A tensorial Lax pair equation and integrable systems in relativity and classical mechanics

TL;DR

This paper introduces a tensorial interpretation of the Lax pair equation for finite-dimensional quadratic Hamiltonians, connecting it to tensors on configuration space and exploring implications in relativity and classical mechanics.

Contribution

It presents a novel tensorial framework for Lax pairs, linking them to third rank tensors and connections, and discusses potential solutions to Einstein equations with Lax pair properties.

Findings

Lax matrices can be derived from third rank tensors.

The second Lax matrix relates to a connection in the Hamiltonian system.

Potential existence of Einstein solutions with Lax pair structure.

Abstract

It is shown that the Lax pair equation dL/dt = [L,A] can be given a neat tensorial interpretation for finite-dimensional quadratic Hamiltonians. The Lax matrices L and A are shown to arise from third rank tensors on the configuration space. The second Lax matrix A is related to a connection which characterizes the Hamiltonian system. The Toda lattice system is used to motivate the definition of the Lax pair tensors. The possible existence of solutions to the Einstein equations having the Lax pair property is discussed.