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- Title
Dynamical Systems Related to Fractional Hamiltonians on the Two-Dimensional Sphere.
- Authors
Eleonskii, V. M.; Korolev, V. G.; Kulagin, N. E.
- Abstract
We consider a class of “fractional” Hamiltonian systems generalizing the classical problem of motion in a central field. Our analysis is based on transforming an integrable Hamiltonian system with two degrees of freedom on the plane into a dynamical system that is defined on the sphere and inherits the integrals of motion of the original system. We show that in the four-dimensional space of structural parameters, there exists a one-dimensional manifold (containing the case of the planar Kepler problem) along which the closedness of the orbits of all finite motions and the third Kepler law are preserved. Similarly, there exists a one-dimensional manifold (containing the case of the two-dimensional isotropic harmonic oscillator) along which the closedness of the orbits and the isochronism of oscillations are preserved. Any deformation of orbits on these manifolds does not violate the hidden symmetry typical of the two-dimensional isotropic oscillator and the planar Kepler problem. We also consider two-dimensional manifolds on which all systems are characterized by the same rotation number for the orbits of all finite motions.
- Subjects
HAMILTONIAN systems; ORBITAL mechanics; DIFFERENTIABLE dynamical systems; KEPLER'S laws; INTEGRAL calculus; PLANETARY theory
- Publication
Theoretical & Mathematical Physics, 2003, Vol 136, Issue 2, p1131
- ISSN
0040-5779
- Publication type
Article
- DOI
10.1023/A:1025018005746