Euler-Calogero-Moser system from SU(2) Yang-Mills theory

TL;DR

This paper explores the connection between SU(2) Yang-Mills mechanics and Euler-Calogero-Moser models through Hamiltonian reduction, revealing new integrable systems and their equations of motion in specific limits.

Contribution

It demonstrates how different Hamiltonian reductions of SU(2) Yang-Mills mechanics lead to specific Euler-Calogero-Moser models with external potentials.

Findings

Derived the ID_3 Euler-Calogero-Moser model from gauge invariance reduction.

Connected the IA_6 Euler-Calogero-Moser model to discrete symmetry reduction.

Presented the equations of motion in Lax form for zero coupling limit.

Abstract

The relation between SU(2) Yang-Mills mechanics, originated from the 4-dimensional SU(2) Yang-Mills theory under the supposition of spatial homogeneity of the gauge fields, and the Euler-Calogero-Moser model is discussed in the framework of Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: due to the gauge invariance and due to the discrete symmetry. In the former case, it is shown that after elimination of the gauge degrees of freedom from the SU(2) Yang-Mills mechanics the resulting unconstrained system represents the ID_3 Euler-Calogero-Moser model with an external fourth-order potential. Whereas in the latter, the IA_6 Euler-Calogero-Moser model embedded in an external potential is derived whose projection onto the invariant submanifold through the discrete symmetry coincides again with the SU(2) Yang-Mills mechanics. Based on this connection, the equations of motion of the SU(2) Yang-Mills mechanics in the limit of the zero coupling constant are presented in the Lax form.