INVARIANTS AT FIXED AND ARBITRARY ENERGY - A UNIFIED GEOMETRIC APPROACH

Invariants at arbitrary and fixed energy (strongly and weakly conserve d quantities) for two-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometri zation of the dynamics. Using Killing tensors we obtain an integrabili ty condition for quadratic invariants which involves an arbitrary anal ytic function S(z). For invariants at arbitrary energy the function S( z) is a second-degree polynomial with real second derivative. The inte grability condition then reduces to Darboux's condition for quadratic invariants at arbitrary energy. The four types of classical quadratic invariants for positive-definite two-dimensional Hamiltonians are show n to correspond to certain conformal transformations. We derive the ex plicit relation between invariants in the physical and Jacobi time gau ges. In this way knowledge about the invariant in the physical time ga uge enables one to write down directly the components of the correspon ding Killing tenser for the Jacobi metric. We also discuss the possibi lity of searching for linear and quadratic invariants at fixed energy and its connection with the problem of the third integral in galactic dynamics. In our approach linear and quadratic invariants at fixed ene rgy can be found by solving a linear ordinary differential equation of the first or second degree, respectively.