Integrable natural systems in pseudoeuclidean plane.
type: Item
instance of: scholarly article
title: Integrable natural systems in pseudoeuclidean plane.
zbMATH Open document ID: 1041.37030
DOI: 10.1016/S0034-4877(01)80071-9
author: T. Koreshova
author: A. A. Zaitsev
published in: Reports on Mathematical Physics
publication date: 2002-09-12T00:00:00Z
review text: A Hamiltonian system is called natural if its Hamiltonian is a sum of a positive definite kinetic energy and a potential. Natural Hamiltonian systems have been the object of much attention classically; if a natural Hamiltonian system admits a real-analytic integral, then it admits an integral which is polynomial in the momenta [cf. \textit{E. T. Whittaker}, A treatise on the analytical dynamics of particles and rigid bodies. With an introduction to the problem of three bodies. With a foreword by Sir William McCrea. Reprint of the fourth edition, Cambridge Mathematical Library. Cambridge (UK) etc.: Cambridge University Press (1988; Zbl 0665.70002)]. The authors replace the kinetic energy \({1\over 2}(p_1^2+p_2^2)\) in the 2-dimensional Hamiltonian, with a conic determined by a symmetric \(2\times 2\) matrix \(G\) (the ``pseudoeuclidean'' metric of the title), and classify those systems that admit an additional integral, quadratic in the momenta. By solving for the Darboux-Nijenhuis tensor, the authors give (bi-Hamiltonian) recursion formulas that generate all the polynomial potentials, as well as the solution by quadratures. In the last section, they provide examples related to the celebrated Garnier system, the Hénon-Heiles system, and the two-gap Schrödinger equation (which they deem to be new in the literature).
reviewed by: Emma Previato
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 70H15
Mathematics Subject Classification ID: 35F20
zbMATH DE Number: 1799977
zbMATH Keywords: Natural Hamiltonian system
zbMATH Keywords: Darboux-Nijenhuis tensor
MaRDI profile type: Publication
cites work: Q5838030
cites work: Q4692312
cites work: Concerning natural Hamiltonian systems integrable in elliptic coordinates
cites work: Q4692311
cites work: When is a Hamiltonian system separable?
cites work: Degenerate Poisson Pencils on Curves: New Separability Theory
cites work: Q4209234
full work available at URL: s0034-4877(01)80071-9
OpenAlex ID: W2082941578
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description: scientific article; zbMATH DE number 1799977