Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 109, 31 pages      arXiv:1106.2950      http://dx.doi.org/10.3842/SIGMA.2011.109

Routh Reduction by Stages

Bavo Langerock a, b, c, Tom Mestdag a and Joris Vankerschaver a, d
a) Department of Mathematics, Ghent University, Krijgslaan 281, S22, B9000 Ghent, Belgium
b) Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium
c) Department of Mathematics, K.U. Leuven, Celestijnenlaan 200 B, B3001 Leuven, Belgium
d) Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, San Diego CA 92093-0112, USA

Received June 16, 2011, in final form November 22, 2011; Published online November 29, 2011

Abstract
This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group.

Key words: symplectic reduction; Routh reduction; Lagrangian reduction; reduction by stages.

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References

  1. Abraham R., Marsden J.E., Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
  2. Adamec L., A route to Routh - the classical setting, J. Nonlinear Math. Phys. 18 (2011), 87-107.
  3. Arnold V.I., Kozlov V.V., Neishtadt A.I., Mathematical aspects of classical and celestial mechanics, Springer-Verlag, Berlin, 1997.
  4. Cendra H., Marsden J.E., Ratiu T.S., Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001), no. 722.
  5. Cortés J., de León M., Marrero J.C., Martín de Diego D., Martínez E., A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys. 3 (2006), 509-558, math-ph/0511009.
  6. Crampin M., Mestdag T., Routh's procedure for non-Abelian symmetry groups, J. Math. Phys. 49 (2008), 032901, 28 pages, arXiv:0802.0528.
  7. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Román-Roy N., Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys. 11 (1999), 1209-1247, math-ph/9911008.
  8. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Interscience Publishers, New York - London, 1963.
    Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, Interscience Publishers, New York - London, 1969.
  9. Lamb H., Hydrodynamics, Reprint of the 1932 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993.
  10. Langerock B., Cantrijn F., Vankerschaver J., Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys. 51 (2010), 022902, 20 pages, arXiv:0912.0863.
  11. Langerock B., Castrillón Lopéz M., Routh reduction for singular Lagrangians, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1451-1489, arXiv:1007.0325.
  12. Marsden J.E., Misioek G., Ortega J.P., Perlmutter M., Ratiu T.S., Hamiltonian reduction by stages, Lecture Notes in Mathematics, Vol. 1913, Springer, Berlin, 2007.
  13. Marsden J.E., Montgomery R., Ratiu T.S., Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc. 88 (1990), no. 436.
  14. Marsden J.E., Ratiu T.S., Scheurle J., Reduction theory and the Lagrange-Routh equations, J. Math. Phys. 41 (2000), 3379-3429.
  15. Mestdag T., Crampin M., Invariant Lagrangians, mechanical connections and the Lagrange-Poincaré equations, J. Phys. A: Math. Theor. 41 (2008), 344015, 20 pages, arXiv:0802.0146.
  16. Mikityuk I.V., Stepin A.M., A sufficient condition for stepwise reduction: proof and applications, Sb. Math. 199 (2008), 663-671.
  17. Milne-Thomson L., Theoretical hydrodynamics, 5th ed., MacMillan, London, 1968.
  18. Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, RI, 2002.
  19. Pars L.A., A treatise on analytical dynamics, Heinemann Educational Books Ltd., London, 1965.
  20. Vankerschaver J., Kanso E., Marsden J.E., The dynamics of a rigid body in potential flow with circulation, Regul. Chaotic Dyn. 15 (2010), 606-629, arXiv:1003.0080.

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