Hamiltonian formalism of periodic collisionless motion of particles in a field

Abstract

The paper presents a Hamiltonian formalism of the motion of gravitational particles without collisions in a flat and self-consistent field. The field of the system is represented as a potential well, the depth and length of which are expressed by energy and pressure. It was shown that the particles in the well accumulate in the boundary, forming separate concentric regions. The two-stream periodic motion of non-relativistic gravitating particles of the system was considered on the basis of the continuity equation. Using the Poisson equation, it was proved that the concentration of non-colliding particles is directly proportional to the gravitational potential. It was also shown that the static equilibrium between the two-flow periodic motion of particles and a self-consistent field can be described using the Poisson equation. Based on the conservation law, it was proved that the Hamiltonian function for particle interactions with a field is the total pressure as the first integral, and the mathematical nature of the Bernoulli force was determined. According to the conditions of the state and the spatial scale of the system, integral curves passing through the origin and the law of spatial potential distribution are determined. It is proved that the law of variation of velocity, pressure and concentration of particles along the length of the system is fulfilled at certain values of the state parameter. The results of theoretical calculations were presented in a simplified form. It was shown that, depending on the values of the state parameter of a two-stream system, the field of the system has the forms of a potential pit and a gap.