Goldfish geodesics and Hamiltonian reduction of matrix dynamics

TL;DR

This paper connects free vector dynamics with eigenvalue motion of symmetric matrices and offers a geometric interpretation of Ruijsenaars-Schneider models, advancing understanding of matrix dynamics and integrable systems.

Contribution

It introduces a novel geometric perspective on matrix eigenvalue dynamics and links it to well-known integrable models, providing new insights into their structure.

Findings

Eigenvalue motion corresponds to geodesics on a specific manifold.

A geometric interpretation of Ruijsenaars-Schneider models is established.

The approach unifies vector dynamics with matrix eigenvalue behavior.

Abstract

We relate free vector dynamics to the eigenvalue motion of a time-dependent real-symmetric NxN matrix, and give a geodesic interpretation to Ruijsenaars Schneider models.