Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator. (English) Zbl 1168.70316
Summary: The existence of bi-Hamiltonian structures for the rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analysed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained.
MSC:
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
| 70H05 | Hamilton’s equations |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
