Reductions of Dynamics on Second Iterated Bundles of Lie Groups

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