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

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 and for explicit dissipative behavior, we solve the example of a unidimensional damped oscillator. (paper)