Bundle-theoretic methods for higher-order variational calculus

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March 2014, 6(1): 99-120. doi: 10.3934/jgm.2014.6.99

Bundle-theoretic methods for higher-order variational calculus

Michał Jóźwikowski ^1, and Mikołaj Rotkiewicz ^1,

1.	Institute of Mathematics. Polish Academy of Sciences, Śniadeckich 8, PO box 21, 00-956 Warsaw, Poland, Poland

Received August 2013 Revised February 2014 Published April 2014

Abstract
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We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.

Keywords: higher-order Euler-Lagrange equations, integration by parts, variational calculus, geometric mechanics., Higher tangent bundles, graded bundles.

Mathematics Subject Classification: 58A20, 58E30, 70H5.

Citation: Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99

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]	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. 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]	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. 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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