Archive for Rational Mechanics and Analysis

, Volume 208, Issue 1, pp 255–274 | Cite as

Completeness of the Trajectories of Particles Coupled to a General Force Field

  • Anna Maria Candela
  • Alfonso Romero
  • Miguel Sánchez
Article

Abstract

We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold (M, g), which is defined by a general class of forces (both prescribed on M or depending on the velocity). The results include the general time-dependent anholonomic case, and further refinements for autonomous systems or forces derived from a potential are obtained. These extend classical results for Lagrangian and Hamiltonian systems. Several examples show the optimality of the assumptions as well as the utility of the results, including an application to relativistic pp-waves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abraham R., Marsden J.: Foundations of Mechanics (6th printing), 2nd edn. Addison-Wesley Publishing Co, Boston (MA) (1987)Google Scholar
  2. 2.
    Abraham R., Marsden J., Ratiu T.: Manifolds, Tensor Analysis and Applications, 2nd edn. Springer, New York (1988)MATHCrossRefGoogle Scholar
  3. 3.
    Barros M., Cabrerizo J.L., Fernández M., Romero A.: The Gauss–Landau–Hall problem on Riemannian surfaces. J. Math. Phys. 46((112905), 1–15 (2005)Google Scholar
  4. 4.
    Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry, 2nd edn. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 202. Marcel Dekker Inc., New York, 1996Google Scholar
  5. 5.
    Candela A., Flores J.L., Sánchez M.: On general plane fronted waves. Geodesics. Gen. Relativ. Gravit. 35, 631–649 (2003)ADSMATHCrossRefGoogle Scholar
  6. 6.
    Candela, A., Romero, A., Sánchez, M.: Remarks on the completeness of plane waves and the trajectories of accelerated particles in Riemannian manifolds. Proceedings of the International Meeting on Differential Geometry (Córdoba, November 15–17, 2010), University of Córdoba, 27–38, 2012Google Scholar
  7. 7.
    Candela, A., Sánchez, M.: Geodesics in semi-Riemannian manifolds: geometric properties and variational tools. Recent Developments in Pseudo-Riemannian Geometry. (Eds. Alekseevsky D.V. and Baum H.), Special Volume in the ESI-Series on Mathematics and Physics, EMS Publ. House, Zürich, 359–418, 2008Google Scholar
  8. 8.
    Curtis, W.D., Miller, F.R.: Differential Manifolds and Theoretical Physics. Pure Applied Mathematics, Vol. 116, Academic Press Inc., Orlando (FL), 1985Google Scholar
  9. 9.
    Ebin D.G.: Completeness of Hamiltonian vector fields. Proc. Am. Math. Soc. 26, 632–634 (1970)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Flores J.L., Sánchez M.: Causality and conjugate points in general plane waves. Class. Quantum Gravity 20, 2275–2291 (2003)MATHCrossRefGoogle Scholar
  11. 11.
    Flores, J.L., Sánchez, M.: The causal boundary of wave-type spacetimes. J. High Energy Phys. 3, 036 (2008)Google Scholar
  12. 12.
    Gordon W.B.: On the completeness of Hamiltonian vector fields. Proc. Am. Math. Soc. 26, 329–331 (1970)MATHCrossRefGoogle Scholar
  13. 13.
    Gordon, W.B.: An analytical criterion for completeness of Riemannian manifolds. Proc. Am. Math. Soc. 37, 221–225 (1973) [Corrected in Proc. Am. Math. Soc. 45, 130–131 (1974)]Google Scholar
  14. 14.
    Landau, L.D., Lifschitz, E.M.: Course of Theoretical Physics, Mechanics, Vol. 1, 3rd edn., Butterworth–Heinemann Ltd, Oxford, 1976Google Scholar
  15. 15.
    Müller O.: A note on closed isometric embeddings. J. Math. Anal. Appl. 349, 297–298 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Pure Applied Mathematics, Vol. 103, Academic Press Inc., New York, 1983Google Scholar
  17. 17.
    Romero A., Sánchez M.: On the completeness of certain families of semi-Riemannian manifolds. Geom. Dedicata. 69, 103–117 (1994)CrossRefGoogle Scholar
  18. 18.
    Sánchez M.: On the geometry of generalized Robertson–Walker spacetimes: geodesics. Gen. Relativity Gravitation 30, 915–932 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Teschl, G.: Ordinary differential equations and dynamical systems. Grad. Stud. Math., Vol. 140. Amer. Math. Soc., Providence, 2012Google Scholar
  20. 20.
    Weinstein A., Marsden J.: A comparison theorem for Hamiltonian vector fields. Proc. Am. Math. Soc. 26, 629–631 (1970)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anna Maria Candela
    • 1
  • Alfonso Romero
    • 2
  • Miguel Sánchez
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di Bari “A. Moro”BariItaly
  2. 2.Departamento de Geometría y Topología, Facultad de CienciasUniversidad de GranadaGranadaSpain

Personalised recommendations