A unified treatment of quartic invariants at fixed and arbitrary energy

Two-dimensional Hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the Jacobi geometrization of the dynamics. This approach allows for a unified treatment of invariants at both arbitrary and fixed energy. In the differential geometric picture, the quartic invariant corresponds to the existence of a fourth rank Killing tensor. Expressing the Jacobi metric in terms of a Kähler potential, the integrability condition for the existence of the Killing tensor at fixed energy is a non-linear equation involving the Kähler potential. At arbitrary energy, further conditions must be imposed which lead to an overdetermined system with isolated solutions. We obtain several new integrable and superintegrable systems in addition to all previously known examples. PACS numbers: 03.20.+i 11.30.-j e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]