The connection between quasi-invariants (invariants of a Hamiltonian system defined only on a single constant energy hypersurface) and generalized Killing vector fields associated with the corresponding Jacobi metric is investigated. The results are used to deduce a generalised form of the classical Whittaker problem in two degrees of freedom.

We consider here the problem of the existence of a quasi-invariant which is linear in the momenta for Hamiltonians in three degrees of freedom. We show that such quasi-invariants are more constrained in their structure than in the two degrees of freedom case. We also show that some of these quasi-invariants have to be interpreted as ‘pseudo-translations’, i.e., as translations in a non-orthogonal system of coordinates.

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.

In this communication, we reanalyze the causes of the singularities of canonical perturbation theory and show that some of these singularities can be removed by using time-dependent canonical perturbation theory. A study of the local and global properties (in terms of the perturbation parameter) is also undertaken.

In this communication, we report the results of the application of time-dependent perturbation theory to the Henon-Heiles system. We show that the predictions of the perturbation theory hold good for short times, and try to explain the increase of error in the predicted results with the increase in energy.

In this communication, we report the results of the application of time dependent perturbation theory to a non-integrable Hamiltonian which is a perturbation on a Hamiltonian with nonconstant frequencies. The theory provides good time dependent local constants of motion and also gives good approximation for mapping of solutions for a time limit determined by the nearest singularity in complexε plane for fixed real time and the order of calculation.

In this communication, we investigate the behavior of the derivatives of invariants for Hamiltonian systems, using information derived from an analysis of the Liapunov exponents of the system. We show that under certain conditions on the analyticity properties of the solutions of the equations of motion, it is possible to construct 2n invariants of motion which are guaranteed to beC^∞ as functions of phase space and time in a suitably defined domainD.