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The Journal of Geometric Mechanics (JGM)
 

Bundle-theoretic methods for higher-order variational calculus
Pages: 99 - 120, Issue 1, March 2014

doi:10.3934/jgm.2014.6.99      Abstract        References        Full text (566.0K)           Related Articles

Michał Jóźwikowski - Institute of Mathematics. Polish Academy of Sciences, Śniadeckich 8, PO box 21, 00-956 Warsaw, Poland (email)
Mikołaj Rotkiewicz - Institute of Mathematics. Polish Academy of Sciences, Śniadeckich 8, PO box 21, 00-956 Warsaw, Poland (email)

1 C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.       
2 F. Cantrijn, M. Crampin, W. Sarlet and D. Saunders, The canonical isomorphism between $T^k T^{*} M$ and $T^{*} T^k M$, C. R. Acad. Sci. Paris, 309 (1989), 1509-1514.       
3 M. Crampin, Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics, Lett. Math. Phys., 19 (1990), 53-58.       
4 M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Phillos. Soc., 99 (1986), 565-587.       
5 F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. Vialard, Invariant higher-order variational problems, Comm. Math. Phys., 309 (2012), 413-458.
6 K. Grabowska, private communication, 2012.
7 K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204.       
8 J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2011), 21-36.       
9 X. Gracia, J. Martin-Solano and M. Munoz-Lecenda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148.
10 M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with application to variational calculus, preprint arXiv:1306.3379.
11 M. Jóźwikowski and W. Respondek, A comparison of vakonomic and nonholonomic variational problems with applications to systems on Lie groups, preprint arXiv:1310.8528.
12 I. Kolar, Weil bundles as generalized jet spaces, in Handbook of Global Analysis, Elsevier Sci. B. V., Amsterdam, 1214 (2008), 625-664.       
13 I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry, Springer, Berlin, 1993.       
14 M. de Leon and E. Lacomba, Lagrangian submanifolds and higher-order mechanical systems, J. Phys. A, 22 (1989), 3809-3820.
15 M. de Leon and P. R. Rodrigues, Higher order almost tangent geometry and non-autonomous Lagrangian dynamics, in Proceedings of the Winter School 'Geometry and Physics', Circolo Matematico di Palermo, Palermo (1987), 157-171.
16 K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, CUP, Cambridge, 2005.       
17 A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.       
18 L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved surfaces, IMA J. Math. Control Inform., 6 (1989), 465-473.       
19 D. J. Saunders, The Geometry of Jet Bundles, Lecture Notes Math., 142, CUP, 1989.       
20 W. Tulczyjew, Sur la différentiele de Lagrange, C. R. Acad. Sci. Paris Serie A, 280 (1975), 1295-1298.       
21 W. Tulczyjew, The Lagrange differential, Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.       
22 W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris Serie A, 283 (1976), 15-18.       
23 W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris, 283 (1976), 675-678.       
24 W. Tulczyjew, Evolution of Ehresmann's jet theory, in Geometry and topology of manifolds: The mathematical legacy of Charles Ehresmann, Banach Centre Publications, 76, Warsaw, 2007, 159-176.       
25 L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories, J. Geom. Phys., 60 (2010), 857-873.       
26 A. Weil, Théorie des points proches sur les varietes différentiables, in Colloque de géometrie différentielle, CNRS, Strasbourg (1953), 111-117.       

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