Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 016, 11 pages      arXiv:0802.0751
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

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

Roman Ya. Matsyuk
Institute for Applied Problems in Mechanics and Mathematics, 15 Dudayev Str., L'viv, Ukraine

Received October 31, 2007, in final form January 18, 2008; Published online February 06, 2008; Some errors are corrected March 27, 2008

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.

Key words: covariant Ostrohrads'kyj mechanics; spin; concircular geometry; uniform acceleration.

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  1. Hill E.L., On the kinematics of uniformly accelerated motions and classical magnetic theory, Phys. Rev. 72 (1947), 143-149.
  2. Yano K., Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Jap. 16 (1940), 195-200.
  3. Matsyuk R.Ya., Variational principle for uniformly accelerated motion, Mat. Metody Fiz.-Mekh. Polya 16 (1982), 84-88 (in Russian).
  4. Arodz H., Sitarz A., Wegrzyn P., On relativistic point particles with curvature-dependent actions, Acta Phys. Polon. B 20 (1989), 921-939.
  5. Matsyuk R.Ya., Poincaré-invariant equations of motion in Lagrangian mechanics with higher derivatives, PhD Thesis, Institute for Applied Problems in Mechanics and Mathematics, L'viv, 1984 (in Russian).
  6. Logan J.D., Invariant variational principles, Academic Press, New York, 1977.
  7. Kawaguchi M., An introduction to the theory of higher order spaces. II. Higher order spaces in multiple parameters, RAAG Memoirs 4 (1968), 578-592.
  8. Matsyuk R.Ya., Autoparallel variational description of the free relativistic top third order dynamics, in Proceedings of Eight International Conference "Differential Geometry and Its Applications" (August 27-31, 2001, Opava, Czech Republic), Editors O. Kowalski et al., Silesian Univ., Opava, 2002, 447-452.
  9. de Léon M., Rodrigues P. R., Generalized classical mechanics and field theory, Elsevier, Amsterdam, 1985.
  10. Matsyuk R.Ya., On the existence of a Lagrangian for a system of ordinary differential equations, Mat. Metody Fiz.-Mekh. Polya 13 (1981), 34-38, 113 (in Russian).
  11. Matsyuk R.Ya., Lagrangian analysis of invariant third-order equations of motion in relativistic classical particle mechanics, Dokl. Akad. Nauk SSSR 282 (1985), 841-844 (English transl.: Soviet Phys. Dokl. 30 (1985), 458-460).
  12. Arreaga G., Capovilla R., Guven J., Frenet-Serret dynamics, Classical Quantum Gravity 18 (2001), 5065-5083, hep-th/0105040.
  13. Matsyuk R.Ya., The variational principle for geodesic circles, in Boundary Value Problems of Mathematical Physics, Naukova Dumka, Kiev, 1981, 79-81 (in Russian).
  14. Acatrinei C.S., A path integral leading to higher order Lagrangians, J. Phys. A: Math. Gen. 40 (2007), F929-F933, arXiv:0708.4351.
  15. Dixon W.G., Dynamics of extended bodies in general relativity I. Momentum and angular momentum, Proc. Roy. Soc. London. Ser. A. 314 (1970), 499-527.
  16. Yano K., Ishihara Sh., Tangent and cotangent bundles, Marcel Dekker, New York, 1973.
  17. Ehresmann Ch., Les prolongements dúne variété différentiable I. Calcul des jets, prolongement principal, C. R. Math. Acad. Sci. Paris 233 (1951), 598-600.
  18. Kruprová O., The geometry of ordinary variational equations, Lect. Notes in Math., Vol. 1678, Springer-Verlag, Berlin, 1997.
  19. Tulczyjew W.M., Sur la différentielle de Lagrange, C. R. Math. Acad. Sci. Paris 280 (1975), 1295-1298.

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