Algebraic integrability of confluent Neumann system

TL;DR

This paper proves the complete algebraic integrability of the confluent Neumann system, showing its flow linearizes on a generalized Jacobian of a singular algebraic curve and relating it to the Rosochatius system via symplectic reduction.

Contribution

It establishes the algebraic integrability of the confluent Neumann system and connects it to the Rosochatius system through symplectic quotient analysis.

Findings

Proves algebraic integrability of the confluent Neumann system.

Shows the flow linearizes on a generalized Jacobian torus.

Relates the system to the Rosochatius system via symplectic reduction.

Abstract

In this paper we study the Neumann system, which describes the harmonic oscillator (of arbitrary dimension) constrained to the sphere. In particular we will consider the confluent case where two eigenvalues of the potential coincide, which implies that the system has S^{1} symmetry. We will prove complete algebraic integrability of confluent Neumann system and show that its flow can be linearized on the generalized Jacobian torus of some singular algebraic curve. The symplectic reduction of S^{1} action will be described and we will show that the general Rosochatius system is a symplectic quotient of the confluent Neumann system, where all the eigenvalues of the potential are double. This will give a new mechanical interpretation of the Rosochatius system.