Canonical and canonoid transformations for Hamiltonian systems on (co)symplectic and (co)contact manifolds

Abstract

In this paper we present canonical and canonoid transformations considered as global geometrical objects for Hamiltonian systems. Under the mathematical formalisms of symplectic, cosymplectic, contact and cocontact geometry, the canonoid transformations are defined for (co)symplectic, (co)contact Hamiltonian systems, respectively. The local characterizations of these transformations is derived explicitly and it is demonstrated that for a given canonoid transformation there exist constants of motion associated with it.

Lie integrability for time-independent and time-dependent Hamiltonian systems

“…The aim of this paper is to present under the formalisms of symplectic geometry, cosymplectic geometry, contact geometry and cocontact geometry the theorem of Lie integrability for Hamiltonian systems. Contact and cocontact Hamiltonian mechanics is a subject of active research this days, see for example [15][16][17][18]. This paper is organized as follows: In section 2 we present a brief review of Hamiltonian sistems on symplectic, cosymplectic, contact and cocontact manifolds; in section 3 we present a brief review of the Lie's theorem on integrability of a dynamical system defined by a smooth vector field on R n .…”