Complex sentence: The difference between the Lagrangian, Hamiltonian, and Routhian are the variables they are functions of-for a given set of generalized coordinates, the split is determined by which coordinates are to have corresponding generalized velocities, or corresponding generalized momenta.
Split into simpler sentences: The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables. For a given set of generalized coordinates, the split is determined by which coordinates are to have corresponding generalized velocities, or corresponding generalized momenta.