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]

Int. J. Geom. Methods Mod. Phys., 11 (2014), 1450038, 8pp. doi: 10.1142/S0219887814500388.  Google Scholar

[2]

International Journal of Geometric Methods in Modern Physics, 8 (2011), 835-851. 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]

IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.  Google Scholar

[5]

Acta Applicandae Mathematicae, 25 (1991), 127-151.  Google Scholar

[6]

in Lie Algebroids and related topics in differential geometry, Banach Center Publications, 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 201-215.  Google Scholar

[7]

Journal of Physics A: Mathematical and Theoretical, 40 (2007), 10031-10048. doi: 10.1088/1751-8113/40/33/008.  Google Scholar

[8]

Mem. Amer. Math. Soc., 152 (2001), x+108 pp. doi: 10.1090/memo/0722.  Google Scholar

[9]

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

[10]

Notes of the talk delivered at XIII Encuentro de Invierno, Zaragoza, (2011). Available from: http://andres.unizar.es/ ei/2011/Contribuciones/LeoColombo.pdf. 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]

J. Math. Phys., 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.  Google Scholar

[13]

Discrete and Continuous Dynamical Systems, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.  Google Scholar

[14]

Ann. of Math. (2), 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[15]

Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.  Google Scholar

[16]

J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[17]

North-Holland Math. Stu., 112, Amsterdam, 1985.  Google Scholar

[18]

J. Braz. Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.  Google Scholar

[19]

Comm. Math. Phys., 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.  Google Scholar

[20]

J. Geometric Mechanics, 3 (2011), 41-79. doi: 10.3934/jgm.2011.3.41.  Google Scholar

[21]

J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[22]

Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.  Google Scholar

[23]

Ann. Global Anal. Geom., 24 (2003), 101-130. doi: 10.1023/A:1024457728027.  Google Scholar

[24]

SIAM J. Control Optim., 49 (2011), 1306-1357. 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]

Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120.  Google Scholar

[27]

in Proceedings of the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, 1988, 364-369. doi: 10.1109/ROBOT.1988.12075.  Google Scholar

[28]

J. Dyn. and Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.  Google Scholar

[29]

Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[30]

Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.  Google Scholar

[31]

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.  Google Scholar

[32]

Rep. Math. Phys., 53 (2004), 79-90. doi: 10.1016/S0034-4877(04)90005-5.  Google Scholar

[33]

SIGMA, 3 (2007), Paper 050, 17pp. doi: 10.3842/SIGMA.2007.050.  Google Scholar

[34]

J. Phys. A: Mat. Gen., 38 (2005), 7145-7160. doi: 10.1088/0305-4470/38/32/005.  Google Scholar

[35]

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.  Google Scholar

[36]

J. Geom. Phys., 44 (2002), 70-95. doi: 10.1016/S0393-0440(02)00114-6.  Google Scholar

[37]

ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.  Google Scholar

[38]

IMA Journal of Mathematical Control & Information, 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.  Google Scholar

[39]

Interscience Publishers John Wiley & Sons, Inc., New York-London, 1964.  Google Scholar

[40]

J. Phys. A: Math. Theor., 44 (2011), 385203, 35pp. doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[41]

Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.  Google Scholar

[42]

Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

[43]

Rep. Math. Phys., 60 (2007), 381-426. doi: 10.1016/S0034-4877(08)00004-9.  Google Scholar

show all references

References:
[1]

Int. J. Geom. Methods Mod. Phys., 11 (2014), 1450038, 8pp. doi: 10.1142/S0219887814500388.  Google Scholar

[2]

International Journal of Geometric Methods in Modern Physics, 8 (2011), 835-851. 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]

IMA J. Math. Control Info., 12 (1995), 399-410. doi: 10.1093/imamci/12.4.399.  Google Scholar

[5]

Acta Applicandae Mathematicae, 25 (1991), 127-151.  Google Scholar

[6]

in Lie Algebroids and related topics in differential geometry, Banach Center Publications, 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 201-215.  Google Scholar

