Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems

Abstract

In this paper, we apply the geometric Hamilton-Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms.The interest of a geometric Hamilton-Jacobi equation is the primordial observation that if a hamiltonian vector field X H can be projected into a configuration manifold by means of a 1-form dW , then the integral curves of the projected vector field X dW H can be transformed into integral curves of X H provided that W is a solution of the Hamilton-Jacobi equation. In this way, we use the geometric Hamilton-Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton-Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz-Smorodinsky oscillators. For explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.

A contact geometry framework for field theories with dissipation

“…Furthermore, in recent years, there has been a growing interest in studying a geometric framework to describe dissipative or damped systems, specifically using contact geometry [4,15,18]. The efforts have been focused mainly in the study of mechanical systems [5,7,9,10,14,19]. All of them are described by ordinary differential equations to which some terms that account for the dissipation or damping have been added.…”

Contact Hamiltonian Dynamics: The Concept and Its Use

“…For instance, contact transformations generalize canonical transformations, and one can find a contact version of the Hamilton-Jacobi theory, equivalent to the dynamics (7)-(9) (cf. [10,46]). …”

Geometry of the discrete Hamilton–Jacobi equation: applications in optimal control

“…A different approach but equivalent to the usual Hamilton-Jacobi theory relying on the projection of a Hamiltonian vector field via γ = dW is here substituted by the projection of discrete flows. We propose an analogue for the geometric diagram as follows [19,20]. Consider the discrete flow F H d : T * Q → T * Q and a discrete section…”