Convex conjugate

I wasn’t aware that convex conjugate is just Legendre transformation. That is,

$$latex f^{*}(p)=\sup _{\tilde {x}}\{\langle p,{\tilde {x}}\rangle -f({\tilde {x}})\}\geq \langle p,x\rangle -f(x)$$

 

Note that it is also known as the Legendre-Fenchel transformation. Btw, we have the Fenchel inequality

$latex \langle p,x\rangle \le f(x)+f^*(p)$

directly from the definition since

$$latex f^{*}(p)=\sup _{\tilde {x}}\{\langle p,{\tilde {x}}\rangle -f({\tilde {x}})\}\geq \langle p,x\rangle -f(x)$$

Examples

Lagrangian and Hermiltonian

I was not aware that Lagrangian and Hermiltonian are just Legendre transformation of one another. Take

$latex L(v,q) = \frac{1}{2} \langle v, Mv \rangle – V(q)$.

The convex conjugate is simply

$latex \sup_v \langle p,v \rangle – L(v,q)=\sup_v \langle p,v \rangle -\frac{1}{2} \langle v, Mv \rangle + V(q)$

$latex =\sup_v \frac{1}{2} \langle p,M^{-1} p\rangle – \frac{1}{2} \langle p,M^{-1} p\rangle +\langle p,v \rangle -\frac{1}{2} \langle v, Mv \rangle + V(q)$

$latex =\sup_v \frac{1}{2} \langle p,M^{-1} p\rangle – \frac{1}{2} \langle M^{-\frac{1}{2}} p – M^{\frac{1}{2}}v, M^{-\frac{1}{2}} p – M^{\frac{1}{2}}v \rangle + V(q)$

$latex =\frac{1}{2} \langle p,M^{-1} p\rangle + V(q)\triangleq H(p,q)$

Thermodynamics

Legendre transformation is widely used in non-canonical form. The internal energy $latex U(S,V)$ can be convert to the enthalpy $latex H(S,P) \triangleq PV + U(S,V)$.

We can verify that the definition “makes sense” as $latex dU = TdS – PdV$ and $latex dU + d\langle P,V \rangle = TdS – PdV + PdV=TdS$ is independent of $latex dV$ as desired since we want an extremum w.r.t. $latex V$.

Moreover, the Helmholtz (free) energy $latex A(T,V)=U-TS$ and the Gibbs (free) energy $latex G(T,P) = H – TS$.

An interesting remark is that the conjugate variables $latex S$ and $latex T$, and $latex V$ and $latex P$ are all with one as extensive variable (double with system size double) and the other as intensive variable (unchanged with system size increase).