Classical and relativistic functional mechanics

The report will consider the approach to solving the problem of the foundations of thermodynamics proposed by I.V.Volovich – the functional mechanics. The main idea of functional mechanics is the postulation of time–irreversible equations as fundamental laws of microscopic dynamics. Thus, Hamiltonian mechanics is reformulated in terms of the Liouville or Fokker-Planck equations, and the variances of the observables are assumed to be nonzero. Such a reformulation of classical mechanics leads to the appearance of corrections to the equations of dynamics of observables, since the trajectory averaged by initial data differs from the trajectory with average initial data. Averaging over the initial data destroys almost periodicity of the trajectories of a dynamical system obeying the conditions of the Poincare return theorem. The general formula for the evolution of the moments of the distribution function under the action of the phase flow is presented in the report. The questions of infinite motion in functional mechanics, including scattering problems, are also considered. Small inaccuracies in the measurement of the initial data taken into account by functional mechanics significantly change the asymptotic behavior of the trajectories of motion, eliminating singularities in part of the observed ones. The Liouville equation for geodesic flow on manifolds is discussed as a relativistic formulation of functional mechanics and the behavior of observables is described for some special cases.