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).  doi: 10.1142/S0219887814500388.  Google Scholar

[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.  doi: 10.1142/S0219887811005427.  Google Scholar

[3]

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

[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.  doi: 10.1093/imamci/12.4.399.  Google Scholar

[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.   Google Scholar

[6]

J. F. Cariñena and E. Martínez, Lie algebroid generalization of geometric mechanics,, in Lie Algebroids and related topics in differential geometry, (2001), 201.   Google Scholar

[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.  doi: 10.1088/1751-8113/40/33/008.  Google Scholar

[8]

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

[9]

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

[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, (2011).   Google Scholar

[11]

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

[12]

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

[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.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[14]

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

[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.  doi: 10.1017/S0305004100064501.  Google Scholar

[16]

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

[17]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Stu., (1985).   Google Scholar

[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.  doi: 10.1007/s00574-011-0030-7.  Google Scholar

[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.  doi: 10.1007/s00220-011-1313-y.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  doi: 10.1007/s10883-010-9080-1.  Google Scholar

[29]

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

[30]

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

[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), 209.   Google Scholar

[32]

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

[33]

E. Martínez, Lie algebroids in classical mechanics and optimal control,, SIGMA, 3 (2007).  doi: 10.3842/SIGMA.2007.050.  Google Scholar

[34]

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

[35]

E. Martínez, The momentum equation,, in Groups, (2006), 187.   Google Scholar

[36]

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

[37]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356.  doi: 10.1051/cocv:2007056.  Google Scholar

[38]

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

[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, (1964).   Google Scholar

[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).  doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[41]

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

[42]

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

[43]

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

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).  doi: 10.1142/S0219887814500388.  Google Scholar

[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.  doi: 10.1142/S0219887811005427.  Google Scholar

[3]

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

[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.  doi: 10.1093/imamci/12.4.399.  Google Scholar

[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.   Google Scholar

[6]

J. F. Cariñena and E. Martínez, Lie algebroid generalization of geometric mechanics,, in Lie Algebroids and related topics in differential geometry, (2001), 201.   Google Scholar

[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.  doi: 10.1088/1751-8113/40/33/008.  Google Scholar

[8]

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

[9]

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

[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, (2011).   Google Scholar

[11]

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

[12]

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

[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.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[14]

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

[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.  doi: 10.1017/S0305004100064501.  Google Scholar

[16]

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

[17]

M. de León and P. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Stu., (1985).   Google Scholar

[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.  doi: 10.1007/s00574-011-0030-7.  Google Scholar

[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.  doi: 10.1007/s00220-011-1313-y.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  doi: 10.1007/s10883-010-9080-1.  Google Scholar

[29]

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

[30]

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

[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), 209.   Google Scholar

[32]

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

[33]

E. Martínez, Lie algebroids in classical mechanics and optimal control,, SIGMA, 3 (2007).  doi: 10.3842/SIGMA.2007.050.  Google Scholar

[34]

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

[35]

E. Martínez, The momentum equation,, in Groups, (2006), 187.   Google Scholar

[36]

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

[37]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 (2008), 356.  doi: 10.1051/cocv:2007056.  Google Scholar

[38]

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

[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, (1964).   Google Scholar

[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).  doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[41]

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

[42]

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

[43]

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

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