Variational Calculus on Lie Groups

  • Gregory S. Chirikjian
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The calculus of variations is concerned with finding extremal paths of functionals in analogy with the way that classical calculus seeks to find critical points of functions. Variational calculus plays a central role in classical mechanics, connecting the “Principle of Least Action” and Lagrange’s equations of motion (also called the Euler– Lagrange equations). In that setting, generalized coordinates are introduced to describe the geometric configuration of a mechanical system. In this chapter, classical variational calculus is reviewed and extended to describe systems on Lie groups. Of course, the introduction of coordinates such as Euler angles to describe the orientation of a rigid body can be used to formulate classical variational problems at the expense of introducing singularities. However, it is possible to formulate variational problems on Lie groups without coordinates. This results in the Euler’Poincaré equations.


Variational Problem Lagrange Equation Variational Calculus Frenet Frame Minimal Energy Conformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, R., Marsden, J.E., Foundations of Mechanics, Benjamin/Cummings, San Mateo, CA, 1978.Google Scholar
  2. 2.
    Amari, S., “Natural gradient works efficiently in learning,” Neural Comput., 10(2), pp. 251– 276, 1998.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balaeff, A., Mahadevan, L., Schulten, K., “Modeling DNA loops using the theory of elasticity,” E-print archive (, 2003).Google Scholar
  4. 4.
    Benham, C.J., Mielke, S.P.,“DNA mechanics,” Ann. Rev. Biomed. Engin., 7, pp. 21–53, 2005.CrossRefGoogle Scholar
  5. 5.
    Bloch, A.M., Crouch, P.E., Sanyal, A.K., “A variational problem on Stiefel manifolds,” Nonlinearity, 19(10), pp. 2247–2276, 2006.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brechtken-Manderscheid, U., Introduction to the Calculus of Variations, Chapman & Hall, New York, 1991.Google Scholar
  7. 7.
    Bloch, A., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S., “The Euler–Poincar´e equations and double bracket dissipation,” Commun. Math. Phys., 175, p. 1, 1996.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bluman, G.W., Anco, S.C., Symmetry and Integration Methods for Differential Equations, Applied Mathematical Sciences, Vol. 154, Springer, New York, 2002.Google Scholar
  9. 9.
    Boscain, U., Rossi, F., “Invariant Carnot–Carath´eodory metrics on S3, SO(3), SL(2) and Lens Spaces,” SIAM J. Control Optimiz., 47(4), pp. 1851–1878, 2009.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Calin, O., Chang, D.-C., Sub-Riemannian Geometry: General Theory and Examples, Cambridge University Press, Cambridge, 2009.Google Scholar
  11. 11.
    Carath´eodory, C., “Investigations into the foundations of thermodynamics,” Math. Ann. 67, pp. 355–386, 1909.Google Scholar
  12. 12.
    Chirikjian, G.S., “The stochastic elastica and excluded-volume perturbations of DNA conformational ensembles,” Int. J. Non-Linear Mech., 43(10), pp. 1108–1120, 2008.MATHCrossRefGoogle Scholar
  13. 13.
    Chirikjian, G.S., Theory and Applications of Hyper-Redundant Robotic Manipulators, Division of Engineering and Applied Science, California Institute of Technology, June 1992. Available at Chirikjian gs 1992.pdfGoogle Scholar
  14. 14.
    Chirikjian, G.S., Burdick, J.W., “Kinematically optimal hyper-redundant manipulator configurations,”IEEE Trans. Robot. Autom., 11, p. 794, 1995.CrossRefGoogle Scholar
  15. 15.
    Chow, W.L., “Systeme von linearen partiellen differential Gleichungen erster Ordnung,” Math. Ann., 117, pp. 98–105, 1939.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Coleman, B.D., Olson, W.K., Swigon, D., “Theory of sequence-dependent DNA elasticity,” J. Chem. Phys., 118, pp. 7127–7140, 2003.CrossRefGoogle Scholar
  17. 17.
    Crouch, P.E., Grossman, R., “Numerical integration of ordinary differential equations on manifolds,” J. Nonlinear Sci., 3, pp. 1–33, 1993.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Dieci, L., Russell, R.D., van Vleck, E.S., “Unitary integrators and applications to continuous orthogonalization techniques,” SIAM J. Num. Anal., 31, pp. 261–281, 1994.MATHCrossRefGoogle Scholar
  19. 19.
    Dungey, N., ter Elst, A.F.M., Robinson, D.W., Analysis on Lie Groups with Polynomial Growth, Birkh¨auser, Boston, 2003.Google Scholar
  20. 20.
    Edelman, A., Arias, T.A., Smith, S.T., “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl., 20, pp. 303–353, 1998.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ewing, G.M., Calculus of Variations with Applications, W.W. Norton and Co., New York, 1969.Google Scholar
  22. 22.
    Fiori, S., “Formulation and integration of learning differential equations on the Stiefel manifold,” IEEE Trans. Neural Networks 16(6), pp. 1697–1701, 2005.CrossRefGoogle Scholar
  23. 23.
