March  2015, 7(1): 81-108. doi: 10.3934/jgm.2015.7.81

Higher-order variational calculus on Lie algebroids

1. 

IUMA and Department of Applied Mathematics, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain

Received  August 2014 Revised  January 2015 Published  March 2015

The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincaré and Lagrange Poincaré type equations is studied. Reduction and reconstruction results for such systems are established.
Citation: 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
References:
[1]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems from the point of view of Lie algebroids, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1450038, 8pp. doi: 10.1142/S0219887814500388.

[2]

L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control, International Journal of Geometric Methods in Modern Physics, 8 (2011), 835-851. doi: 10.1142/S0219887811005427.

[3]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles,, preprint, (). 

[4]

M. Camarinha, F. Silva-Leite and P. Crouch, Splines of class $C^k$ on non-Euclidean spaces, IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.

[5]

J. F. Cariñena, C. López and E. Martínez, Sections along a map applied to higher-order Lagrangian mechanics. Noether's theorem, Acta Applicandae Mathematicae, 25 (1991), 127-151.

[6]

J. F. Cariñena and E. Martínez, Lie algebroid generalization of geometric mechanics, in Lie Algebroids and related topics in differential geometry, Banach Center Publications, 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 201-215.

[7]

J. F. Cariñena, J. M. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid structures, Journal of Physics A: Mathematical and Theoretical, 40 (2007), 10031-10048. doi: 10.1088/1751-8113/40/33/008.

[8]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp. doi: 10.1090/memo/0722.

[9]

L. Colombo, Lagrange-Poincaré reduction for optimal control of underactuated mechanical systems,, preprint, (). 

[10]

L. Colombo and D. Martín de Diego, A variational and geometric approach for the second-order Euler-Poincaré equations, Notes of the talk delivered at XIII Encuentro de Invierno, Zaragoza, (2011). Available from: http://andres.unizar.es/ ei/2011/Contribuciones/LeoColombo.pdf.

[11]

L. Colombo and D. Martín de Diego, On the geometry of higher-order variational problems on Lie groups,, preprint, (). 

[12]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.

[13]

J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Continuous Dynamical Systems, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.

[14]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.

[15]

M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian mechanics, Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.

[16]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01.

[17]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Math. Stu., 112, Amsterdam, 1985.

[18]

F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, J. Braz. Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.

[19]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. X. Vialard, Invariant higher-order variational problems, Comm. Math. Phys., 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[20]

F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics, J. Geometric Mechanics, 3 (2011), 41-79. doi: 10.3934/jgm.2011.3.41.

[21]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[22]

K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[23]

K. Grabowska, J. Grabowski and P. Urbański, Lie brackets on affine bundles, Ann. Global Anal. Geom., 24 (2003), 101-130. doi: 10.1023/A:1024457728027.

[24]

J. Grabowski and M. Jóźwikowski, Pontryagin maximum principle on almost Lie algebroids, SIAM J. Control Optim., 49 (2011), 1306-1357. doi: 10.1137/090760246.

[25]

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

[26]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120.

[27]

K. J. Kyriakopoulos and G. N. Saridis, Minimum jerk path generation, in Proceedings of the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, 1988, 364-369. doi: 10.1109/ROBOT.1988.12075.

[28]

L. Machado, F. Silva-Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. and Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.

[29]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.

[30]

E. Martínez, Lagrangian Mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[31]

E. Martínez, Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics (Medina del Campo, 1999), Publicaciones de la RSME, 2, R. Soc. Mat. Esp., Madrid, 2001, 209-222.

[32]

E. Martínez, Reduction in optimal control theory, Rep. Math. Phys., 53 (2004), 79-90. doi: 10.1016/S0034-4877(04)90005-5.

[33]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA, 3 (2007), Paper 050, 17pp. doi: 10.3842/SIGMA.2007.050.

[34]

E. Martínez, Classical field theory on Lie algebroids: Variational aspects, J. Phys. A: Mat. Gen., 38 (2005), 7145-7160. doi: 10.1088/0305-4470/38/32/005.

[35]

E. Martínez, The momentum equation, in Groups, Geometry and Physics, Monografías de la Real Academia de Ciencias de Zaragoza, 29, Acad. Cienc. Exact. Fs. Qum. Nat. Zaragoza, Zaragoza, 2006, 187-196.

[36]

E. Martínez, T. Mestdag and W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44 (2002), 70-95. doi: 10.1016/S0393-0440(02)00114-6.

[37]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[38]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control & Information, 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.

[39]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1964.

[40]

P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A: Math. Theor., 44 (2011), 385203, 35pp. doi: 10.1088/1751-8113/44/38/385203.

