I have just read a mention about a variant of Legendre transformation that instead of producing the Hamiltonian produces "Routh function", and that the some of this coordinates in this function are interpreted as extra coordinates producing the potential energy. It sound very like Kaluza Klein, where the extra coordinates produce the force fields (and thus the potential energy). How valid is this analogy? Will KK theory produce Routh Theory in the non relativistic limit? Generically, is there some modern information on this formalism?