What are Legendre transformations

The transition from Lagrangian to Hamiltonian formulism corresponds to changing the variables in our mechanical functions from $(q,\stackrel{}{\stackrel{.}{q}},t)\stackrel{}{\mathrm{to}(q,p,t)}$

where , p is related to q and $\stackrel{.}{q}$ by the equation

$p_i=\frac{\partial L(q_j,{\dot{q}}_j,t)\stackrel{}{{}_{}}}{\partial {\dot{q}}_i}$

The procedures for switching variables in this manner is provided by the legendre transformations.

Consider a function of only two variables f(x,y), so that differential of f has the form

df=udx+vdy

where , $u=\frac{\mathrm{df}}{\mathrm{dx}}$ and $v=\frac{\mathrm{df}}{\mathrm{dy}}$                      (1)

To change the basis of description from x,y to a new set of variables u,y , so that differential quantities are expressed in terms of differential du and dy. Let g be the function of u and y defined by the equation

g=f-ux

differential of g is given as

dg=df-udx-xdu

or,

dg=vdy-xdu

which is exactly in the desired form. The quantities x and v are now functions of variables u and y given by the relations

$x=-\frac{\partial g}{\partial u},v=\frac{\partial g}{\partial y}$

which are exactly converse of equation 1