On superintegrability of the Manev problem and its real Hamiltonian form.

*(English)* Zbl 1136.70318
Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 262-275 (2005).

Summary: We construct Ermanno-Bernoulli type invariants for the Manev model dynamics which may be viewed as remnants of the Laplace-Runge-Lenz vector in the Kepler model. If the orbits are bounded, these invariants exist only when a certain rationality condition is met, and thus we have superintegrability only on a subset of initial values. The dynamics of the Manev model is demonstrated to be bi-Hamiltonian and a recursion operator is constructed. We analyze the ‘real form dynamics’ of the Manev model and establish that it is always superintegrable. We also discuss the symmetry algebras of the Manev model and its real Hamiltonian form.

##### MSC:

70H05 | Hamilton’s equations |

37J35 | Completely integrable systems, topological structure of phase space, integration methods |

70F05 | Two-body problems |

70H06 | Completely integrable systems and methods of integration (mechanics of particles and systems) |

70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc. |