Previous |  Up |  Next

Article

Keywords:
Lie groupoid; Lie algebroid; constrained mechanical system; Legendre transformation
Summary:
We give a formulation of certain types of mechanical systems using the structure of groupoid of the tangent and cotangent bundles to the configuration manifold $M$; the set of units is the zero section identified with the manifold $M$. We study the Legendre transformation on Lie algebroids.
References:
[A.M] R. Abraham, J. E. Marsden: Foundations of mechanics. Benjamin Reading, 1978. MR 0515141
[A.D] C. Albert, P. Dazord: Groupoides de Lie et groupoides symplectiques. Séminaire Sud-Rhodanien. Publications Dept. Math. Lyon (1-partie 1989, 2-partie 1990 ). MR 1104916
[A] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt: Mathematical aspects of classical and celestial mechanics. in “Dynamical Systems III (V. I. Arnold ed.), Springer-Verlag Berlin, Heidelberg, 1988. MR 2269239
[D] P. Dazord: Mécanique hamiltonienne en présence de contraintes. Illinois Journal of Mathematics, 38, 1, 1994, 148-175. MR 1245839
[D.S] P. Dazord, D. Sondaz: Variétés de Poisson. Algébroides de Lie, Séminaire Sud-Rhodanien, Publications Dept Math. Lyon, 1988 1/B. MR 1040867
[E] C. Ehresmann: Oeuvres complètes. Parties I. 1 et I.2 “Topologie algébrique et Geométrie Différentielle". Zbl 0561.01027
[G] C. Godbillon: Geométrie différentielle et mécanique analytique. Hermann, Paris, 1969. MR 0242081 | Zbl 0653.53001
[H] R. Hermann: Geometry, Physics and Systems. Marcel Dekker, New York, 1973. MR 0494183 | Zbl 0285.58001
[Kl] J. Klein: Espaces variationnels et mécanique. Ann. Inst. Fourier, Grenoble 12 (1962), 1-124. MR 0215269 | Zbl 0281.49026
[Ko] I. Kolář: On the Euler-Lagrange differential in fibered manifolds. Reports on mathematical physics, vol 12 (1977). MR 0501101
[K$_1$] Y. Kosmann-Schwarzbach: Exact Gerstenhaber algebras and Lie bialgebroids. Acta Applicandae-Mathematicae, 41, 1990, 35-81. MR 1362125 | Zbl 0837.17014
[K$_2$] Y. Kosmann-Schwarzbach: The Lie bialgebroid of a Poisson-Nijenhuis manifold. Letters Math. Phys., 1996. MR 1421686 | Zbl 1005.53060
[L$_1$] P. Libermann: Pseudogroupes infinitésimaux. Bull. Soc. Math. 87 (1959), 409-425. MR 0123279 | Zbl 0198.26801
[L$_2$] P. Libermann: Prolongements des fibrés principaux et groupoides différentiables. Sem. Analyse Globale Montreal (1969), 7-108. MR 0356117
[L$_3$] P. Libermann: Groupoides différentiables et presque parallélisme. Symp. Math. X (convegno di Geom. Diff.) Roma. 1971, 59-93. MR 0370668
[L$_4$] P. Libermann: Parallélismes. Jour. of Diff. Geometry, 8, No. 4, Dec, 1973, 511-539. MR 0353189 | Zbl 0284.53023
[L$_5$] P. Libermann: On symplectic and contact groupoids. Proc. Conf. Opava, 1992, Silesian University, Opava, 1993, 29-45. MR 1255524 | Zbl 0805.53034
[L.M] P. Libermann, C. M. Marle: Symplectic geometry and analytical mechanics. Math. Appl., Reidel, Publ. Comp. Dordrecht, 1987. MR 0882548
[L.W.X] Z.J. Liu, A. Weinstein, P.Xu: Manin triples for Lie bialgebroids. preprint 1995. MR 1472888
[Mac] K. C. H. Mackenzie: Lie algebroids and Lie pseudoalgebras. Bull. London Math. Soc. 27 (1995), 97-147. MR 1325261 | Zbl 0829.22001
[M.X] K. C. H. Mackenzie, P. Xu: Lie bialgebroids and Poisson groupoids. Duke Math. Jour. 73, 415-452, 1994. MR 1262213 | Zbl 0844.22005
[M] C. M. Marle: Reduction of Constrained Mechanical Systems and Stability of Relative Equilibria. Comm. Math. Phys. 174, 1995, 295-318. MR 1362167 | Zbl 0859.70012
[P] J. Pradines: Théorie de Lie pour les groupoides différentiables. C.R. Acad. Sc. Paris, 263, 1966, 907-910; 264, 1967, 245-248 Troisième théorème de Lie pour les groupoides différentiables C.R. Acad. Sc. Paris, 267, 1968, 21-23. Sur le groupoide cotangent de Weinstein-Dazord C.R. Acad. Sc. Paris 306, 1988, 557-560. Zbl 0232.22028
[S] J. M. Souriau: Structure des Systèmes Dynamiques. Dunod, Paris, 1969. MR 0260238 | Zbl 0186.58001
[T] W. M. Tulczyjew: Sur la différentielle de Lagrange. C.R. Acad. Sc. Paris 280, A, 1975, 1295-1298. MR 0377987 | Zbl 0314.58018
[V] I. Vaisman: Second order hamiltonian vector fields on tangent bundles. E.S.I 49, 1993, Vienna. Zbl 0822.58021
[W] A. Weinstein: Lagrangian mechanics and groupoids. preprint U. of Cal., Berkeley 1992. MR 1365779 | Zbl 0844.22007
Partner of
EuDML logo