Advertisement

The Optimal Momentum Map

  • Juan-Pablo Ortega
  • Tudor S. Ratiu

Abstract

The presence of symmetries in a Hamiltonian system usually simplies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden-Weinstein reduction procedure takes advantage of this and associates to the original system a new Hamiltonian system with fewer degrees of freedom. However, in a large number of situations, this standard approach does not work or is not e cient enough, in the sense that it does not use all the information encoded in the symmetry of the system. In this work, a new momentum map will be defined that is capable of overcoming most of the problems encountered in the traditional approach.

Keywords

Hamiltonian System Poisson Bracket Symplectic Manifold Invariant Function Hamiltonian Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abraham, R., and Marsden, J. E. [1978], Foundations of Mechanics. Second edition, Addison-Wesley.Google Scholar
  2. Abraham, R., Marsden, J. E., and Ratiu, T. S. [1988], Manifolds, Tensor Analysis, and Applications. Volume 75 of Applied Mathematical Sciences, Springer-Verlag.Google Scholar
  3. Alekseev, A., Malkin, A., and Meinrenken, E. [1997], Lie group valued momentum maps. Preprint, dg-ga/9707021.Google Scholar
  4. Arms, J. M., Cushman, R., and Gotay, M. J. [1991], A universal reduction procedure for Hamiltonian group actions. In The Geometry of Hamiltonian Systems. T. S. Ratiu ed. pages 33–51. Springer Verlag.Google Scholar
  5. Bates, L., and Lerman, E. [1997], Proper group actions and symplectic stratified spaces. Pacific J. Math., 181(2):201–229.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Bredon, G. E. [1972], Introduction to Compact Transformation Groups. Academic Press.Google Scholar
  7. Camacho, C., and Lins Neto, A. [1985] Geometric Theory of Foliations. Birkhäuser.Google Scholar
  8. Gotay, M. J., and Tuynman, G. M. [1991], A symplectic analogue of the Mostow-Palais Theorem. Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., 20:173–182, Springer-Verlag.MathSciNetGoogle Scholar
  9. Kempf, G. [1987], Computing invariants. Springer Lecture Notes in Mathematics, volume 1278, 62–80. Springer-Verlag.MathSciNetCrossRefGoogle Scholar
  10. Kirillov, A. A. [1976], Elements of the Theory of Representations. Grundlehren der mathematischen Wissenschaften, volume 220. Springer-Verlag.Google Scholar
  11. Lerman, E. [1995], Symplectic cuts. Mathematical Research Letters, 2:247–258.zbMATHMathSciNetGoogle Scholar
  12. Libermann, P., and Marle, C.-M. [1987], Symplectic Geometry and Analytical Mechanics. Reidel.Google Scholar
  13. Marsden, J. E., Misiolek, G., Ortega, J.-P., Perlmutter, M., and Ratiu, T. S. [2001], Symplectic Reduction by Stages. Preprint.Google Scholar
  14. Marsden, J. E. and Ratiu, T. S. [1999], Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, volume 17. Second Edition. Springer-Verlag.Google Scholar
  15. Marsden, J. E., and Weinstein, A. [1974], Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 5(1):121–130.CrossRefMathSciNetzbMATHGoogle Scholar
  16. Mather, J. [1977], Differentiable invariants. Topology, 16:145–156.CrossRefzbMATHMathSciNetGoogle Scholar
  17. McDuff, D. [1988], The moment map for circle actions on symplectic manifolds. J. Geom. Phys., 5:149–160.CrossRefzbMATHMathSciNetGoogle Scholar
  18. Meyer, K. R. [1973], Symmetries and integrals in mechanics. In Dynamical Systems, pp. 259–273. M.M. Peixoto, ed. Academic Press.Google Scholar
  19. Noether, E. [1918], Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen Mathematisch-physikalische Klasse, pp. 235–258.Google Scholar
  20. Ortega, J.-P. [1998], Symmetry, Reduction, and Stability in Hamiltonian Systems. Ph.D. Thesis. University of California, Santa Cruz. June, 1998.Google Scholar
  21. Ortega, J.-P. [2001a], Optimal reduction. In preparation.Google Scholar
  22. Ortega, J.-P. [2001b], Singular dual pairs. Preprint, INLN.Google Scholar
  23. Ortega, J.-P. and Ratiu, T. S. [1998], Singular reduction of Poisson manifolds. Letters in Mathematical Physics, 46:359–372.CrossRefMathSciNetzbMATHGoogle Scholar
  24. Ortega, J.-P. and Ratiu, T. S. [2002], Hamiltonian Singular Reduction. To appear in Birkhäuser, Progress in Mathematics.Google Scholar
  25. Otto, M. [1987] A reduction scheme for phase spaces with almost Kähler symmetry. Regularity results for momentum level sets. J. Geom. Phys., 4:101–118.CrossRefzbMATHMathSciNetGoogle Scholar
  26. Palais, R. [1961], On the existence of slices for actions of non-compact Lie groups. Ann. Math., 73:295–323.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Paterson, A. L. T. [1999] Grupoids, Inverse Semigroups, and their Operator Algebras. Progress in Mathematics, volume 170. Birkhäuser.Google Scholar
  28. Poènaru, V. [1976], Singularités C en préesence de syméetrie. Lecture Notes in Mathematics, volume 510. Springer-Verlag.Google Scholar
  29. Schwarz, G. W. [1974], Smooth functions invariant under the action of a compact Lie group. Topology, 14:63–68.CrossRefGoogle Scholar
  30. Sjamaar, R. and Lerman, E. [1991], Stratified symplectic spaces and reduction. Ann. of Math., 134:375–422.CrossRefMathSciNetGoogle Scholar
  31. Stefan, P. [1974a], Accessibility and foliations with singularities. Bull. Amer. Math. Soc., 80:1142–1145.zbMATHMathSciNetCrossRefGoogle Scholar
  32. Stefan, P. [1974b], Accessible sets, orbits and foliations with singularities. Proc. Lond. Math. Soc., 29:699–713.zbMATHMathSciNetGoogle Scholar
  33. Sussman, H. [1973], Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc., 180:171–188.CrossRefMathSciNetGoogle Scholar
  34. Trotter, H. F. [1958], Approximation of semi-groups of operators. Pacific J. Math., 8:887–919.zbMATHMathSciNetGoogle Scholar
  35. Warner, F. W. [1983], Foundation of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer-Verlag.Google Scholar
  36. Weinstein, A. [1976], Lectures on Symplectic Manifolds. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. Regional Conference Series in Mathematics, number 29. American Mathematical Society.Google Scholar
  37. Weinstein, A. [1983], The local structure of Poisson manifolds. J. Differential Geometry, 18:523–557.zbMATHMathSciNetGoogle Scholar
  38. Weitsman, J. [1993], A Duistermaat-Heckman formula for symplectic circle actions. Internat. Math. Res. Notices, 12:309–312.CrossRefzbMATHMathSciNetGoogle Scholar
  39. Weyl, H. [1946], The Classical Groups. Second Edition. Princeton University Press.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • Juan-Pablo Ortega
  • Tudor S. Ratiu

There are no affiliations available

Personalised recommendations