Hamiltonian mechanics and its generalizations

Abstract

The basics of Hamiltonian mechanics and its generalizations are analyzed to find the most general laws of motion. The specific features of the variational principle in Hamiltonian mechanics (the problems of covariant formulation and boundary conditions) and the Maupertuis principle are discussed. The connection between Hamiltonian mechanics and statistical physics (Hamiltonian equations of motion preserve the Gibbs distribution and the evolution of nonequilibrium states of the harmonic oscillator in the thermal bath is described by the probability amplitudes) is underlined. The most well known generalizations, the Birkhoff and Nambu mechanics, are considered from this point of view. The Ostrogradsky mechanics, in which the Lagrangian depends on higher derivatives, theories not on symplectic manifolds, theories not on manifolds, and theories with complex variables are discussed. The simplest generalization of Poisson brackets for description of the evolution of nonequilibrium states results in the cosmological constant that arises in gravitational equations.