So, in my mechanics class, the teacher mentioned there is a special function which is kind of a midpoint between the Lagrangian and the Hamiltonian, called the Routhian. Now, I wanted to give it a look, but I can't find any trace of it in Google, and my mathematical physics teacher never heard of it (the reason I didn't ask my mechanics teacher being that he would probably give me an informal view, whereas I would like to see the mathematical formalism too). Has anyone ever heard of this function, and know where I find some reference of it?
This approach to classical mechanics is named after Edward Routh (with ou rather than just u). In this approach the Legendre transform is only carried out for the cyclic coordinates. Goldstein covers it in section 8.3 (Routh's procedure).