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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 710274, 26 pages
http://dx.doi.org/10.1155/2011/710274
Research Article

Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

1Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
2Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
3School of Automation, Beijing Institute of Technology, Beijing 100081, China

Received 3 December 2010; Revised 19 March 2011; Accepted 12 April 2011

Academic Editor: Heinrich Begehr

Copyright © 2011 Linyu Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. Casetti, M. Pettini, and E. G. D. Cohen, “Geometric approach to Hamiltonian dynamics and statistical mechanics,” Physics Reports A, vol. 337, no. 3, pp. 237–341, 2000. View at Publisher · View at Google Scholar
  2. A. Yahalom, J. Levitan, M. Lewkowicz, and L. Horwitz, “Lyapunov vs. geometrical stability analysis of the Kepler and the restricted three body problems,” Physics Letters A, vol. 375, no. 21, pp. 2111–2117, 2011. View at Publisher · View at Google Scholar
  3. T. Iwai and H. Yamaoka, “Stratified dynamical systems and their boundary behaviour for three bodies in space, with insight into small vibrations,” Journal of Physics A, vol. 38, no. 25, pp. 5709–5730, 2005. View at Publisher · View at Google Scholar
  4. G. Blankenstein, “A joined geometric structure for hamiltonian and gradient control systems,” in Lagrangian and Hamiltonian Methods for Nonlinear Control 2003, pp. 51–56, IFAC, Laxenburg, 2003. View at Google Scholar
  5. E. V. Ferapontov, “Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type,” Funktsional Analiz i ego Prilozheniya, vol. 25, no. 3, pp. 37–49, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. E. V. Ferapontov, “Hamiltonian systems of hydrodynamic type and their realizations on hypersurfaces of a pseudo-Euclidean space,” in Problems in Geometry, vol. 22 of Itogi Nauki i Tekhniki, pp. 59–96, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, Russia, 1990. View at Google Scholar
  7. T. Kambe, Geometrical Theory of Dynamical Systems and Fluid Flows, vol. 23 of Advanced Series in Nonlinear Dynamics, World Scientific, Hackensack, NJ, USA, 2005.
  8. S. Tanimura and T. Iwai, “Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics,” Journal of Mathematical Physics, vol. 41, no. 4, pp. 1814–1842, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. T. Uzer, C. Jaffé, J. Palacián, P. Yanguas, and S. Wiggins, “The geometry of reaction dynamics,” Nonlinearity, vol. 15, no. 4, pp. 957–992, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Yu. Boldin, A. A. Bronnikov, V. V. Dmitrieva, and R. A. Sharipov, “Complete normality conditions for dynamical systems on riemannian manifolds,” Rossiĭskaya Akademiya Nauk. Teoreticheskaya i Matematicheskaya Fizika, vol. 103, no. 2, pp. 267–275, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. P. Boyland and C. Golé, “Lagrangian systems on hyperbolic manifolds,” Ergodic Theory and Dynamical Systems, vol. 19, no. 5, pp. 1157–1173, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. R. A. Sharipov, “Newtonian normal shift in multidimensional Riemannian geometry,” Matematicheskiĭ Sbornik, vol. 192, no. 6, pp. 105–144, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser, Boston, Mass, USA, 1992.
  14. P. Petersen, Riemannian Geometry, vol. 171 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 2006.
  15. L. Todjihounde, “Ricci deformation and conformal change of metrics,” IC/99/89, International Centre for Theoretical Physics, Trieste, Italy, 1999. View at Google Scholar
  16. L. Peng, H. Sun, D. Sun, and J. Yi, “The geometric structures and instability of entropic dynamical models,” Advances in Mathematics, vol. 227, no. 1, pp. 459–471, 2011. View at Publisher · View at Google Scholar
  17. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, Calif, USA, 1973.
  18. R. S. Hamilton, “Three-manifolds with positive Ricci curvature,” Journal of Differential Geometry, vol. 17, no. 2, pp. 255–306, 1982. View at Google Scholar · View at Zentralblatt MATH
  19. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1994.
  20. L. P. Eisenhart, “Dynamical trajectories and geodesics,” Annals of Mathematics(Princeton), vol. 30, pp. 591–606, 1929. View at Google Scholar
  21. V. G. Gurzadyan, “Chaotic phenomena in astrophysics and cosmology,” in Proceedings of the 10th Brazilian School of Cosmology and Gravitation, July-August 2002.
  22. H. E. Kandrup, “Geometric interpretation of chaos in two-dimensional hamiltonian systems,” Physical Review E, vol. 57, no. 3, part A, pp. 2722–2732, 1997. View at Publisher · View at Google Scholar