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Differential Structure of Orbit Spaces

Published online by Cambridge University Press:  20 November 2018

Richard Cushman
Affiliation:
Mathematics Institute, University of Utrecht, Budapestlaan 6, 3508TA Utrecht, The Netherlands email: cushman@math.uu.nl
Jędrzej Śniatycki
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Dr. N.W., Calgary, Alberta, T2N 1N4 email: sniat@math.ucalgary.ca
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Abstract

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We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds.

Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Arms, J. M., Cushman, R. and Gotay, M. J., A universal reduction procedure for Hamiltonian group actions. In: The geometry of Hamiltonian systems (ed. Ratiu, T. S.), Birkhäuser, Boston, 1991, 3151.Google Scholar
[2] Arms, J., Marsden, J. E. and Moncrief, V., Symmetry and bifurcation of momentum mappings. Commun. Math. Phys. 78(1981), 455478.Google Scholar
[3] Bates, L. and Lerman, E., Proper group actions and symplectic stratified spaces. Pacific J.Math. 191(1997), 201229.Google Scholar
[4] Cendra, H., Holm, D. D., Marsden, J. E. and Ratiu, T. S., Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products. Trans. Amer.Math. Soc. 186(1998), 125.Google Scholar
[5] Cushman, R. and Bates, L., Global aspects of classical integrable systems. Birkhäuser, Basel, 1997.Google Scholar
[6] Cushman, R. and Sjamaar, R., On singular reduction of Hamiltonian systems. In: Symplectic geometry and mathematical physics (ed. Donato, P.), Birkhäuser, Boston, 1991, 114128.Google Scholar
[7] Cushman, R. and Śniatycki, J., Hamiltonian systems on principal bundles. C. R. Math. Rep. Acad. Sci. Canada 21(1999), 6064.Google Scholar
[8] Duistermaat, J. J. and Kolk, J. A. C., Lie groups. Springer-Verlag, New York, 1999.Google Scholar
[9] Goresky, M. and MacPherson, R., Stratified Morse theory. Ergeb. der Math. 14, Springer Verlag, New York, 1988.Google Scholar
[10] Guillemin, V. and Sternberg, S., Symplectic techniques in physics. Cambridge University Press, Cambridge, 1984.Google Scholar
[11] Liebermann, P. and Marle, C., Symplectic geometry and analytical mechanics. D. Reidel, Dordrecht, 1987.Google Scholar
[12] Marsden, J. E. and Ratiu, T. S., Reduction of Poisson manifolds. Lett. Math. Phys. 11(1986), 161169.Google Scholar
[13] Marsden, J. E. and Weinstein, A., Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1974), 121130.Google Scholar
[14] Meyer, K., Symmetries and integrals in mechanics. In: Dynamical systems (ed. Piexoto, M.), Academic Press, New York, 1973, 259272.Google Scholar
[15] Michor, P., Manifolds of differentiable mappings. Shiva Mathematical Series 3, Shiva Publishing, Nantwich, 1980.Google Scholar
[16] Ortega, J.-P., Symmetry, reduction and stability in Hamiltonian systems. Ph.D Thesis, University of California at Santa Cruz, 1998.Google Scholar
[17] Ortega, J.-P. and Ratiu, T. S., Singular reduction of Poisson manifolds. Lett. Math. Phys. 46(1998), 359372.Google Scholar
[18] Ortega, J.-P. and Ratiu, T. S., Hamiltonian Singular Reduction. Manuscript.Google Scholar
[19] Palais, R., On the existence of slices for actions of noncompact Lie groups. Ann. of Math. 73(1961), 295323.Google Scholar
[20] Pukanski, L., Unitary representations of solvable groups. Ann. Sci. École Norm. Sup. 4(1971), 457608.Google Scholar
[21] Schwarz, G., Smooth functions invariant under the action of a compact Lie group. Topology 14(1975), 6368.Google Scholar
[22] Sikorski, R., Abstract covariant derivative. Colloq. Math. 18(1967), 151172.Google Scholar
[23] Sikorski, R., Wst. ep do geometrii różniczkowej. PWN, Warszawa, 1972. MR 57 7400.Google Scholar
[24] Sjamaar, R. and Lerman, E., Stratified symplectic spaces and reduction. Ann. of Math. 134(1991), 375422.Google Scholar
[25] Śniatycki, J., Schwarz, G. and Bates, L., Yang-Mills and Dirac fields in a bag, constraints and reduction. Commun. Math. Phys. 168(1995), 441453.Google Scholar
[26] Stefan, P., Accessible sets, orbits and foliations with singularities. Proc. LondonMath. Soc. 29(1974), 699713.Google Scholar
[27] Sussmann, H., Orbits of families of vector fields and integrability of distributions. Trans. Amer.Math. Soc. 180(1973), 171188.Google Scholar
[28] Whitney, H., Analytic extensions of differentiable functions defined on closed sets. Trans. Amer.Math. Soc. 36(1934), 6389.Google Scholar