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.