arXiv subjects

Roman Ya. Matsyuk

Publications and source records attributed to Roman Ya. Matsyuk.

4 recordsLinked to original sources

arXiv2016

Inverse Variational Problem and Symmetry in Action: The Relativistic Third Order Dynamics

Tools of the intrinsic analysis on manifolds, helpful in solving the invariant inverse problem of the calculus of variations are being presented comprising a combined approach which consists in the simultaneous imposition of symmetry principles and the inverse variational problem considerations in terms of vector-valued differential forms. In three-dimensional space-time we obtain a unique (covector) third-order Poincar\'e-invariant variational equation, which then is identified with the motion of a free relativistic top in flat three-dimensional space-time.

math.GM↗

arXiv2014

Integration by parts and vector differential forms in higher order variational calculus on fibred manifolds

Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by parts' procedure for this variation is then described in purely formal language and is shown to consist in application of nonlinear Green formula to the vertical differential of the Lagrangian. Euler-Lagrange expressions and the Green operator are calculated by simple pull-backs of certain vector bundle valued differential forms associated with the given variational problem.

math.DG↗

arXiv2011

Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in General Relativity

We show how the Dixon's system of first order equations of motion for the particle with inner dipole structure together with the side Mathisson constraint follows from rather general construction of the 'Hamilton system' developed by Weyssenhoff, Rund and Gr\"asser to describe the phase space counterpart of the evolution under the ordinary Euler-Poisson differential equation of the parameter-invariant variational problem with second derivatives. One concrete expression of the 'Hamilton function' leads to the General Relativistic form of the fourth order equation of motion known to describe the quasi-classical 'quiver' particle in Special Relativity. The corresponding Lagrange function including velocity and acceleration coincides in the flat space of Special Relativity with the one considered by Bopp in an attempt to give an approximate variational formulation of the motion of self-radiating electron, when expressed in terms of geometric quantities.

math.DG↗

arXiv2008

The Variational Principle for the Uniform Acceleration and Quasi-Spin in Two Dimensional Space-Time

The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic particle is emphasized. The known form of spin-curvature interaction emerges due to the presence of second order derivatives in the expression for the Lagrange function. The variational equation itself reduces to the unique invariant variational equation of constant Frenet curvature in two dimensional (pseudo)-Euclidean geometry.

math-ph↗