Abstract
Natural affine analogs of Lie brackets on affine bundles are studied.In particular, a close relation to Lie algebroids and a duality withcertain affine analog of Poisson structure is established as well asaffine versions of complete lifts and Cartan exterior calculi.
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Benenti, S.: Fibrés affines canoniques et mécanique newtonienne, Séminaire Sud-Rhodanien de Géometrie, Journées S.M.F. Lyon, 2–30 May, 1986.
Grabowski, J. and Urbański, P.: Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom. 15 (1997), 44–486.
Grabowski, J. and Urbański, P.: Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys. 40 (1997), 19–208.
Grabowski, J. and Urbański, P.: Algebroids - General differential calculi on vector bundles, J. Geom. Phys. 31 (1999), 11–141.
Grabowski, J. and Marmo, G., Jacobi structures revisited, J. Phys. A 34 (2001), 1097–10990.
Iglesias, D. and Marrero, J. C.: Generalized Lie bialgebroids and Jacobi structures J. Geom. Phys. 40 (2001), 17–199.
Konieczna, K.: Affine formulation of the Newtonian analytical mechanics, Thesis, University of Warsaw, 1995 [in Polish].
Kosmann-Schwarzbach, Y.: Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 15–165.
Libermann, P.: Lie algebroids and mechanics, Arch. Math. (Brno) 32 (1996), 14–162.
Mackenzie, K. C. H.: Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 9–147.
Mackenzie, K. C. H. and Xu, P.: Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford 49(2) (1998), 5–85.
Martínez, E.: Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67 (2001), 29–320.
Martínez, E., Mestdag, T. and Sarlet, W.: Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys. 44 (2002), 7–95.
Menzio, M. R. and Tulczyjew, W. M.: Infinitesimal symplectic relations and generalized Hamiltonian dynamics, Ann. Inst. H. Poincaré 29 (1978), 34–367.
Pidello, G.: Una formulazione intrinseca della meccanica newtoniana Tesi di dottorato di Ricerca in Matematica, Consorzio Interuniversitario Nord-Ovest, 1987/1988.
Sarlet, W., Mestdag, T. and Martínez, E.: Lie algebroid structures on a class of affine bundles, J. Math. Phys. 43 (2002), 565–5675.
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 Internat. Conf. (Opava 2001), to appear.
Tulczyjew, W. M.: Frame independence of analytical mechanics, Atti Accad. Sci. Torino 119 (1985).
Tulczyjew, W. M. and Urba´nski, P.: An affine framework for the dynamics of charged particles, Atti Accad. Sci. Torino, Suppl. n. 2, 126 (1992), 25–265.
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. 12–129.
Weinstein, A.: A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2 (1978), 41–420.
Weinstein, A.: Lagrangian mechanics and groupoids, Fields Inst. Comm. 7 (1996), 20–231.
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Grabowska, K., Grabowski, J. & Urbański, P. Lie Brackets on Affine Bundles. Annals of Global Analysis and Geometry 24, 101–130 (2003). https://doi.org/10.1023/A:1024457728027
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DOI: https://doi.org/10.1023/A:1024457728027