Invariant critical sets of conserved quantities

TL;DR

This paper explores how conserved quantities in dynamical systems can be used to construct invariant sets and identify solutions that also satisfy simpler systems, with applications to the Toda lattice.

Contribution

It introduces conditions under which solutions of nonlinear systems are also solutions of simpler or linear systems, expanding understanding of invariant sets in dynamical systems.

Findings

Conditions for solutions to be shared between nonlinear and linear systems

Construction of invariant sets from conserved quantities

Application to Toda lattice example

Abstract

For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice.