Constructing a complete integral of the Hamilton-Jacobi equation on pseudo-Riemannian spaces with simply transitive groups of motions

TL;DR

This paper introduces an efficient method for constructing complete integrals of the Hamilton-Jacobi equation on pseudo-Riemannian spaces with transitive symmetry groups, demonstrated on a non-separable spacetime example.

Contribution

The paper presents a novel approach using canonical coordinates on coadjoint orbits to solve the Hamilton-Jacobi equation in complex pseudo-Riemannian manifolds.

Findings

Method successfully constructs complete integrals in non-separable cases

Applied to McLenaghan-Tariq-Tupper spacetime example

Demonstrates effectiveness on spaces with simply transitive groups of motions

Abstract

In this work, an efficient method for constructing a complete integral of the geodesic Hamilton-Jacobi equation on pseudo-Riemannian manifolds with simply transitive groups of motions is suggested. The method is based on using a special transition to canonical coordinates on coadjoint orbits of the group of motion. As a non-trivial example, we consider the problem of constructing a complete integral of the geodesic Hamilton-Jacobi equation in the McLenaghan-Tariq-Tupper spacetime. An essential feature of this example is that the Hamilton-Jacobi equation is not separable in the corresponding configuration space.