Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems

TL;DR

This paper develops geometric Hamilton--Jacobi equations on cosymplectic and contact manifolds to solve classical Hamiltonian systems with time dependence and dissipation, providing explicit solutions for physical examples.

Contribution

It introduces new geometric Hamilton--Jacobi equations tailored for cosymplectic and contact structures, addressing time-dependent and dissipative Hamiltonian systems.

Findings

Derived explicit Hamilton--Jacobi equations on cosymplectic and contact manifolds.

Solved physical systems with explicit time dependence, including oscillators.

Provided solutions for dissipative systems like damped oscillators.

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 $X_{H}^{dW}$ 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.