Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1998-11-09
Nonlinear Sciences
Exactly Solvable and Integrable Systems
16 pages, LaTeX2e, slightly revised version. To appear in J. Math. Phys. vol 41, pp 370-384 (2000)
Scientific paper
10.1063/1.533137
Cubic invariants for two-dimensional Hamiltonian systems are investigated using the Jacobi geometrization procedure. This approach allows for a unified treatment of invariants at both fixed and arbitrary energy. In the geometric picture the invariant generally corresponds to a third rank Killing tensor, whose existence at a fixed energy value forces the metric to satisfy a nonlinear integrability condition expressed in terms of a Kahler potential. Further conditions, leading to a system of equations which is overdetermined except for singular cases, are added when the energy is arbitrary. As solutions to these equations we obtain several new superintegrable cases in addition to the previously known cases. We also discover a superintegrable case where the cubic invariant is of a new type which can be represented by an energy dependent linear invariant. A complete list of all known systems which admit a cubic invariant at arbitrary energy is given.
Karlovini Max
Rosquist Kjell
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