Editing Legendre transformation (section)

===Thermodynamics===
The strategy behind the use of Legendre transforms in thermodynamics is to shift from a function that depends on a variable to a new (conjugate) function that depends on a new variable, the conjugate of the original one. The new variable is the partial derivative of the original function with respect to the original variable. The new function is the difference between the original function and the product of the old and new variables. Typically, this transformation is useful because it shifts the dependence of, e.g., the energy from an [[Intensive and extensive properties|extensive variable]] to its conjugate intensive variable, which can often be controlled more easily in a physical experiment.

For example, the [[internal energy]] {{mvar|U}} is an explicit function of the ''[[extensive quantity|extensive variables]]'' [[entropy]] {{mvar|S}}, [[volume]] ''{{mvar|V}}'', and [[chemical composition]] {{mvar|N<sub>i</sub>}} (e.g., <math> i = 1, 2, 3, \ldots</math>)
<math display="block"> U = U \left (S,V,\{N_i\} \right ),</math>
which has a total differential
<math display="block"> dU = T\,dS - P\,dV + \sum \mu_i \,dN _i</math>

where <math> T = \left. \frac{\partial U}{\partial S} \right \vert _{V, N_{i\ for\ all\  i\ values}}, P = \left. -\frac{\partial U}{\partial V} \right \vert _{S, N_{i\ for\ all\  i\ values}}, \mu_i = \left. \frac{\partial U}{\partial N_i} \right \vert _{S,V, N_{j\ for\ all\ j \ne i}}</math>.

(Subscripts are not necessary by the definition of partial derivatives but left here for clarifying variables.) Stipulating some common reference state, by using the (non-standard) Legendre transform of the internal energy {{mvar|U}} with respect to volume {{mvar|V}}, the [[enthalpy]] {{mvar|H}} may be obtained as the following.

To get the (standard) Legendre transform <math display="inline">U^*</math> of the internal energy {{mvar|U}} with respect to volume {{mvar|V}}, the function <math display="inline">u\left( p,S,V,\{{{N}_{i}}\} \right)=pV-U</math> is defined first, then it shall be maximized or bounded by {{mvar|V}}. To do this, the condition <math display="inline">\frac{\partial u}{\partial V} = p - \frac{\partial U}{\partial V} = 0 \to p = \frac{\partial U}{\partial V}</math> needs to be satisfied, so <math display="inline">U^* = \frac{\partial U}{\partial V}V - U</math> is obtained. This approach is justified because {{mvar|U}} is a linear function with respect to {{mvar|V}} (so a convex function on {{mvar|V}}) by the definition of [[Intensive and extensive properties|extensive variables]]. The non-standard Legendre transform here is obtained by negating the standard version, so <math display="inline">-U^* = H = U - \frac{\partial U}{\partial V}V = U + PV</math>.

{{mvar|H}} is definitely a [[state function]] as it is obtained by adding {{mvar|PV}} ({{mvar|P}} and {{mvar|V}} as [[State variable|state variables]]) to a state function <math display="inline"> U = U \left (S,V,\{N_i\} \right )</math>, so its differential is an [[exact differential]]. Because of <math display="inline"> dH = T\,dS + V\,dP + \sum \mu_i \,dN _i</math> and the fact that it must be an exact differential, <math> H = H(S,P,\{N_i\})</math>.

The enthalpy is suitable for description of processes in which the pressure is controlled from the surroundings.

It is likewise possible to shift the dependence of the energy from the extensive variable of entropy, {{mvar|S}}, to the (often more convenient) intensive variable {{mvar|T}}, resulting in the [[Helmholtz energy|Helmholtz]] and [[Gibbs energy|Gibbs]] [[thermodynamic free energy|free energies]]. The Helmholtz free energy {{mvar|A}}, and Gibbs energy {{mvar|G}}, are obtained by performing Legendre transforms of the internal energy and enthalpy, respectively,
<math display="block"> A = U - TS ~,</math><math display="block"> G = H - TS = U + PV - TS ~.</math>

The Helmholtz free energy is often the most useful thermodynamic potential when temperature and volume are controlled from the surroundings, while the Gibbs energy is often the most useful when temperature and pressure are controlled from the surroundings.