shall solely assume that W 1 is not a nilpotent element of X div (M).…”