(1+1)-dimensional separation of variables

TL;DR

This paper investigates conditions for separability of geodesic flows on 2D indefinite signature manifolds, revealing three distinct structures, including new complex and null separation types, based on conformal Killing tensors.

Contribution

It classifies separability structures in 2D indefinite metrics, identifying new types like complex/harmonic and null separations, and derives conditions for weak and strong integrability.

Findings

Three types of conformal Killing tensors identified.

New complex/harmonic and null separation structures discovered.

Conditions for weak and strong integrability derived.

Abstract

In this paper we explore general conditions which guarantee that the geodesic flow on a 2-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a 2-dimensional natural Hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a 2nd-rank Killing tensor. We examine the possibility that the integral is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary values of the Hamiltonian (strong integrability). Using null coordinates, we show that the leading-order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability. These functions are quadratic polynomials in the coordinates in the case of strong integrability. We show that for $(1+1)$-dimensional systems there are three possible types of conformal Killing tensors, and therefore, three distinct separability structures in contrast to the single standard Hamilton-Jacobi type separation in the positive definite case. One of the new separability structures is the complex/harmonic type which is characterized by complex separation variables. The other new type is the linear/null separation which occurs when the conformal Killing tensor has a null eigenvector.