As many of you, I studied Lagrangian Mechanics and Hamiltonian Mechanics, with the so famous functions called Lagrangian $\mathcal{L}$ and Hamiltonian $\mathcal{H}$ related by:
$$\mathcal{H}(q_i, p_i, t) = \sum_i p_i\ \dot{q}_i - \mathcal{L}(q, \dot{q}, t)$$
Recently I found also another strange function, called the Routh function, or Routhian $\mathcal{R}$ which I didn't get at all. It's considered to be a sort of hybrid formulation between Lagrangian and Hamiltonian mechanics, and I also have read the Wikipedia page about it (cfr. https://en.wikipedia.org/wiki/Routhian_mechanics) but I didn't really get its whole meaning.
May anyone explain its use and meaning with a small example? Also I was wondering why it's not that used. For example in Quantum Field Theory we are always almost satisfied with Lagrangians and rarely with Hamiltonians, for what I studied until now. What about a Routhian formulation of field theory? Would it be a redundant useless question?
Thank you for every hint and your time.
