Lie Brackets on Affine Bundles

Annals of Global Analysis and Geometry

Volumn 24, Issue 2, 2003, Pages 101-130

  Grabowska, Katarzyna ^a   Grabowski, Janusz ^b   Urbański, Paweł ^a  

^a UNIVERSITY OF WARSAW   (Poland)

^b INSTITUTE OF MATHEMATICS   (Poland)

Author keywords

Affine bundle; Lie algebroid; Lie bracket; Poisson structure

Indexed keywords

EID: 0038759654     PISSN: 0232704X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1024457728027     Document Type: Article

References (22)

1. Benenti, S.: Fibrés affines canoniques et mécanique newtonienne, Séminaire Sud-Rhodanien de Géometrie, Journées S.M.F. Lyon, 26-30 May, 1986.
2. Grabowski, J. and Urbański, P.: Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom. 15 (1997), 447-486.
3. Grabowski, J. and Urbański, P.: Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys. 40 (1997), 195-208.
4. Grabowski, J. and Urbański, P.: Algebroids - General differential calculi on vector bundles, J. Geom. Phys. 31 (1999), 111-141.
5. Grabowski, J. and Marmo, G., Jacobi structures revisited, J. Phys. A 34 (2001), 10975-10990.
6. Iglesias, D. and Marrero, J. C.: Generalized Lie bialgebroids and Jacobi structures J. Geom. Phys. 40 (2001), 176-199.
7. Konieczna, K.: Affine formulation of the Newtonian analytical mechanics, Thesis, University of Warsaw, 1995 [in Polish].
8. Kosmann-Schwarzbach, Y.: Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153-165.
9. Libermann, P.: Lie algebroids and mechanics, Arch. Math. (Brno) 32 (1996), 147-162.
10. Mackenzie, K. C. H.: Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97-147.
11. Mackenzie, K. C. H. and Xu, P.: Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford 49(2) (1998), 59-85.
12. Martínez, E.: Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67 (2001), 295-320.
13. Martínez, E., Mestdag, T. and Sarlet, W.: Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys. 44 (2002), 70-95.
14. Menzio, M. R. and Tulczyjew, W. M.: Infinitesimal symplectic relations and generalized Hamiltonian dynamics, Ann. Inst. H. Poincaré 29 (1978), 349-367.
15. Pidello, G.: Una formulazione intrinseca della meccanica newtoniana Tesi di dottorato di Ricerca in Matematica, Consorzio Interuniversitario Nord-Ovest, 1987/1988.
16. Sarlet, W., Mestdag, T. and Martínez, E.: Lie algebroid structures on a class of affine bundles, J. Math. Phys. 43 (2002), 5654-5675.
17. Sarlet, W., Mestdag, T. and Martínez, E.: Lagrangian equations on affine Lie algebroids, in: D. Krupka et al. (eds), Differential Geometry and Its Applications, Proc. 8th Internal. Conf. (Opava 2001), to appear.
18. Tulczyjew, W. M.: Frame independence of analytical mechanics, Atti Accad. Sci. Torino 119 (1985).
19. Tulczyjew, W. M. and Urbański, P.: An affine framework for the dynamics of charged particles, Atti Accad. Sci. Torino, Suppl. n. 2, 126 (1992), 257-265.
20. Urbański, P.: Affine Poisson structure in analytical mechanics, in: J.-P. Antoine et al. (eds), Quantization and Infinite-Dimensional Systems, Plenum Press, New York, 1994, pp. 123-129.
21. Weinstein, A.: A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2 (1978), 417-420.
22. Weinstein, A.: Lagrangian mechanics and groupoids, Fields Inst. Comm. 7 (1996), 207-231.