Analytic integrability of Hamiltonian systems with exceptional potentials
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Departamento de Matemática)

Date:	2015
Abstract:	We study the existence of analytic first integrals of the complex Hamiltonian systems of the form H = 1 2 ∑ 2 i=1 p 2 i + Vl(q1, q2) with the homogeneous polynomial potential Vl(q1, q2) = α(q2 − iq1) L (q2 + iq1) k−l , l = 0, . . . , k, α ∈ C \ {0} of degree k called exceptional potentials. In Remark 2. 1 of J. Math. Phys. 46 (2005), 062901, the authors state: The exceptional potentials V0, V1, Vk−1, Vk and Vk/2 when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials. Here we prove that the exceptional potentials with k even diferent from V0, V1, Vk−1, Vk and Vk/2, have no independent analytic first integral different from the Hamiltonian one. Additionally in the cases V2 and Vk−2 with k either even or odd we show that they do not have rational first integrals independent of the Hamiltonian.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 316338 European Commission 318999
Note:	Agraïments: UNAB13-4E-1604. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Exceptional potential ; Hamiltonian system with 2 degrees of freedom ; Homogeneous potentials of degree k ; Integrability
Published in:	Physics Letters. A, Vol. 379 (2015) , p. 2295-2299, ISSN 0375-9601

DOI: 10.1016/j.physleta.2015.07.034

Postprint 9 p, 770.9 KB

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