    Forest, E., “Sixth-order Lie group integrators,” J. Comput. Phys. 99, pp. 209–213, 1992.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ge, Z., Marsden, J.E., “Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,” Phys. Lett. A, 133, pp. 134–139, 1988.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gonzalez, O., Maddocks, J.H., “Extracting parameters for base-pair level models of DNA from molecular dynamics simulations,” Theor. Chem. Acc., 106(1–2), pp. 76–82, 2001.Google Scholar
  26. 26.
    Goyal, S., Perkins, N.C., Lee, C.L., “Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables,” J. Comp. Phys., 209, pp. 371– 389, 2005.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Gromov, M., “Groups of polynomial growth and expanding maps,” Inst. Hautes ´ Etudes Sci. Publ. Math., 53(1), pp. 53–78, 1981.Google Scholar
  28. 28.
    Gruver, W.A., Sachs, E., Algorithmic Methods in Optimal Control, Pitman Publishing, Boston, 1980.Google Scholar
  29. 29.
    Hairer, E., Lubich, C., Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, New York, 2006.Google Scholar
  30. 30.
    Hermann, R., Geometry, Physics, and Systems, Marcel Dekker, New York, 1973.Google Scholar
  31. 31.
    Holm, D.D., Marsden, J.E., Ratiu, T.S., “The Euler–Poincar´e equations and semidirect products with applications to continuum theories,” Adv. Math., 137 p. 1, 1998.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    H¨ormander, L., “Hypoelliptic second-order differential equations,” Acta Math., 119, pp. 147–171, 1967.Google Scholar
  33. 33.
    Iserles, A., Munthe-Kaas, H.Z., Norsett, S.P., Zanna, A., “Lie group methods,” Acta Numerica, 9, pp. 215–365, 2000.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Junkins, J.L., Optimal Spacecraft Rotational Maneuvers, Studies in Astronautics Vol. 3, Elsevier, Amsterdam, 1986.Google Scholar
  35. 35.
    Kamien, M.I., Schwartz, N.L., Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, North-Holland, New York, 1991.Google Scholar
  36. 36.
    Kim, J.-S., Chirikjian, G.S., “Conformational analysis of stiff chiral polymers with end constraints,” Mol. Simul., 32(14), pp. 1139–1154, 2006.MATHCrossRefGoogle Scholar
  37. 37.
    Koh, S., Chirikjian, G.S., Ananthasuresh, G.K., “A Jacobian-based algorithm for planning attitude maneuvers using forward and reverse rotations,” ASME J. Comput. Nonlinear Dynam., 4(1), pp. 1–12, 2009.Google Scholar
  38. 38.
    Le Donne, E., Lecture Notes on sub-Riemannian Geometry. Available at Scholar
  39. 39.
    Marko, J.F., Siggia, E.D., “Bending and twisting elasticity of DNA,” Macromolecules, 27, pp. 981–988, 1994CrossRefGoogle Scholar
  40. 40.
    Marsden, J.E., Pekarsky, S., Shkoller, S., “Discrete Euler–Poincar´e and Lie–Poisson equations,” Nonlinearity, 12, pp. 1647–1662, 1999.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Marsden, J.E.,West, M., “Discrete mechanics and variational integrators,” Acta Numerica, 10, pp. 357–514, 2001.MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    McLachlan, R.I., “Explicit Lie–Poisson integration and the Euler equations,” Phys. Rev. Lett., 71, pp. 3043–3046, 1993.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    McLachlan, R.I., Zanna, A., “The discrete Moser–Veselov algorithm for the free rigid body, revisited,” Found. Comput. Math., 5, pp. 87–123, 2005.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Mitchell, J., “On Carnot–Carath´eodory metrics,” J. Diff. Geom., 21, pp. 35–45, 1985.MATHGoogle Scholar
  45. 45.
    Montgomery, R., A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, Math Surveys and Monographs Vol. 91, American Mathematical Society, Providence, RI, 2002.Google Scholar
  46. 46.
    Neuenschwander, D.E., Emmy Noether’s Wonderful Theorem, Johns Hopkins University Press, Baltimore, 2010.Google Scholar
  47. 47.
    Oja, E., “Neural networks, principal components, and subspaces,” Int. J. Neural Syst., 1, pp. 61–68, 1989.MathSciNetCrossRefGoogle Scholar
  48. 48.
    Poincar´e, H. “Sur une forme nouvelle des equations de la mechanique,” Cr. Hebd. Acad. Sci., 132, p. 369, 1901.Google Scholar
  49. 49.
    Rucker, C.,Webster, R.J., III, Chirikjian, G.S., Cowan, N.J., “Equilibrium conformations of concentric-tube continuum robots,” Int. J. Robot. Res., 29(10), pp. 1263–1280, 2010.CrossRefGoogle Scholar
  50. 50.
    Stephani, H., Differential Equations: Their Solution Using Symmetries, M. Maccallum, ed., Cambridge University Press, Cambridge, 1989.Google Scholar
  51. 51.
    Varopoulos, N. Th., “Sobolev inequalities on Lie groups and symmetric spaces,” J. Funct. Anal., 86, pp. 19–40, 1989.MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Varopoulos, N. Th., Saloff-Coste, L., Coulhon, T., Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.Google Scholar
  53. 53.
    Wiggins, P.A., Phillips, R., Nelson, P.C., “Exact theory of kinkable elastic polymers,” E-print archive (arXiv:cond-mat/0409003 v1. 31 Aug. 2004.)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations