Invariants at fixed and arbitrary energy. A unified geometric approach

摘要: Invariants at arbitrary and fixed energy (strongly weakly conserved quantities) for two-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of dynamics. Using Killing tensors we obtain an integrability condition quadratic invariants which involves analytic function S(z). For S(z) second-degree polynomial with real second derivative. The then reduces to Darboux's energy. four types classical positive-definite Hamiltonians shown correspond certain conformal transformations. We derive explicit relation between physical time gauges. In this way knowledge about invariant gauge enables one write down directly components corresponding tensor metric. also discuss possibility searching linear its connection problem third integral galactic our approach can be found solving ordinary differential equation first or degree, respectively.