Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3
Copyright policy )Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3
Agraïments: The third author is partially supported by FCT through CAMGDS, Lisbon. 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.
- AGH University of Science and Technology Poland
- Autonomous University of Barcelona Spain
- University of North Carolina at Chapel Hill United States
- University of Lisbon Portugal
- Technical University of Lisbon Portugal
Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Hamiltonian system with two degrees of freedom, homogeneous potential of degree \(-3\), Homogeneous potential of degree -3, Explicit solutions, first integrals of ordinary differential equations, Hamiltonian system with 2-degrees of freedom, Symmetries, invariants of ordinary differential equations, polynomial integrability, Polynomial integrability
Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Hamiltonian system with two degrees of freedom, homogeneous potential of degree \(-3\), Homogeneous potential of degree -3, Explicit solutions, first integrals of ordinary differential equations, Hamiltonian system with 2-degrees of freedom, Symmetries, invariants of ordinary differential equations, polynomial integrability, Polynomial integrability
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