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Journal of Lie Theory 27 (2017), No. 2, 329--356
Copyright Heldermann Verlag 2017

Tulczyjew's Triplet for Lie Groups. II: Dynamics

Ogul Esen
Dept. of Mathematics, Gebze Technical University, 41400 Gebze -- Kocaeli, Turkey

Hasan Gümral
Dept. of Mathematics, Australian College, 13015 Safat, Kuwait

Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplectic space. Tulczyjew's generalized Legendre transformations for trivialized and reduced dynamics are constructed.

Keywords: Trivialized Euler-Lagrange equations, trivialized Hamilton's equations, Euler-Poincaré equations, Lie-Poisson equations, Morse families, Tulczyjew's triplet, Legendre transformation, Lagrangian submanifold, diffeomorphisms group.

MSC: 22E65, 22E60, 22E70, 37E65, 70K65, 70H03, 70H05

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