Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2008-04-17
Journal of Math. Phys., Vol. 46, 052902 (2005)
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Some references updated with respect to the printed version
Scientific paper
10.1063/1.1888565
In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary energy (strong integrability). Using null complex coordinates, we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one. We review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants.
Pucacco Giuseppe
Rosquist Kjell
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