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

SIGMA 11 (2015), 009, 10 pages      arXiv:1408.3019      https://doi.org/10.3842/SIGMA.2015.009

Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

Cornelia Vizman
Department of Mathematics, West University of Timişoara, Romania

Received August 14, 2014, in final form January 22, 2015; Published online January 29, 2015

Abstract
We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance properties of the associated Euler-Poincaré equations with advected parameters.

Key words: Lagrangian; homogeneous space; Euler-Poincaré equation.

pdf (321 kb)   tex (15 kb)

References

  1. Arnold V., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), 319-361.
  2. Cendra H., Holm D.D., Marsden J.E., Ratiu T.S., Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, in Geometry of Differential Equations, Amer. Math. Soc. Transl., Vol. 186, Amer. Math. Soc., Providence, RI, 1998, 1-25, chao-dyn/9906004.
  3. Gay-Balmaz F., Holm D.D., Ratiu T.S., Variational principles for spin systems and the Kirchhoff rod, J. Geom. Mech. 1 (2009), 417-444, arXiv:0904.1428.
  4. Gay-Balmaz F., Ratiu T.S., The geometric structure of complex fluids, Adv. in Appl. Math. 42 (2009), 176-275, arXiv:0903.4294.
  5. Gay-Balmaz F., Tronci C., Reduction theory for symmetry breaking with applications to nematic systems, Phys. D 239 (2010), 1929-1947, arXiv:0909.2165.
  6. Holm D.D., Marsden J.E., Ratiu T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), 1-81, chao-dyn/9801015.
  7. Khesin B., Lenells J., Misiołek G., Preston S.C., Geometry of diffeomorphism groups, complete integrability and geometric statistics, Geom. Funct. Anal. 23 (2013), 334-366, arXiv:1105.0643.
  8. Khesin B., Misiołek G., Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003), 116-144, math.SG/0210397.
  9. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in Applied Mathematics, Vol. 17, 2nd ed., Springer-Verlag, New York, 1999.
  10. Tiğlay F., Vizman C., Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys. 97 (2011), 45-60, arXiv:1008.4377.
  11. Vizman C., Geodesic equations on diffeomorphism groups, SIGMA 4 (2008), 030, 22 pages, arXiv:0803.1678.
  12. Vizman C., Invariant variational problems on homogeneous spaces, J. Geom. Phys., to appear.

Previous article  Next article   Contents of Volume 11 (2015)