2018/09/30 by Stephen C. Anco, Ángel Ballesteros, Angel Ballesteros +2 · 4 citations
Mathematics · Physics and Astronomy · #Classical mechanics #Constant curvature #Curvature #Differential equation #Differential geometry #Geometry #Hamiltonian (control theory) #Hamiltonian system #Homogeneous space #Lie algebra #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Ordinary differential equation #Physics #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Symmetry (geometry) #math-ph #math.MP
paper · pdf · doi:10.1016/j.physleta.2018.12.007
published in Physics Letters A 383(9), 801-807 (Elsevier BV) · 13 pages; in press, Phys. Lett. A
openalex publication_date 2018/12/05 · arxiv created 2019/01/23 · arxiv updated 2020/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.