arXiv subjects
Publications and source records attributed to R. Ya. Matsyuk.
We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of particles.
Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous utilization of symmetry principles along with the inverse variational problem considerations in terms of vector-valued differential forms. Next, some simple algorithm of transition between the autonomous variational problem and the variational problem in parametric form is established. The example definitely solved shows no-existence of a globally and intrinsically defined Lagrangian for the Poincar\'e-invariant and well defined unique variational equation in the case in hand. Hamiltonian counterpart is briefly discussed in terms of Poisson bracket. The model appears to provide a generalized canonical description of the quasi-classical spinning particle governed by the Mathisson-Papapetrou equations in flat space-time.
This note treats the notion of Lagrange derivative for the third order mechanics in the context of covariant Riemannian geometry. The variational differential equation for geodesic circles in two dimensions is obtained. The influence of the curvature tensor on the Lagrange derivative leads to the emergence of the notion of quasiclassical spin in the pseudo-Riemannian case.
Dynamics of classical spinning particle in special relativity with Pirani constraint is a typical example of the generalized Hamilton theory developed by O. Krupkov\'a and discovers some characteristic features of the latter.
A variational equation of the third order in three-dimensional space is proposed which describes autoparallel curves of some connection.
Symmetries of variational problems are considered as symmetries of vector bundle valued exterior differential systems. This approach is then applied to third order ordinary variational equations of motion of the semi-classical spinning particle.