Supersymmetry and Integrability in Planar Mechanical Systems

TL;DR

This paper explores how N=2 supersymmetry influences integrability in a two-degree-of-freedom mechanical system derived from SU(2) Yang-Mills theory, highlighting the restrictive role of supersymmetry on potential interactions.

Contribution

It introduces a supersymmetric mechanical model from Yang-Mills reduction and analyzes how supersymmetry constrains integrable potentials using the Painleve test.

Findings

Supersymmetry reduces the parameter space for integrable potentials.

Parity invariance affects the integrability of the model.

Supersymmetric constraints limit the possibilities for chaotic behavior.

Abstract

We present an N=2-supersymmetric mechanical system whose bosonic sector, with two degrees of freedom, stems from the reduction of an SU(2) Yang-Mills theory with the assumption of spatially homogeneous field configurations and a particular ansatz imposed on the gauge potentials in the dimensional reduction procedure. The Painleve test is adopted to discuss integrability and we focus on the role of supersymmetry and parity invariance in two space dimensions for the attainment of integrable or chaotic models. Our conclusion is that the relationships among the parameters imposed by supersymmetry seem to drastically reduce the number of possibilities for integrable interaction potentials of the mechanical system under consideration.