Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

  • Lígia Abrunheiro
  • Leonardo Colombo


The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems, and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew’s triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de León, J.C. Marrero and E. Martínez on “Lagrangian submanifolds and dynamics on Lie algebroids”.


Mechanics on Lie algebroids Higher order mechanics Lagrangian mechanics Lagrangian submanifolds 

Mathematics Subject Classification

Primary 53D12 70H50 53D17 Secondary 70H03 37J15 53D05 


  1. 1.
    Abrunheiro, L., Camarinha, M., Carinena, J., Clemente-Gallardo, J., Martínez, E., Santos, P.: Some applications of quasi-velocities in optimal control. Int. J. Geometr. Methods Mod. Phys. 8, 835–851 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bruce, A.J., Grabowska, K., Grabowski, J.: Higher order mechanics on graded bundles. J. Phys. A 48, 205203 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruce, A.J., Grabowska, K., Grabowski, J., Urbanski, P.: New developments in geometric mechanics. In: Conference proceedings “Geometry of Jets and Fields” (Bedlewo, 10–16 May, 2015), Banach Center Publication, vol. 110, pp. 57–72, 2016. Polish Academy of Sciences, Warsaw (2016)Google Scholar
  4. 4.
    Campos, C.M., Guzmán, E., Marrero, J.C.: Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. J. Geom. Mech. 4(1), 1–26 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cariñena, J., Nunes da Costa, J., Santos, P.: Quasi-coordinates from the point of view of Lie algebroid. J. Phys. A 40, 10031 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colombo, L.: Second-order constrained variational problems on Lie algebroids: applications to optimal control. J. Geom. Mech 9(1), 1–45 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colombo, L., Martín de Diego, D., Zuccalli, M.: Optimal control of underactuated mechanical systems: a geometric approach. J. Math. Phys. 51, 083519 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colombo, L., Martín de Diego, D.: Higher-order variational problems on Lie groups and optimal control applications. J. Geom. Mech 6(4), 451–478 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Colombo, L.: Geometric and numerical methods for optimal control of mechanical systems. PhD thesis, Instituto de Ciencias Matemáticas, ICMAT (CSIC-UAM-UCM-UC3M) (2014)Google Scholar
  10. 10.
    Esen, O., Gumral, H.: Tulczyjew’s triplet for Lie groups I: Trivializations and reduction. J. Lie Theory 24(4), 1115–1160 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Esen, O., Gumral, H.: Tulczyjew’s triplet for Lie groups II: Dynamics. J. Lie Theory 27(2), 329–356 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    García-Toraño Andrés, E., Guzmán, E., Marrero, J.C., Mestdag, T.: Reduced dynamics and Lagrangian submanifolds of symplectic manifolds. J. Phys. A 47, 225203 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grabowska, K.: The Tulczyjew triple for classical fields. J. Phys. A 45, 145207–145242 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grabowska, K., Grabowski, J.: Dirac algebroids in Lagrangian and Hamiltonian mechanics. J. Geom. Phys. 61(11), 2233–2253 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grabowski, J., Urbański, P.: Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids. Ann. Glob. Anal. Geom. 15, 447–486 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iglesias, D., Marrero, J.C., Martín de Diego, D., Sosa, D.: Singular Lagrangian systems and variational constrained mechanics on Lie algebroids. Dyn. Syst. 23, 351–397 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jozwikowski, M.: Prolongations vs. Tulczyjew triples in Geometric Mechanics. arXiv:1712.09858 (preprint)
  18. 18.
    Jozwikowski, M., Rotkiewicz, M.: Bundle-theoretic methods for higher-order variational calculus. J. Geom. Mech. 6, 99–120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jozwikowski, M., Rotkiewicz, M.: Models for higher algebroids. J. Geom. Mech. 7, 317–359 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    de León, M., Lacomba, E.: Lagrangian submanifolds and higher-order mechanical systems. J. Phys. A. 22, 3809–3820 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    de León, M., Marrero, J.C., Martínez, E.: Lagrangian submanifolds and dynamics on Lie algebroids. J. Phys. A. 38, R241–R308 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    de León, M., Marrero, J.C., Martín de Diego, D.: Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. J. Geom. Mech. 2, 159–198 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Machado, L., Silva-Leite, F., Krakowski, K.: Higher-order smoothing splines versus least squares problems on Riemannian manifolds. J. Dyn. Control Syst. 16, 121–148 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mackenzie, K.: General Theory of Lie Groupoids and Lie Algebroids London Mathematical Society Lecture Notes, vol. 213. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  25. 25.
    Martín de Diego, D., Sato Martín de Almagro, R.: Variational order for forced Lagrangian systems. arXiv preprint arXiv:1712.09377
  26. 26.
    Marrero, J.C.: Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. J. Geom. Mech. 2, 243–263 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Marmo, G., Mendella, G., Tulczyjew, W.: Constrained Hamiltonian systems as implicit differential equations. J. Phys. A 30(1), 277293 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Martínez, E.: Lagrangian Mechanics on Lie algebroids. Acta Appl. Math. 67, 295–320 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Martínez, E.: Higher-order variational calculus on Lie algebroids. J. Geom. Mech. 7, 81–108 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Meier, D.: Invariant higher-order variational problems: reduction, geometry and applications. PhD thesis, Imperial College London (2013)Google Scholar
  31. 31.
    Sniatycki, J., Tulczyjew, W.M.: Generating forms of Lagrangian submanifolds. Indiana Univ. Math. J. 22(3), 267 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tulczyjew, W.M.: Les sous-variétés lagrangiennes et la dynamique hamiltonienne. C. R. Acad. Sci. Paris A 283, 15–18 (1976)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Tulczyjew, W.M.: Les sous-variétés lagrangiennes et la dynamique lagrangienne. C. R. Acad. Sci. Paris A-B 283(8), A675–A678 (1976)zbMATHGoogle Scholar
  34. 34.
    Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329–346 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Weinstein, A.: Lagrangian mechanics and groupoids. Fields Inst. Commun. 7, 207–231 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zajac, M., Grabowska, K.: The Tulczyjew triple in mechanics on a Lie group. J. Geom. Mech. 8(4), 413–435 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Research and Development in Mathematics and ApplicationsISCA, Universidade de AveiroAveiroPortugal
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain

Personalised recommendations