[Paper Review] Algebraically Linearizable Dynamical Systems

Raquel Caseiro, Jean–Pierre Françoise|ArXiv.org|Oct 25, 1999

Nonlinear Waves and Solitons12 references3 citations

TL;DR

This paper demonstrates explicit algebraic linearization for Ruijsenaars-Schneider-type dynamical systems and their Calogero-type perturbations, showing all orbits are periodic with the same period. The authors establish that this linearization applies to Calogero-Moser and Calogero-Sutherland systems as well, linking it to maximal superintegrability through an algebraic framework that simplifies the analysis of integrable systems with high symmetry.

ABSTRACT

The main result of this paper is the evidence of an explicit linearization of dynamical systems of Ruijsenaars-Schneider type and of the perturbations introduced by F. Calogero of these systems with all orbits periodic of same period. Several other systems share the existence of this explicit linearization, among them, the Calogero-Moser system (with and without external potential) and the Calogero-Sutherland system. This explicit linearization is compared with the notion of maximal superintegrability which has been discussed in several articles.

Motivation & Objective

• To establish a general algebraic framework for linearizing integrable dynamical systems with periodic orbits.
• To analyze the structural properties of Ruijsenaars-Schneider systems and their Calogero-type perturbations.
• To demonstrate that systems like Calogero-Moser and Calogero-Sutherland also admit explicit algebraic linearization.
• To connect this linearization to the concept of maximal superintegrability in classical mechanics.
• To provide a unified algebraic approach that simplifies the study of periodic behavior in many-body integrable systems.

Proposed method

• The authors employ algebraic geometry and symmetric function theory to construct explicit linearizing transformations for the phase space variables.
• They identify a set of conserved quantities that are algebraically related to the system's action-angle variables.
• The method relies on the existence of a Lax pair representation, enabling the use of spectral invariants for linearization.
• The analysis is extended to systems with external potentials by preserving the algebraic structure of the Lax matrix.
• The approach uses polynomial and rational functions of the dynamical variables to map nonlinear flows to linear ones in an auxiliary space.
• The framework is validated by applying it to known integrable systems, confirming consistency with existing results on periodicity and superintegrability.

Experimental results

Research questions

• RQ1 Can Ruijsenaars-Schneider-type systems and their Calogero perturbations be linearized via an algebraic transformation?
• RQ2 What algebraic structure underlies the periodicity of all orbits in these systems?
• RQ3 How does the proposed linearization relate to the concept of maximal superintegrability?
• RQ4 Do the Calogero-Moser and Calogero-Sutherland systems also admit such algebraic linearization?
• RQ5 What role do symmetric functions and spectral invariants play in constructing the linearizing map?

Key findings

• The Ruijsenaars-Schneider system and its Calogero-type perturbations admit explicit algebraic linearization, mapping nonlinear dynamics to linear flows in an auxiliary space.
• All orbits in the perturbed systems are periodic with the same period, a feature confirmed through the algebraic structure of the conserved quantities.
• The Calogero-Moser system with and without external potential is shown to be algebraically linearizable, extending the scope of the method.
• The Calogero-Sutherland model also satisfies the conditions for algebraic linearization, indicating broad applicability across integrable systems.
• The linearization mechanism is deeply connected to maximal superintegrability, as the number of independent conserved quantities matches the system's degrees of freedom.
• The algebraic framework provides a systematic way to construct action-angle variables and verify periodicity without solving the equations of motion explicitly.