To uncover the most general laws of mechanics, the Hamiltonian mechanics and its generalizations are analyzed. The peculiarities of variational principle in the Hamiltonian mechanics (problems of covariant formulation, the boundary conditions) and the Mopertuis principle are discussed. Connection of the Hamiltonian mechanics with the statistical physics (the Hamiltonian equations of motion preserve the Gibbs distribution, and the evolution of nonequilibrium states of a harmonic oscillator in a thermal bath is described by probability amplitudes) is stressed. The most known generalizations - the Birkhoff and Nambu mechanics are considered from this point of view. Theories not on symplectic manifolds, not on manifolds, theories with complex variables and the Ostrogradsky mechanics (a theory with the Lagrangian depending on higher derivatives) are discussed. The simplest generalization of the Poisson brackets describing evolution of nonequilibrium states leads to appearance of the cosmological constant in the gravitational equations.

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