Legendre Transformation

#Physics

Source

• Basically just another way to find the differential of a variable defined as a function of other variables of the same units

Examples

$\displaystyle U(\sigma,V)\rightarrow H(\sigma,P)$

• $\displaystyle \mathrm{d}U(\sigma,V)=\tau\mathrm{d}\sigma-P\mathrm{d}V$
• $\displaystyle H(\sigma,P)=U+PV$
• $\displaystyle \mathrm{d}H=\mathrm{d}U+P\mathrm{d}V+V\mathrm{d}P=\tau \mathrm{d}\sigma+V\mathrm{d}P$

Same can be done for $\displaystyle F(\tau,V)$ and $\displaystyle G(\tau,P)$

$\displaystyle \left(\mathrm{d}f(x,y)=u\mathrm{d}x+w\mathrm{d}y\right)\land(g\equiv f-wy)\rightarrow g(x,y)=-y^{2}\left( \frac{\partial }{\partial y} \frac{f}{y} \right)_{x}$

• $\displaystyle u=\left( \frac{\partial f }{\partial x} \right)_{y}$
• $\displaystyle w=\left( \frac{\partial f }{\partial y} \right)_{x}$