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

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

Received  August 2013 Revised  February 2014 Published  April 2014

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.
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, (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.

[3]

M. Crampin, Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics,, Lett. Math. Phys., 19 (1990), 53. 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. 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. 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). 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. 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. doi: 10.1016/S0034-4877(03)80006-X.

[10]

M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with application to variational calculus,, preprint , ().

[11]

M. Jóźwikowski and W. Respondek, A comparison of vakonomic and nonholonomic variational problems with applications to systems on Lie groups,, preprint , ().

[12]

I. Kolar, Weil bundles as generalized jet spaces,, in Handbook of Global Analysis, 1214 (2008), 625. doi: 10.1016/B978-044452833-9.50013-9.

[13]

I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry,, Springer, (1993).

[14]

M. de Leon and E. Lacomba, Lagrangian submanifolds and higher-order mechanical systems,, J. Phys. A, 22 (1989), 3809. 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', (1987), 157.

[16]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, CUP, (2005).

[17]

A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order,, Nagoya Math. J., 40 (1970), 99.

[18]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved surfaces,, IMA J. Math. Control Inform., 6 (1989), 465. doi: 10.1093/imamci/6.4.465.

[19]

D. J. Saunders, The Geometry of Jet Bundles,, Lecture Notes Math., 142 (1989). doi: 10.1017/CBO9780511526411.

[20]

W. Tulczyjew, Sur la différentiele de Lagrange,, C. R. Acad. Sci. Paris Serie A, 280 (1975), 1295.

[21]

W. Tulczyjew, The Lagrange differential,, Bull. Acad. Polon. Sci., 24 (1976), 1089.

[22]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sci. Paris Serie A, 283 (1976), 15.

[23]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sci. Paris, 283 (1976), 675.

[24]

W. Tulczyjew, Evolution of Ehresmann's jet theory,, in Geometry and topology of manifolds: The mathematical legacy of Charles Ehresmann, 76 (2007), 159. doi: 10.4064/bc76-0-6.

[25]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories,, J. Geom. Phys., 60 (2010), 857. doi: 10.1016/j.geomphys.2010.02.003.

[26]

A. Weil, Théorie des points proches sur les varietes différentiables,, in Colloque de géometrie différentielle, (1953), 111.

show all references

References:
[1]

C. de Boor, A Practical Guide to Splines,, Springer-Verlag, (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.

[3]

M. Crampin, Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics,, Lett. Math. Phys., 19 (1990), 53. 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. 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. 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). 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. 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. doi: 10.1016/S0034-4877(03)80006-X.

[10]

M. Jóźwikowski and M. Rotkiewicz, Prototypes of higher algebroids with application to variational calculus,, preprint , ().

[11]

M. Jóźwikowski and W. Respondek, A comparison of vakonomic and nonholonomic variational problems with applications to systems on Lie groups,, preprint , ().

[12]

I. Kolar, Weil bundles as generalized jet spaces,, in Handbook of Global Analysis, 1214 (2008), 625. doi: 10.1016/B978-044452833-9.50013-9.

[13]

I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry,, Springer, (1993).

[14]

M. de Leon and E. Lacomba, Lagrangian submanifolds and higher-order mechanical systems,, J. Phys. A, 22 (1989), 3809. 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', (1987), 157.

[16]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, CUP, (2005).

[17]

A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order,, Nagoya Math. J., 40 (1970), 99.

[18]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved surfaces,, IMA J. Math. Control Inform., 6 (1989), 465. doi: 10.1093/imamci/6.4.465.

[19]

D. J. Saunders, The Geometry of Jet Bundles,, Lecture Notes Math., 142 (1989). doi: 10.1017/CBO9780511526411.

[20]

W. Tulczyjew, Sur la différentiele de Lagrange,, C. R. Acad. Sci. Paris Serie A, 280 (1975), 1295.

[21]

W. Tulczyjew, The Lagrange differential,, Bull. Acad. Polon. Sci., 24 (1976), 1089.

[22]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sci. Paris Serie A, 283 (1976), 15.

[23]

W. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sci. Paris, 283 (1976), 675.

[24]

W. Tulczyjew, Evolution of Ehresmann's jet theory,, in Geometry and topology of manifolds: The mathematical legacy of Charles Ehresmann, 76 (2007), 159. doi: 10.4064/bc76-0-6.

[25]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories,, J. Geom. Phys., 60 (2010), 857. doi: 10.1016/j.geomphys.2010.02.003.

[26]

A. Weil, Théorie des points proches sur les varietes différentiables,, in Colloque de géometrie différentielle, (1953), 111.

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