  We study new Legendre transforms in classical mechanics and investigate some
of their general properties. The behaviour of the new functions is analyzed
under coordinate transformations.When invariance under different kinds of
transformations are considered, the new formulation is found to be completly
equivalent to the usual Lagrangian formulation, recovering well established
results like the conservation of the angular momentum. Furthermore, a natural
generalization of the Poisson Bracket is found to be inherent to the formalism
introduced. On the other hand, we find that with a convenient redefinition of
the Lagrangian, $\mathcal{L}^{\prime}=-\mathcal{L}$, it is possible to
establish an exact one-to-one mathematical correspondence between the
thermodynamic potentials and the new potentials of classical mechanics
