Geometric Hamilton–Jacobi theory on Nambu–Poisson manifolds

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 origi...