Geometric Hamilton–Jacobi theory on Nambu–Poisson manifolds

Abstract

The Hamilton–Jacobi theory is a formulation of classical mechanics equivalent to other formulations as Newtonian, Lagrangian, or Hamiltonian mechanics. The primordial observation of a geometric Hamilton–Jacobi theory is that if a Hamiltonian vector field XH can be projected into the configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field XHdWcan be transformed into integral curves of XH provided that W is a solution of the Hamilton–Jacobi equation. Our aim is to derive a geometric Hamilton–Jacobi theory for physical systems that are compatible with a Nambu–Poisson structure. For it, we study Lagrangian submanifolds of a Nambu–Poisson manifold and obtain explicitly an expression for a Hamilton–Jacobi equation on such a manifold. We apply our results to two interesting examples in the physics literature: the third-order Kummer–Schwarz equations and a system of n copies of a first-order differential Riccati equation. From the first example, we retrieve the original Nambu bracket in three dimensions and from the second example, we retrieve Takhtajan’s generalization of the Nambu bracket to n dimensions.