| Title:
|
Lie algebroids and mechanics (English) |
| Author:
|
Libermann, Paulette |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
32 |
| Issue:
|
3 |
| Year:
|
1996 |
| Pages:
|
147-162 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Lie groupoid |
| Keyword:
|
Lie algebroid |
| Keyword:
|
constrained mechanical system |
| Keyword:
|
Legendre transformation |
| MSC:
|
22E70 |
| MSC:
|
58F05 |
| MSC:
|
58H05 |
| MSC:
|
70F25 |
| MSC:
|
70H03 |
| MSC:
|
70H35 |
| idZBL:
|
Zbl 0912.70009 |
| idMR:
|
MR1421852 |
| . |
| Date available:
|
2008-06-06T21:30:40Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107570 |
| . |
| Reference:
|
[A.M] R. Abraham, J. E. Marsden: Foundations of mechanics.Benjamin Reading, 1978. MR 0515141 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[D] P. Dazord: Mécanique hamiltonienne en présence de contraintes.Illinois Journal of Mathematics, 38, 1, 1994, 148-175. MR 1245839 |
| Reference:
|
[D.S] P. Dazord, D. Sondaz: Variétés de Poisson.Algébroides de Lie, Séminaire Sud-Rhodanien, Publications Dept Math. Lyon, 1988 1/B. MR 1040867 |
| Reference:
|
[E] C. Ehresmann: Oeuvres complètes.Parties I. 1 et I.2 “Topologie algébrique et Geométrie Différentielle". Zbl 0561.01027 |
| Reference:
|
[G] C. Godbillon: Geométrie différentielle et mécanique analytique.Hermann, Paris, 1969. Zbl 0653.53001, MR 0242081 |
| Reference:
|
[H] R. Hermann: Geometry, Physics and Systems.Marcel Dekker, New York, 1973. Zbl 0285.58001, MR 0494183 |
| Reference:
|
[Kl] J. Klein: Espaces variationnels et mécanique.Ann. Inst. Fourier, Grenoble 12 (1962), 1-124. Zbl 0281.49026, MR 0215269 |
| Reference:
|
[Ko] I. Kolář: On the Euler-Lagrange differential in fibered manifolds.Reports on mathematical physics, vol 12 (1977). MR 0501101 |
| Reference:
|
[K$_1$] Y. Kosmann-Schwarzbach: Exact Gerstenhaber algebras and Lie bialgebroids.Acta Applicandae-Mathematicae, 41, 1990, 35-81. Zbl 0837.17014, MR 1362125 |
| Reference:
|
[K$_2$] Y. Kosmann-Schwarzbach: The Lie bialgebroid of a Poisson-Nijenhuis manifold.Letters Math. Phys., 1996. Zbl 1005.53060, MR 1421686 |
| Reference:
|
[L$_1$] P. Libermann: Pseudogroupes infinitésimaux.Bull. Soc. Math. 87 (1959), 409-425. Zbl 0198.26801, MR 0123279 |
| Reference:
|
[L$_2$] P. Libermann: Prolongements des fibrés principaux et groupoides différentiables.Sem. Analyse Globale Montreal (1969), 7-108. MR 0356117 |
| Reference:
|
[L$_3$] P. Libermann: Groupoides différentiables et presque parallélisme.Symp. Math. X (convegno di Geom. Diff.) Roma. 1971, 59-93. MR 0370668 |
| Reference:
|
[L$_4$] P. Libermann: Parallélismes.Jour. of Diff. Geometry, 8, No. 4, Dec, 1973, 511-539. Zbl 0284.53023, MR 0353189 |
| Reference:
|
[L$_5$] P. Libermann: On symplectic and contact groupoids.Proc. Conf. Opava, 1992, Silesian University, Opava, 1993, 29-45. Zbl 0805.53034, MR 1255524 |
| Reference:
|
[L.M] P. Libermann, C. M. Marle: Symplectic geometry and analytical mechanics.Math. Appl., Reidel, Publ. Comp. Dordrecht, 1987. MR 0882548 |
| Reference:
|
[L.W.X] Z.J. Liu, A. Weinstein, P.Xu: Manin triples for Lie bialgebroids.preprint 1995. MR 1472888 |
| Reference:
|
[Mac] K. C. H. Mackenzie: Lie algebroids and Lie pseudoalgebras.Bull. London Math. Soc. 27 (1995), 97-147. Zbl 0829.22001, MR 1325261 |
| Reference:
|
[M.X] K. C. H. Mackenzie, P. Xu: Lie bialgebroids and Poisson groupoids.Duke Math. Jour. 73, 415-452, 1994. Zbl 0844.22005, MR 1262213 |
| Reference:
|
[M] C. M. Marle: Reduction of Constrained Mechanical Systems and Stability of Relative Equilibria.Comm. Math. Phys. 174, 1995, 295-318. Zbl 0859.70012, MR 1362167 |
| Reference:
|
[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 |
| Reference:
|
[S] J. M. Souriau: Structure des Systèmes Dynamiques.Dunod, Paris, 1969. Zbl 0186.58001, MR 0260238 |
| Reference:
|
[T] W. M. Tulczyjew: Sur la différentielle de Lagrange.C.R. Acad. Sc. Paris 280, A, 1975, 1295-1298. Zbl 0314.58018, MR 0377987 |
| Reference:
|
[V] I. Vaisman: Second order hamiltonian vector fields on tangent bundles.E.S.I 49, 1993, Vienna. Zbl 0822.58021 |
| Reference:
|
[W] A. Weinstein: Lagrangian mechanics and groupoids.preprint U. of Cal., Berkeley 1992. Zbl 0844.22007, MR 1365779 |
| . |
