Extended Hamilton-Jacobi Theory, symmetries and integrability by quadratures

TL;DR

This paper extends Hamilton-Jacobi theory to systems with symmetries, providing methods to solve the equations via reconstruction or quadratures, with explicit formulas for exponential curves in Lie groups.

Contribution

It develops a framework for solving Hamilton-Jacobi equations in symmetric systems, including explicit quadrature solutions and new formulas for exponential curves in Lie groups.

Findings

Complete solutions relate to reconstruction equations.

Solutions enable integration of vector fields up to quadratures.

Explicit exponential curve formulas for Lie groups.

Abstract

In this paper, we study the extended Hamilton-Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group $G$ on a manifold $M$ and a $G$-invariant vector field $X$ on $M$, we construct complete solutions of the Hamilton-Jacobi equation (HJE) related to $X$ (and a given fibration on $M$). We do that along each open subset $U\subseteq M$ such that $\pi\left(U\right)$ has a manifold structure and $\pi_{\left|U\right.}:U\rightarrow\pi\left(U\right)$, the restriction to $U$ of the canonical projection $\pi:M\rightarrow M/G$, is a surjective submersion. If $X_{\left|U\right.}$ is not vertical with respect to $\pi_{\left|U\right.}$, we show that such complete solutions solve the "reconstruction equations" related to $X_{\left|U\right.}$ and $G$, i.e., the equations that enable us to write the integral curves of $X_{\left|U\right.}$ in terms of those of its projection on $\pi\left(U\right)$. On the other hand, if $X_{\left|U\right.}$ is vertical, we show that such complete solutions can be used to construct (around some points of $U$) the integral curves of $X_{\left|U\right.}$ up to quadratures. To do that we give, for some elements $\xi$ of the Lie algebra $\mathfrak{g}$ of $G$, an explicit expression up to quadratures of the exponential curve $\exp\left(\xi\,t\right)$, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of $\exp\left(\xi\,t\right)$ is valid for all $\xi$ inside an open dense subset of $\mathfrak{g}$.