Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree -3
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mahdi, Adam (University of North Carolina at Charlotte. Mathematics Department)
Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Departamento de Matemática)

Date:	2011
Abstract:	In this paper we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k = −2, −1, . . . , 3, 4 their polynomial integrability has been characterized. Here we have two main results. First we characterize the polynomail integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian system with homogeneous polynomial potentials. Finally we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410
Note:	Agraïments: The third author is partially supported by FCT through CAMGDS, Lisbon.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Hamiltonian system with 2-degrees of freedom ; Homogeneous potential of degree -3 ; Polynomial integrability
Published in:	Physica D. Nonlinear phenomena, Vol. 240 (2011) , p. 1928-1935, ISSN 0167-2789

DOI: 10.1016/j.physd.2011.09.003

Postprint 15 p, 852.9 KB

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