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We consider trivializations of second iterated bundles of a Lie group thatpreserve lifted group structures. With such a trivialization, we elaborateHamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and,both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space whichis tangent of cotangent bundle of Lie group. We present all possible Poisson,symplectic and Lagrangian reductions of spaces and corresponding dynamics onthem. In particular, reduction of Lagrangian dynamics on second iteratedtangent bundle includes reduction of dynamics on second order tangent bundle.

Reductions of Dynamics on Second Iterated Bundles of Lie Groups