By making use of the Weyl-Wigner-Groenewold-Moyal association rules, a commutative product and a new quantum bracket are constructed in the ring of operators \cal{F}(H). In this way, an isomorphism between Lie algebra of classical observables (with Poisson bracket) and the Lie algebra of quantum observables with this new bracket is established. By these observations, a formulation of the classical mechanics in \cal{F}(H} is obtained and is shown to be \hbar\to 0 limit of the Heisenberg picture formulation of the quantum mechanics.
Based on the Bohr's correspondence principle it is shown that relativistic mechanics and quantum mechanics may be considered as generalizations of classical mechanics. A comparative description of relativistic and classical mechanics is given using three main principles: the definition of momentum, the main law of dynamics (Newton's second law) and the law of conservation of energy. The difference of relativistic mechanics from classical mechanics is due to the new definition of momentum that is proportional to energy and velocity. It is shown that new relativistic laws of dynamics make it necessary to change kinematical relations of classical mechanics such as the law of velocity composition, coordinate transformation etc. It is demonstrated that quantum mechanics may be considered as a rational statistical generalization of classical mechanics. Statistical conformities in quantum mechanics are fundamental and are not due to the incomplete information about the system. From all possible multi-parametric statistical models the root model is most remarkable. Constructing of a root multi-parametric statistical model leads to obtaining such frequencies and base functions in Fourier decomposition so that classical laws of motion are fulfilled in average. Root model leads to a consistent condition that connects eigenvectors and eigenvalues of a mechanical system that is described by a matrix equation of Heisenberg. Matrix equation of Heisenberg leads to an operator equation. The solution of the operator equation may be considered as the construction of Hamiltonian of the system and a move to the Schrodinger representation. The considered approach naturally leads to the notion of momentum operator, fundamental commutation relations, density matrix and Liouville equation construction etc.