Legendre Transformations (II, An example)

Consider a system whose energy has the form:

E = 2S2 + 4 + V2 = E(S,V)

where E is an explicit function of S and V (as required in elementary thermo). At constant V, we have

We wish to perform a Legendre Transformation on E(S,V) to change the S variable to something else (it's going to be the temperature). We wish

 (1)

Since the slope is 4S (see above) we have S=slope/4, so that, substituting into equation 1 we have

so that

which is, substituting (slope/4) for S on the left hand side

which is an equation for the intercept as a function of the slope (and V, which is being carried along here):

If we call the intercept A, the Helmholtz Free Energy, and the slope we call T, the temperature, then we have

which is the desired result. That is, we have started with a function of S and V and ended up with a function of T and V.

Just to prove that we know what we're doing, we will do it again, this time transforming A(T,V) into E(S,V). We start with desiring a representation of A(T,V) of the form:

 (2)

Then

i.e., . Substituting for T in equation 2

which is, substituting on the l.h.s.

which is, again substituting for T,

which rearranges to

i.e.

which is what we started with (sorry about the sentence construction).