Tulczyjew’s triplet for lie groups III: Higher order dynamics and reductions for iterated bundles

Given a Lie group 𝐺, we elaborate the dynamics on 𝑇*𝑇*𝐺 and 𝑇*𝑇𝐺, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space 𝑇𝑇*𝐺, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.