Geometry of Real Forms of the Complex Neumann System

DOI

10.1080/14029251.2016.1135643How to use a DOI?

Keywords

integrable systems; Neumann system; Arnold-Liouville level sets; spectral curves; real structures; real forms

Abstract

In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair (L^ℂ(λ), M^ℂ(λ)) of 2 × 2 matrices, where Lℂ(λ)=[Vℂ(λ)Wℂ(λ)Uℂ(λ)-Vℂ(λ)] and U^ℂ(λ), V^ℂ(λ), W^ℂ(λ) are suitable polynomials. The topology of a regular level set of the moment map of a real form is determined by the positions of the roots of the suitable real form of U^ℂ(λ), with respect to the position of the values of suitable parameters of the system. For two families of the real forms of the complex Neumann system, we describe the topology of the regular level set of the moment map. For one of these two families the level sets are noncompact.

In the paper, we also give the formula which provides the relation between two systems of the first integrals in involution of the Neumann system. One of these systems is obtained from the Lax pair of the Mumford type, while the second is obtained from the Lax pair whose matrices are of dimension (n+1) × (n+1).

Copyright

© 2016 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Tina Novak
PY  - 2021
DA  - 2021/01/06
TI  - Geometry of Real Forms of the Complex Neumann System
JO  - Journal of Nonlinear Mathematical Physics
SP  - 74
EP  - 91
VL  - 23
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2016.1135643
DO  - 10.1080/14029251.2016.1135643
ID  - Novak2021
ER  -