Quantization in classical mechanics and its relation to the Bohmian Ψ-field

Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of Hamilton systems in the form of the Schrödinger equation.It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with the Bohm “quantum” potential. Within the frame-work of Bohmian quantum mechanics supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Schrödinger equation is equivalent semantically and syntactically to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters. The conditions for the correctness of trajectory interpretation of quantum mechanics are discussed.

Research highlights
► The Schrödinger equation is derived from the classical Hamiltonian mechanics.
► This derivation is based on the Chetaev theorem on stable dynamical trajectories.
► The conditions for correctness of trajectory quantum mechanics are discussed.