Search

For the Toda lattice and the Holt system we consider properties of canonical transformations of the extended phase space, which preserve integrability. The separated variables are invariant under change of the time. On the other hand, mapping of the time induces transformations of the action-angles variables and a shift of the generating function of the B\"{a}cklund transformation.

We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map completely integrable system into other completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and R-matrices. As an example, we consider canonical transformations of the extended phase space for the Toda lattices and the Stackel systems., Comment: LaTeX2e + Amssymb, 22pp

The semiclassical Einstein-Brillouin Keller vortex tube quantization for the quasi-energies in time-periodic Hamiltonians is studied for typical systems, when the vortex tubes of the quantizing tubes may have complicated structures in extended phase space. Numerical application to a periodically driven Duffing (quartic) oscillator yields semiclassical quasi-energies in excellent agreement with the exact quantum mechanical ones for the regular states.

Statistical-mechanical problems of principle associated with long-time correlation are discussed using the example of retarded Brownian motion. The well known 'stochastic' solutions are quoted and the fundamental inadequacies of the Fokker-Planck solutions to the problem demonstrated. An alternative is to adopt Edwards' (1964) concept of a 'Lagrangian' phase space with a Lagrangian Fokker-Planck equation. Within this extended phase space, problems of history dependence, non-separation of time scales and non-exponential decays are solved consistently. Considering the role of the fluctuation-dissipation theorem the character of the asymptotic solution in the Brownian problem is obtained.