Hamilton-Jacobi formalism on locally conformally symplectic manifolds

Abstract

In this article, we provide a Hamilton-Jacobi formalism on locally conformally symplectic (lcs) manifolds. We are interested in the Hamilton-Jacobi as an alternative method for formulating the dynamics, while our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bundles, whose dynamic takes place in open subsets of the general manifold. We present a lcs Hamilton-Jacobi equation in subsets of the general manifold and then provide a global view by using the Lichnerowicz-deRham differential. We show a comparison between the global and local description of a lcs Hamilton-Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton-Jacobi theory. A particular example is the case of Gaussian isokinetic dynamics in which we apply our structure in certain submanifolds of the phase space.