Abstract:

Is classical mechanics reversible? This is one of the fundamental unresolved problems of modern physics, the solution of which can significantly affect the axiomatic of quantum mechanics. However, this question is of paramount importance for mathematics, since it can substantially change our understanding of the differential equations describing dynamical systems. Recall that the dynamical systems theory it is a mathematical theory that draws on analysis, geometry, and topology - areas which in turn had their origins in Newtonian mechanics. As is known, the classical three-body problem with the use of 10 integrals of motion in a symplectic space reduces to a system of the eighth order, which is solved as a Cauchy problem. A. Poincar´e showed that although the system of Hamilton equations is time reversible, nevertheless, it manifests chaotic behavior in phase space on large sections of the phase space, which at first glance contradicts the formulation of the Cauchy problem. Such systems were later called classical dynamical systems or Poincar´e systems. We formulated the general classical three-body problem as a problem of geodesic trajectories flows on the Riemannian manifold. It is shown that a curved space with local coordinate system allows detecting new hidden symmetries of the internal motion of a dynamical system and reduces the three-body problem to the system of 6th order. It is proved that the equivalence of the initial Newtonian three-body problem and the developed representation provides coordinate transformations in combination with the underdetermined system of algebraic equations. The latter makes the system of geodesic equations relative to the evolutionary parameter (internal time) irreversible, which reveals the true causes of the emergence of chaos in symplectic space of the Hamiltonian system. Equations of deviation of close geodesic trajectories describing the characteristic properties of a dynamical system are obtained, depending on the evolutionary parameter. Various systems of stochastic differential equations (SDE) are derived to describe the motion of a three-body system taking into account the influence of external regular and stochastic forces. Using the system of SDE, a partial differential equation of the second order for the joint probability distribution of the momentum and coordinate of dynamical system in the phase space is obtained.

Identifier:

oai:noad.sci.am:135831

Language:

English

URL:

Affiliation:

Institute for Informatics and Automation Problems ; Institute of Chemical Physics

Country:

Armenia

Year:

2018

Time period:

October 22-24

Conference title:

100TH BIRTH ANNIVERSARY OF THE ACADEMICIAN MKHITAR DJRBASHYAN

Place:

Yerevan

Participation type:

oral