Randomizes hamiltonian mechanics

Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamiltonian is the mean value of random Hamiltonian function. The analogs of the above results is obtained for a random quantum system (which is a random infinite dimensional Hamiltonian system). This random quantum Hamiltonians are the part of Hamiltonians of open quantum system.

Hamiltonian System and Symmetries for Scale Invariant Wavefunctions

The connection between scale invariant wavefunctions and solutions of some nonlinear equations (e.g., solitons and compactons) have been studied. Scale invariant functions are shown to have variational properties and a nonlinear algebraic structure. Any two-scale equation follows from Hamilton's equation of an infinite-dimensional Hamiltonian system, providing a self-similar formalism that is useful in studies of hierarchical and nonlinear lattices, soliton and compacton waves. The algebraic structure of any scaling function is shown to be a deformation of the trigonometric series generating algebra.