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

TL;DR

This paper develops a geometric Hamilton--Jacobi theory tailored for Nambu--Jacobi manifolds, enabling reduction of complex multi-Hamiltonian dynamics to simpler lower-dimensional systems, demonstrated through a Riccati equation example.

Contribution

It introduces a novel geometric Hamilton--Jacobi framework specifically for Nambu--Jacobi manifolds, extending existing theories to multi-Hamiltonian systems.

Findings

Derived explicit Hamilton--Jacobi equation for Nambu--Jacobi manifolds

Applied the theory to solve a third-order Riccati differential equation

Showed reduction of dynamics to lower-dimensional manifolds

Abstract

In this paper we propose a geometric Hamilton--Jacobi theory on a Nambu--Jacobi manifold. The advantange of a geometric Hamilton--Jacobi theory is that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a one-form $dW$, then the integral curves of the projected vector field $X_H^{dW}$ can be transformed into integral curves of the vector field $X_H$ provided that $W$ is a solution of the Hamilton--Jacobi 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 Nambu--Jacobi 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 Hamilton--Jacobi equation on a Nambu--Jacobi manifold and apply it to the third-order Riccati differential equation as an example.