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Keywords:
Frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction
Frame bundle; Euler-Lagrange equations; invariant Lagrangian; Euler-Poincaré reduction
Summary:
Let $\mu \colon FX \to X$ be a principal bundle of frames with the structure group ${\rm Gl}_{n}(\mathbb R)$. It is shown that the variational problem, defined by ${\rm Gl}_{n}(\mathbb R)$-invariant Lagrangian on $J^{r} FX$, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.
References:
[1] Brajerčík, J.: $\mathop Gl_{n}(\mathbb R)$-invariant variational principles on frame bundles. Balkan J. Geom. Appl. 13 (2008), 11-19. MR 2395370
[2] Brajerčík, J., Krupka, D.: Variational principles for locally variational forms. J. Math. Phys. 46 (2005), 1-15. DOI 10.1063/1.1901323 | MR 2143003
[3] López, M. Castrillón, Pérez, P. L. García, Ratiu, T. S.: Euler-Poincaré reduction on principal bundles. Lett. Math. Phys. 58 (2001), 167-180. DOI 10.1023/A:1013303320765 | MR 1876252
[4] López, M. Castrillón, Pérez, P. L. García, Rodrigo, C.: Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange. Differ. Geom. Appl. 25 (2007), 585-593. DOI 10.1016/j.difgeo.2007.06.007 | MR 2373936
[5] López, M. Castrillón, Ratiu, T. S., Shkoller, S.: Reduction in principal fiber bundles: Covariant Euler-Poincaré equations. Proc. Am. Math. Soc. 128 (2000), 2155-2164. DOI 10.1090/S0002-9939-99-05304-6 | MR 1662269
[6] Dieudonné, J.: Treatise on Analysis. Vol. I. Academic Press New York-London (1969).
[7] Pérez, P. L. García: Connections and 1-jet fiber bundles. Rend. Sem. Mat. Univ. Padova 47 (1972), 227-242. MR 0315624
[8] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific Singapore (1997). MR 2001723 | Zbl 0913.58001
[9] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol. 1. Interscience Publishers/John Wiley & Sons New York/London (1963). MR 0152974
[10] Kolář, I.: On the torsion of spaces with connection. Czech. Math. J. 21 (1971), 124-136. MR 0293531
[11] Krupka, D.: A geometric theory of ordinary first order variational problems in fibered manifolds. II. Invariance. J. Math. Anal. Appl. 49 (1975), 469-476. DOI 10.1016/0022-247X(75)90190-0 | MR 0362398 | Zbl 0312.58003
[12] Krupka, D.: Local invariants of a linear connection. In: Differential Geometry. Colloq. Math. Soc. János Bolyai, Budapest, 1979, Vol. 31 North Holland Amsterdam (1982), 349-369. MR 0706930 | Zbl 0513.53038
[13] Krupka, D., Janyška, J.: Lectures on Differential Invariants. Folia Fac. Sci. Nat. Univ. Purk. Brunensis, Mathematica 1. University J. E. Purkyně Brno (1990). MR 1108622
[14] Marsden, J. E., Ratiu, T. S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Texts Applied Mathematics, Vol. 17. Springer New York (1994). DOI 10.1007/978-1-4612-2682-6 | MR 1304682
[15] Saunders, D. J.: The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series Vol. 142. Cambridge University Press Cambridge (1989). MR 0989588
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