[41]

W. Tulczyjew, The Lagrange differential, Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.

[42]

A. Weinstein, Lagrangian Mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.

[43]

H. Yoshimura and J. E. Marsden, Reduction of Dirac structures and the Hamilton-Pontryagin principle, Rep. Math. Phys., 60 (2007), 381-426. doi: 10.1016/S0034-4877(08)00004-9.

show all references

References:
[1]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems from the point of view of Lie algebroids, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1450038, 8pp. doi: 10.1142/S0219887814500388.

[2]

L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control, International Journal of Geometric Methods in Modern Physics, 8 (2011), 835-851. doi: 10.1142/S0219887811005427.

[3]

A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles,, preprint, (). 

[4]

M. Camarinha, F. Silva-Leite and P. Crouch, Splines of class $C^k$ on non-Euclidean spaces, IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.

[5]

J. F. Cariñena, C. López and E. Martínez, Sections along a map applied to higher-order Lagrangian mechanics. Noether's theorem, Acta Applicandae Mathematicae, 25 (1991), 127-151.

[6]

J. F. Cariñena and E. Martínez, Lie algebroid generalization of geometric mechanics, in Lie Algebroids and related topics in differential geometry, Banach Center Publications, 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 201-215.

[7]

J. F. Cariñena, J. M. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid structures, Journal of Physics A: Mathematical and Theoretical, 40 (2007), 10031-10048. doi: 10.1088/1751-8113/40/33/008.

[8]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp. doi: 10.1090/memo/0722.

[9]

L. Colombo, Lagrange-Poincaré reduction for optimal control of underactuated mechanical systems,, preprint, (). 

[10]

L. Colombo and D. Martín de Diego, A variational and geometric approach for the second-order Euler-Poincaré equations, Notes of the talk delivered at XIII Encuentro de Invierno, Zaragoza, (2011). Available from: http://andres.unizar.es/ ei/2011/Contribuciones/LeoColombo.pdf.

[11]

L. Colombo and D. Martín de Diego, On the geometry of higher-order variational problems on Lie groups,, preprint, (). 

[12]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.

[13]

J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Continuous Dynamical Systems, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.

[14]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.

[15]

M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian mechanics, Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.

[16]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01.

[17]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Math. Stu., 112, Amsterdam, 1985.

[18]

F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, J. Braz. Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.

[19]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. X. Vialard, Invariant higher-order variational problems, Comm. Math. Phys., 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.

[20]

F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics, J. Geometric Mechanics, 3 (2011), 41-79. doi: 10.3934/jgm.2011.3.41.

[21]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[22]

K. Grabowska, J. Grabowski and P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[23]

K. Grabowska, J. Grabowski and P. Urbański, Lie brackets on affine bundles, Ann. Global Anal. Geom., 24 (2003), 101-130. doi: 10.1023/A:1024457728027.

[24]

J. Grabowski and M. Jóźwikowski, Pontryagin maximum principle on almost Lie algebroids, SIAM J. Control Optim., 49 (2011), 1306-1357. doi: 10.1137/090760246.

[25]

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

[26]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120.

[27]

K. J. Kyriakopoulos and G. N. Saridis, Minimum jerk path generation, in Proceedings of the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, 1988, 364-369. doi: 10.1109/ROBOT.1988.12075.

[28]

L. Machado, F. Silva-Leite and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. and Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.

[29]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.

[30]

E. Martínez, Lagrangian Mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[31]

E. Martínez, Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics (Medina del Campo, 1999), Publicaciones de la RSME, 2, R. Soc. Mat. Esp., Madrid, 2001, 209-222.

[32]

E. Martínez, Reduction in optimal control theory, Rep. Math. Phys., 53 (2004), 79-90. doi: 10.1016/S0034-4877(04)90005-5.

[33]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA, 3 (2007), Paper 050, 17pp. doi: 10.3842/SIGMA.2007.050.

[34]

E. Martínez, Classical field theory on Lie algebroids: Variational aspects, J. Phys. A: Mat. Gen., 38 (2005), 7145-7160. doi: 10.1088/0305-4470/38/32/005.

[35]

E. Martínez, The momentum equation, in Groups, Geometry and Physics, Monografías de la Real Academia de Ciencias de Zaragoza, 29, Acad. Cienc. Exact. Fs. Qum. Nat. Zaragoza, Zaragoza, 2006, 187-196.

[36]

E. Martínez, T. Mestdag and W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44 (2002), 70-95. doi: 10.1016/S0393-0440(02)00114-6.

[37]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[38]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control & Information, 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.

[39]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1964.

[40]

P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A: Math. Theor., 44 (2011), 385203, 35pp. doi: 10.1088/1751-8113/44/38/385203.

[41]

W. Tulczyjew, The Lagrange differential, Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.

[42]

A. Weinstein, Lagrangian Mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.

[43]

H. Yoshimura and J. E. Marsden, Reduction of Dirac structures and the Hamilton-Pontryagin principle, Rep. Math. Phys., 60 (2007), 381-426. doi: 10.1016/S0034-4877(08)00004-9.

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