Global versus local superintegrability of nonlinear oscillators
arXiv:1809.02248 · doi:10.1016/j.physleta.2018.12.007
Abstract
Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.
13 pages; in press, Phys. Lett. A
References in corpus (8)
- Superintegrable Systems in Darboux spaces
- Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
- Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
- A maximally superintegrable system on an n-dimensional space of nonconstant curvature
- Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
- A new exactly solvable quantum model in N dimensions
- On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications
- Some new aspects of first integrals and symmetries for central force dynamics