[7]

Journal of Physics A: Mathematical and Theoretical, 40 (2007), 10031-10048. doi: 10.1088/1751-8113/40/33/008.  Google Scholar

[8]

Mem. Amer. Math. Soc., 152 (2001), x+108 pp. doi: 10.1090/memo/0722.  Google Scholar

[9]

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

[10]

Notes of the talk delivered at XIII Encuentro de Invierno, Zaragoza, (2011). Available from: http://andres.unizar.es/ ei/2011/Contribuciones/LeoColombo.pdf. 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]

J. Math. Phys., 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.  Google Scholar

[13]

Discrete and Continuous Dynamical Systems, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.  Google Scholar

[14]

Ann. of Math. (2), 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[15]

Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.  Google Scholar

[16]

J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[17]

North-Holland Math. Stu., 112, Amsterdam, 1985.  Google Scholar

[18]

J. Braz. Math. Soc., 42 (2011), 579-606. doi: 10.1007/s00574-011-0030-7.  Google Scholar

[19]

Comm. Math. Phys., 309 (2012), 413-458. doi: 10.1007/s00220-011-1313-y.  Google Scholar

[20]

J. Geometric Mechanics, 3 (2011), 41-79. doi: 10.3934/jgm.2011.3.41.  Google Scholar

[21]

J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[22]

Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.  Google Scholar

[23]

Ann. Global Anal. Geom., 24 (2003), 101-130. doi: 10.1023/A:1024457728027.  Google Scholar

[24]

SIAM J. Control Optim., 49 (2011), 1306-1357. 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]

Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120.  Google Scholar

[27]

in Proceedings of the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, 1988, 364-369. doi: 10.1109/ROBOT.1988.12075.  Google Scholar

[28]

J. Dyn. and Control Syst., 16 (2010), 121-148. doi: 10.1007/s10883-010-9080-1.  Google Scholar

[29]

Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[30]

Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.  Google Scholar

[31]

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.  Google Scholar

[32]

Rep. Math. Phys., 53 (2004), 79-90. doi: 10.1016/S0034-4877(04)90005-5.  Google Scholar

[33]

SIGMA, 3 (2007), Paper 050, 17pp. doi: 10.3842/SIGMA.2007.050.  Google Scholar

[34]

J. Phys. A: Mat. Gen., 38 (2005), 7145-7160. doi: 10.1088/0305-4470/38/32/005.  Google Scholar

[35]

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.  Google Scholar

[36]

J. Geom. Phys., 44 (2002), 70-95. doi: 10.1016/S0393-0440(02)00114-6.  Google Scholar

[37]

ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.  Google Scholar

[38]

IMA Journal of Mathematical Control & Information, 6 (1989), 465-473. doi: 10.1093/imamci/6.4.465.  Google Scholar

[39]

Interscience Publishers John Wiley & Sons, Inc., New York-London, 1964.  Google Scholar

[40]

J. Phys. A: Math. Theor., 44 (2011), 385203, 35pp. doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[41]

Bull. Acad. Polon. Sci., 24 (1976), 1089-1096.  Google Scholar

[42]

Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

[43]

Rep. Math. Phys., 60 (2007), 381-426. doi: 10.1016/S0034-4877(08)00004-9.  Google Scholar

[1]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021062

[2]

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

[3]

Muhammad Aslam Noor, Khalida Inayat Noor. Properties of higher order preinvex functions. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 431-441. doi: 10.3934/naco.2020035

[4]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[5]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[6]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020

[7]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[8]

Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic & Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006

[9]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104

[10]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605

[11]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001

[12]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031

[13]

Zhenbing Gong, Yanping Chen, Wenyu Tao. Jump and variational inequalities for averaging operators with variable kernels. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021045

[14]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065

[15]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[16]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[17]

Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021050

[18]

Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021055

[19]

Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021083

[20]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]