A geometric approach to solve time dependent and dissipative Hamiltonian systems

In this paper, we apply the geometric Hamilton--Jacobi theory to obtain solutions of Hamiltonian systems in Classical Mechanics, 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. On the other hand, the interest of a geometric Hamilton--Jacobi equation is the primordial observation that a Hamiltonian vector field $X_{H}$ can be projected into the 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 Hamiltonian formulation. A new expression for a geometric Hamilton Jacobi equation is obtained for time dependent Hamiltonians described with the aid of a cosymplectic structure. Then, another expression for the Hamilton Jacobi equation is retrieved for Hamiltonians with frictional terms described through contact geometry. Both approaches shall be applied to physical examples.