Cosymplectic and contact structures to resolve 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 $X_H^{dW}$ 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.

arXiv commentExtension of our previous preprint on geometric methods for the treatment of time dependent and dissipative Hamiltonians. Examples have been generalized to higher-dimensions. arXiv admin note: substantial text overlap with arXiv:1607.01239

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