Variationality of geodesic circles in two dimensions

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.Comment: 10th International Conference "Differential Geometry and Its Applications" (Olomouc, August 27-31, 2007). MR2462829 (2009k:53087) Corrected after publ.: reference 5 change

Third-order relativistic dynamics: classical spinning particle travelling in a plane

Mathisson's 'new mechanics' of a relativistic spinning particle is shown to follow, in the case of planar motion, from only general requirements of relativistic invariance and of the dependence on third order derivatives along with the 'variationality' feature. The Hamiltonian counterpart ultimately recovers the Dixon system of equations for this case with the Mathisson-Pirani supplementary condition.Comment: 10 pages. Minor misprints corrected. http://www.icmp.lviv.ua/journal/zbirnyk.15/004/art04.pd

Autoparallel variational description of the free relativistic top third order dynamics

A second order variational description of the autoparallel curves of some differential-geometric connection for the third order Mathisson's 'new mechanics' of a relativistic free spinning particle is suggested starting from general requirements of invariance and 'variationality'.Comment: Conf. "Differential Geometry and Its Applicatons" (Opava, Czech Republic, August 27-31, 2001). Corrected after publ.: page 456, second formula. MR1978798(2004d:70025

Towards the physical significance of the (k2+A)∥u∥(k^2+A)\|u\| metric

We offer an example of the second order Kawaguchi metric function the extremal flow of which generalizes the flat space-time model of the semi-classical spinning particle to the framework of the pseudo-Riemannian space-time. The general shape of the variational Euler-Poisson equation of the fourth order in the (pseudo-)Riemannian space is being developed too.Comment: 10-th International Conference of Tensor Society (Constantza, Romania, Sept. 3-7, 2008) After publ.: Typos: pg. 109, 110. References: url links added. MR2761679 (2011k:53109

Symmetries of vector exterior differential systems and the inverse problem in second-order Ostrohrads'kyj mechanics

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.Comment: MR1401574 (97h:58009); Proc. conf. Symmetry in nonlinear mathematical physics (Kyiv, Ukraine, July 3-8, 1995), Vol. 4; Corrected after publ.: Formula (15) and a few minor correction

The next variational prolongation of the Euclidean space

The unique third-order invariant variational equation in three-dimensional (pseudo)Euclidean space is derived.Comment: 8th International Conference of Tensor Society (Varna, Bulgaria, Aug. 22-26, 2005) Corrected after publ.: formula (8), typos pg. 3 MR236366

Second order variational problem and 2-dimensional concircular geometry

It is proved that the set of geodesic circles in two dimensions may be given a variational description and the explicit form of it is presented. In the limit case of the Euclidean geometry a certain claim of uniqueness of such description is proved. A formal notion of 'spin' force is discovered as a by-product of the variation procedure involving the acceleration.Comment: Proceedings of the 8th Conference on Geometry and Topology of Manifolds (Lie algebroids, dynamical systems and applications) (Luxembourg-Poland-Ukraine conference Przemy\'sl (Poland)-L'viv (Ukraine), 30.IV.-6.V. 2007). MR2553650(2010i:53065). Several minor typos correcte

Hamilton-Ostrohrads'kyj approach to relativistic free spherical top dynamics

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.Comment: 7th International Conference Differential Geometry and Applications (Brno, Czech Republic, Aug. 10-14, 1998). MR1712785 (2000f:70022). Corrected after publ.: formulas (9) and succeeding, (17), (B4); reference [4]; typos. Appendices A and B adde

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.Comment: 15 figure

A first order prolongation of the conventional space

A variational equation of the third order in three-dimensional space is proposed which describes autoparallel curves of some connection.Comment: 3 figures. MR1406360 (97e:58061); Zbl 0867.58021. Corrected after publ.: Formulae (24), (28), equation numbering, typos on p. 404, reference