Poisson Brackets with Prescribed Casimirs

Damianou, Pantelis A. ; Petalidou, Fani

Canadian journal of mathematics, Tome 64 (2012) no. 5, pp. 991-1018

Résumé

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$ , while, in the case where $M$ is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.

Détail

DOI : 10.4153/CJM-2011-082-2

Mots-clés : 53D17, 53D15, Poisson bracket, Casimir function, almost symplectic structure, almost cosymplectic structure

Bibliographie

[1] [1] Hernández-Bermejo, B., Characterization, global analysis and integrability of a family of Poisson structures. Phys. Lett. A 372(2008), no. 7, 1009–1017. Google Scholar | DOI

[2] [2] Brylinski, J.-L., A differential complex for Poisson manifolds. J. Differential Geom. 28(1988), no. 1, 93–114. Google Scholar

[3] [3] Cantrijn, F., de León, M., and Martin de Diego, D., On almost-Poisson structures in nonholonomic mechanics. Nonlinearity 12(1999), no. 3, 721–737. Google Scholar | DOI

[4] [4] Damianou, P. A., Nonlinear Poisson Brackets. Ph. D. Dissertation, University of Arizona, 1989. Google Scholar

[5] [5] Damianou, P. A., Transverse Poisson structures of coadjoint orbits. Bull. Sci. Math. 120(1996), no. 2, 195–214. Google Scholar

[6] [6] Damianou, P. A. and Loja Fernandes, R., From the Toda lattice to the Volterra lattice and back. Rep. Math. Phys. 50(2002), no. 3, 361–378. Google Scholar | DOI

[7] [7] Damianou, P. A. and Magri, F., A gentle (without chopping) approach to the full Kostant–Toda lattice. SIGMA Symmetry Integrability Geom. Methods Appl. 1(2005), Paper 010. Google Scholar

[8] [8] Damianou, P. A., Sabourin, H., and Vanhaecke, P., Transverse Poisson structures to adjoint orbits in semi-simple Lie algebras. Pacific J. Math. 232(2007), no. 1, 111–138. Google Scholar | DOI

[9] [9] Dirac, P. A. M., Lectures in quantum mechanics. Yeshida University, 1964. Google Scholar

[10] [10] Dufour, J.-P. and Zung, N. T., Poisson structures and their normal forms. Progress in Mathematics, 242, Birkhäuser Verlag, Basel, 2005. Google Scholar

[11] [11] Flaschka, H., The Toda lattice. I. Existence of integrals. Phys. Rev. B 9(1974), 1924–1925. Google Scholar | DOI

[12] [12] Grabowski, J., Marmo, G., and Perelomov, A. M., Poisson structures: towards a classification. Modern Phys. Lett. A 8(1993), no. 18, 1719–1733. Google Scholar | DOI

[13] [13] Grabowski, J. and Marmo, G., Generalized n-Poisson brackets on a symplectic manifold.Modern Phys. Lett. A 13(1998), no. 39, 3185–3192. Google Scholar | DOI

[14] [14] Kirillov, A. A., Local Lie algebras. (Russian) Uspehi Mat. Nauk 31(1976), no. 4(190), 57–76. Google Scholar

[15] [15] Koon, W. S. and Marsden, J. E., Poisson reduction of nonholonomic mechanical systems with symmetry. Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics (Calgary, AB, 1997). Rep. Math. Phys. 42(1998), no. 1–2, 101–134. Google Scholar | DOI

[16] [16] Koszul, J.-L., Crochet de Schouten-Nijenhuis et cohomologie. In: The mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque 1985, Numéro Hors Série, 257–271. Google Scholar

[17] [17] Kupershmidt, B. A., Discrete Lax equations and differential-difference calculus. Astérisque 123(1985), 1–212. Google Scholar

[18] [18] Libermann, P., Sur le problème d’équivalence de certaines structures infinitésimales régulières. Ann. Mat. Pura Appl. 36(1954), 27–120. Google Scholar | DOI

[19] [19] Libermann, P., Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, 1959, pp. 37–59. Google Scholar

[20] [20] Libermann, P. and Marle, Ch.-M., Symplectic geometry and analytical mechanics. Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. Google Scholar

[21] [21] Lichnérowicz, A., Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geometry 12(1977), no. 2, 253–300. Google Scholar

[22] [22] Lichnérowicz, A., Les variétés de Jacobi et leurs algèbres de Lie associées. J. Math. Pures et Appl. 57(1978), no. 4, 453–488. Google Scholar

[23] [23] Lie, S., Theorie der transformationsgruppen. Zweiter Abschnitt, Teubner, Leipzig, 1890. Google Scholar

[24] [24] Marle, Ch.-M., Various approaches to conservative and nonconservative nonholonomic systems. Pacific Institute of Mathematical Sciences Workshop on Non-Holonomic Constraints in Dynamics (Calgary 1997). Rep. Math. Phys. 42(1998), no. 1–2, 211–229. Google Scholar | DOI

[25] [25] Meucci, A., Toda equations, bi-Hamiltonian systems, and compatible Lie algebroids. Math. Phys. Anal. Geom. 4(2001), no. 2, 131–146. Google Scholar | DOI

[26] [26] Odesskii, A. V. and Rubtsov, V. N., Polynomial Poisson algebras with regular structure of symplectic leaves. (Russian) Teoret. Mat. Fiz. 133(2002), no. 1, 3–23; translation in Theoret. and Math. Phys. 133(2002), no. 1, 1321–1337. Google Scholar

[27] [27] Poisson, S. D., Sur la variation des constantes arbitraires dans les questions de Mécanique. Journal de l’École Polytechnique, quinzième cahier, tome VIII, 266–344, 1809. Google Scholar

[28] [28] Vaisman, I., Lectures on the geometry of Poisson manifolds. Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. Google Scholar

[29] [29] Vaisman, I., Complementary 2-forms of Poisson structures. Compositio Math. 101(1996), no. 1, 55–75. Google Scholar

[30] [30] Van der, A. J. Schaft and Maschke, B. M., On the Hamiltonian formulation of nonholonomic mechanical systems. Rep. Math. Phys. 34(1994), no. 2, 225–233. Google Scholar | DOI

[31] [31] Weinstein, A., The local structure of Poisson manifolds. J. Differential Geom. 18(1983), no. 3, 523–557. Google Scholar