On superintegrability of the Manev problem and its real Hamiltonian form. (English) Zbl 1136.70318

Kyuldjiev, Assen; Gerdjikov, Vladimir; Marmo, Giuseppe; Vilasi, Gaetano

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.
For the entire collection see [Zbl 1066.53003].

MSC:

70H05	Hamilton’s equations
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70F05	Two-body problems
70H06	Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70H33	Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics