# Why are only the Lagrangian and Hamiltonian used in mechanics? [duplicate]

Why is it that we have a closed set of four functions, connected by Legendre transforms, in thermodynamics but nobody ever mentions but two of the corresponding functions in mechanics?

I've read that the Lagrangian L(q, q') corresponds to the tangent bundle of the configuration space in differential geometry, while the Hamiltonian H(p, q) corresponds to the cotangent bundle. If we call the mystery functions in mechanics X(p, p') and Y(p', q'), what do they correspond to in differential geometry?

Note that is order to get an exact one-to-one correspondence with the thermodynamic functions, with all of the plus and minus signs coming out the same, it is necessary to re-define the Lagrangian as L=V-T, instead of the conventianal L=T-V.