A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

Abstract

In this paper, we propose a geometric Hamilton–Jacobi (HJ) theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ theory is that if a Hamiltonian vector field [Formula: see text] can be projected into a configuration manifold by means of a one-form [Formula: see text], then the integral curves of the projected vector field [Formula: see text] can be transformed into integral curves of the vector field [Formula: see text] provided that [Formula: see text] is a solution of the HJ equation. This procedure allows us to reduce the dynamics to a lower-dimensional manifold in which we integrate the motion. On the other hand, the interest of a NJ structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric HJ equation on a NJ manifold and apply it to the third-order Riccati differential equation as an example.