Invariants at arbitrary and fixed energy (strongly and weakly conserved
quantities) for 2-dimensional Hamiltonian systems are treated in a unified way.
This is achieved by utilizing the Jacobi metric geometrization of the dynamics.
Using Killing tensors we obtain an integrability condition for quadratic
invariants which involves an arbitrary analytic function S(z). For invariants
at arbitrary energy the function S(z) is a second degree polynomial with real
second derivative. The integrability condition then reduces to Darboux's
condition for quadratic invariants at arbitrary energy. The four types of
classical quadratic invariants for positive definite 2-dimensional Hamiltonians
are shown to correspond to certain conformal transformations. We derive the
explicit relation between invariants in the physical and Jacobi time gauges. In
this way knowledge about the invariant in the physical time gauge enables one
to directly write down the components of the corresponding Killing tensor for
the Jacobi metric. We also discuss the possibility of searching for linear and
quadratic invariants at fixed energy and its connection to the problem of the
third integral in galactic dynamics. In our approach linear and quadratic
invariants at fixed energy can be found by solving a linear ordinary
differential equation of the first or second degree respectively.Comment: Some misprints corrected with respect to the printed versio
