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arXiv · 1309.0089

Three and four-body systems in one dimension: integrability, superintegrability and discrete symmetries

C. Chanu·L. Degiovanni·G. Rastelli
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Abstract

Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.

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C. Chanu, L. Degiovanni, G. Rastelli. 2013-08-31. Three and four-body systems in one dimension: integrability, superintegrability and discrete symmetries. https://arxiv.org/abs/1309.0089

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