×

Singular reduction of Dirac structures. (English) Zbl 1385.70039

From the text: The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about \(G\)-invariant vector fields and one-forms are shown.
The paper is organized as follows. Dirac structures are reviewed in Section 2 and vector fields on differential spaces, stratifications, and orbit type manifolds in Section 3. Generalized distributions and the integrability of tangent distributions as well as pointwise and smooth annihilators are introduced and discussed in Section 4. The averaging method is presented in Section 5. Using this technique, we show that the strata of the quotient \(\overline M\) correspond to the quotients of the orbit type strata on \(M\) and that the local one-forms on the manifold \(M\) descend to analogous objects on the reduced stratified space \(\overline M\). Then we study the properties of descending sections of the Pontryagin bundle and get many technical results needed in the final reduction proof. Section 6 is devoted to the main result of the paper, namely singular Dirac reduction. First we recall the reduction procedure in the case of conjugated isotropy subgroups. Then the two main theorems of the paper (Theorems 6.4 and 6.5) are proved and the reduced dynamics of implicit Hamiltonian systems is constructed. Several examples are also given.

MSC:

70H45 Constrained dynamics, Dirac’s theory of constraints
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D20 Momentum maps; symplectic reduction
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Judith M. Arms, Richard H. Cushman, and Mark J. Gotay, A universal reduction procedure for Hamiltonian group actions, The geometry of Hamiltonian systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 22, Springer, New York, 1991, pp. 33 – 51. · Zbl 0742.58016
[2] N. Aronszajn. Subcartesian and subriemannian spaces. Notices Amer. Math. Soc., 14:111, 1967.
[3] Edward Bierstone, Lifting isotopies from orbit spaces, Topology 14 (1975), no. 3, 245 – 252. · Zbl 0317.57015
[4] G. Blankenstein. Implicit Hamiltonian Systems: Symmetry and Interconnection. Ph.D.Thesis, University of Twente, 2000.
[5] Guido Blankenstein and Tudor S. Ratiu, Singular reduction of implicit Hamiltonian systems, Rep. Math. Phys. 53 (2004), no. 2, 211 – 260. · Zbl 1065.37040
[6] G. Blankenstein and A. J. van der Schaft, Symmetry and reduction in implicit generalized Hamiltonian systems, Rep. Math. Phys. 47 (2001), no. 1, 57 – 100. · Zbl 0978.37046
[7] K. Buchner, M. Heller, P. Multarzyński, and W. Sasin. Literature on differential spaces. Acta Cosmologica, 19:111-129, 1993.
[8] Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726 – 765. · Zbl 1115.53056
[9] Theodore James Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631 – 661. · Zbl 0850.70212
[10] Ted Courant and Alan Weinstein, Beyond Poisson structures, Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986) Travaux en Cours, vol. 27, Hermann, Paris, 1988, pp. 39 – 49. · Zbl 0698.58020
[11] Richard Cushman and JÈ©drzej Śniatycki, Differential structure of orbit spaces, Canad. J. Math. 53 (2001), no. 4, 715 – 755. · Zbl 1102.53301
[12] Richard Cushman, Reduction, Brouwer’s Hamiltonian, and the critical inclination, Celestial Mech. 31 (1983), no. 4, 401 – 429. · Zbl 0559.70026
[13] J.J. Duistermaat. Dynamical systems with symmetry. Available at http://www.math.uu.nl/ people/duis/. · Zbl 0534.70019
[14] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. · Zbl 0955.22001
[15] Rui Loja Fernandes, Juan-Pablo Ortega, and Tudor S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math. 131 (2009), no. 5, 1261 – 1310. · Zbl 1180.53083
[16] M. Jotz and T.S. Ratiu. Dirac and nonholonomic reduction. Preprint, arXiv:0806.1261v2, 2008. · Zbl 1247.53096
[17] M. Jotz and T. S. Ratiu, Induced Dirac structure on isotropy type manifolds., arXiv:1008.2280v1, to appear in “Transformation groups” (2010). · Zbl 1271.70033
[18] -, Optimal Dirac reduction., arXiv:1008.2283v1 (2010).
[19] M. Jotz, T.S. Ratiu, and M. Zambon, Invariant frames for vector bundles and applications., Preprint (2010). · Zbl 1319.70018
[20] Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, vol. 35, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the French by Bertram Eugene Schwarzbach. · Zbl 0643.53002
[21] T. Lusala and J. Śniatycki. Stratified subcartesian spaces. arXiv:0805.4807v1, To appear in Canad. Math. Bull. · Zbl 1237.58008
[22] Jerrold E. Marsden and Tudor Ratiu, Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), no. 2, 161 – 169. · Zbl 0602.58016
[23] Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121 – 130. · Zbl 0327.58005
[24] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. · Zbl 0441.13001
[25] Juan-Pablo Ortega and Tudor S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics, vol. 222, Birkhäuser Boston, Inc., Boston, MA, 2004. · Zbl 1241.53069
[26] Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295 – 323. · Zbl 0103.01802
[27] Markus J. Pflaum, Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics, vol. 1768, Springer-Verlag, Berlin, 2001. · Zbl 0988.58003
[28] Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63 – 68. · Zbl 0297.57015
[29] Roman Sikorski, Abstract covariant derivative, Colloq. Math. 18 (1967), 251 – 272. · Zbl 0162.25101
[30] Roman Sikorski, Differential modules, Colloq. Math. 24 (1971/72), 45 – 79. · Zbl 0226.53004
[31] Roman Sikorski, Wstȩp do geometrii różniczkowej, Państwowe Wydawnictwo Naukowe, Warsaw, 1972 (Polish). Biblioteka Matematyczna, Tom 42. [Mathematics Library, Vol. 42].
[32] JÈ©drzej Śniatycki, Integral curves of derivations on locally semi-algebraic differential spaces, Discrete Contin. Dyn. Syst. suppl. (2003), 827 – 833. Dynamical systems and differential equations (Wilmington, NC, 2002). · Zbl 1086.53100
[33] JÈ©drzej Śniatycki, Orbits of families of vector fields on subcartesian spaces, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2257 – 2296 (English, with English and French summaries). · Zbl 1048.53060
[34] P. Stefan, Accessibility and foliations with singularities, Bull. Amer. Math. Soc. 80 (1974), 1142 – 1145. · Zbl 0293.57015
[35] P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3) 29 (1974), 699 – 713. · Zbl 0342.57015
[36] P. Stefan, Integrability of systems of vector fields, J. London Math. Soc. (2) 21 (1980), no. 3, 544 – 556. · Zbl 0432.58002
[37] Héctor J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171 – 188. · Zbl 0274.58002
[38] Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. · Zbl 0810.53019
[39] P. Walczak, A theorem on diffeomorphisms in the category of differential spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 325 – 329 (English, with Russian summary). · Zbl 0258.58001
[40] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.