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Bundle-theoretic methods for higher-order variational calculus
1. | Institute of Mathematics. Polish Academy of Sciences, Śniadeckich 8, PO box 21, 00-956 Warsaw, Poland, Poland |
References:
[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.
doi: 10.1007/BF00402260. |
[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.
doi: 10.1017/S0305004100064501. |
[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.
doi: 10.1007/s00220-011-1313-y. |
[6] | |
[7] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204.
doi: 10.1088/1751-8113/41/17/175204. |
[8] |
J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2011), 21-36.
doi: 10.1016/j.geomphys.2011.09.004. |
[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.
doi: 10.1016/S0034-4877(03)80006-X. |
[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.
doi: 10.1016/B978-044452833-9.50013-9. |
[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.
doi: 10.1088/0305-4470/22/18/019. |
[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.
doi: 10.1093/imamci/6.4.465. |
[19] |
D. J. Saunders, The Geometry of Jet Bundles, Lecture Notes Math., 142, CUP, 1989.
doi: 10.1017/CBO9780511526411. |
[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.
doi: 10.4064/bc76-0-6. |
[25] |
L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories, J. Geom. Phys., 60 (2010), 857-873.
doi: 10.1016/j.geomphys.2010.02.003. |
[26] |
A. Weil, Théorie des points proches sur les varietes différentiables, in Colloque de géometrie différentielle, CNRS, Strasbourg (1953), 111-117. |
show all references
References:
[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.
doi: 10.1007/BF00402260. |
[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.
doi: 10.1017/S0305004100064501. |
[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.
doi: 10.1007/s00220-011-1313-y. |
[6] | |
[7] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204.
doi: 10.1088/1751-8113/41/17/175204. |
[8] |
J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2011), 21-36.
doi: 10.1016/j.geomphys.2011.09.004. |
[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.
doi: 10.1016/S0034-4877(03)80006-X. |
[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.
doi: 10.1016/B978-044452833-9.50013-9. |
[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.
doi: 10.1088/0305-4470/22/18/019. |
[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.
doi: 10.1093/imamci/6.4.465. |
[19] |
D. J. Saunders, The Geometry of Jet Bundles, Lecture Notes Math., 142, CUP, 1989.
doi: 10.1017/CBO9780511526411. |
[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.
doi: 10.4064/bc76-0-6. |
[25] |
L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories, J. Geom. Phys., 60 (2010), 857-873.
doi: 10.1016/j.geomphys.2010.02.003. |
[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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