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Based on the Chetaev theorem on stable dynamical trajectories in the presenceof dissipative forces, we obtain the generalized condition for stability ofrelativistic classical Hamiltonian systems (with an invariant evolutionparameter) in the form of the Stueckelberg equation. As is known, this equationis the basis of a competing paradigm known as parametrized relativistic quantummechanics (pRQM). It is shown that the energy of dissipative forces, whichgenerate the Chetaev generalized condition of stability, coincides exactly withBohmian relativistic quantum potential. We show that the squared amplitude of awave function in the Stueckelberg equation is equivalent to the probabilitydensity function for the number of particle trajectories, relative to which thevelocity and the position of the particle are not hidden parameters.  The conditions for reasonableness of trajectory interpretation of pRQM arediscussed. On basis of analysis of a general formalism for vacuum-flavor mixingof neutrino within the context of the standard and pRQM models we show that thecorresponding expressions for the probability of transition from one neutrinoflavor to another differ appreciably, but they are experimentally testable: theestimations of absolute value for neutrino mass based on modern experimentaldata for solar and atmospheric neutrinos show that the pRQM results have apreference. It is noted that the selection criterion of mass solutions relieson proximity between the average size of condensed neutrino clouds, which isdescribed by the Muraki formula (29th ICRC, 2005) and depends on the neutrinomass, and the average size of typical observed void structure (dark matter +hydrogen gas), which plays the role of characteristic dimension of large-scalestructure of the Universe.

Quantization in relativistic classical mechanics: the Stueckelberg equation, neutrino oscillation and large-scale structure of the Universe