American Institute of Mathematical Sciences

Advanced
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

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, (1978).   Google Scholar

[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.   Google Scholar

[3]

M. Crampin, Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics,, Lett. Math. Phys., 19 (1990), 53.  doi: 10.1007/BF00402260.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K., Grabowska,, private communication, (2012).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.   Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

show all references

References:
[1]

C. de Boor, A Practical Guide to Splines,, Springer-Verlag, (1978).   Google Scholar

[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.   Google Scholar

[3]

M. Crampin, Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics,, Lett. Math. Phys., 19 (1990), 53.  doi: 10.1007/BF00402260.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K., Grabowska,, private communication, (2012).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.   Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[1]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[2]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81
[3]

Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493
[4]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[5]

Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131
[6]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
[7]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[8]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[9]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451
[10]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[11]

Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018
[12]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[13]

Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51
[14]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[15]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006
[16]

Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399
[17]

Marco Castrillón López, Pedro Luis García Pérez. The problem of Lagrange on principal bundles under a subgroup of symmetries. Journal of Geometric Mechanics, 2019, 11 (4) : 539-552. doi: 10.3934/jgm.2019026
[18]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
[19]

Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051
[20]

Jibin Li, Weigou Rui, Yao Long, Bin He. Travelling wave solutions for higher-order wave equations of KDV type (III). Mathematical Biosciences & Engineering, 2006, 3 (1) : 125-135. doi: 10.3934/mbe.2006.3.125

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (98)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences