Editing Legendre transformation (section)

==Applications==

===Analytical mechanics===
A Legendre transform is used in [[classical mechanics]] to derive the [[Hamiltonian mechanics|Hamiltonian formulation]] from the [[Lagrangian mechanics|Lagrangian formulation]], and conversely. A typical Lagrangian has the form

<math display="block">L(v,q)=\tfrac{1}2\langle v,Mv\rangle-V(q),</math>
where <math>(v,q)</math> are coordinates on {{math|'''R'''<sup>''n''</sup> × '''R'''<sup>''n''</sup>}}, {{mvar|M}} is a positive definite real matrix, and
<math display="block">\langle x,y\rangle = \sum_j x_j y_j.</math>

For every {{mvar|q}} fixed, <math>L(v, q)</math> is a convex function of <math>v</math>, while <math>V(q)</math> plays the role of a constant.

Hence the Legendre transform of <math>L(v, q)</math> as a function of <math>v</math> is the Hamiltonian function,
<math display="block">H(p,q)=\tfrac {1}{2} \langle p,M^{-1}p\rangle+V(q).</math>

In a more general setting, <math>(v, q)</math> are local coordinates on the [[tangent bundle]] <math>T\mathcal M</math> of a manifold <math>\mathcal M</math>. For each {{mvar|q}}, <math>L(v, q)</math> is a convex function of the tangent space {{math|''V<sub>q</sub>''}}. The Legendre transform gives the Hamiltonian <math>H(p, q)</math> as a function of the coordinates {{math|(''p'', ''q'')}} of the [[cotangent bundle]] <math>T^*\mathcal M</math>; the inner product used to define the Legendre transform is inherited from the pertinent canonical [[symplectic vector space|symplectic structure]]. In this abstract setting, the Legendre transformation corresponds to the [[tautological one-form]].{{Explain|date=April 2023}}

===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.

===Variable capacitor===
As another example from [[physics]], consider a parallel conductive plate [[capacitor]], in which the plates can move relative to one another. Such a capacitor would allow transfer of the electric energy which is stored in the capacitor into external mechanical work, done by the [[force]] acting on the plates. One may think of the electric charge as analogous to the "charge" of a [[gas]] in a [[cylinder (engine)|cylinder]], with the resulting mechanical [[force]] exerted on a [[piston]].

Compute the force on the plates as a function of {{math|'''x'''}}, the distance which separates them. To find the force, compute the potential energy, and then apply the definition of force as the gradient of the potential energy function.

The [[Electric potential energy|electrostatic potential energy]] stored in a capacitor of the [[capacitance]] {{math|''C''('''x''')}} and a positive [[electric charge]] {{math|+''Q''}} or negative charge {{math|-''Q''}} on each conductive plate is (with using the definition of the capacitance as <math display="inline">C = \frac{Q}{V}</math>),

<math display="block"> U (Q, \mathbf{x}) = \frac{1}{2} QV(Q,\mathbf{x}) = \frac{1}{2} \frac{Q^2}{C(\mathbf{x})},~</math>

where the dependence on the area of the plates, the dielectric constant of the insulation material between the plates, and the separation {{math|'''x'''}} are abstracted away as the [[capacitance]] {{math|''C''('''x''')}}. (For a parallel plate capacitor, this is proportional to the area of the plates and inversely proportional to the separation.)

The force {{math|'''F'''}} between the plates due to the electric field created by the charge separation is then
<math display="block"> \mathbf{F}(\mathbf{x}) = -\frac{dU}{d\mathbf{x}} ~. </math>

If the capacitor is not connected to any electric circuit, then the ''[[electric charge|electric charges]]'' on the plates remain constant and the voltage varies when the plates move with respect to each other, and the force is the negative [[gradient]] of the [[electrostatics|electrostatic]] potential energy as
<math display="block"> \mathbf{F}(\mathbf{x}) = \frac{1}{2} \frac{dC(\mathbf{x})}{d\mathbf{x}} \frac{Q^2}{{C(\mathbf{x})}^2}
= \frac{1}{2} \frac{dC(\mathbf{x})}{d\mathbf{x}}V(\mathbf{x})^2 </math>

where <math display="inline"> V(Q,\mathbf{x}) = V(\mathbf{x})  </math> as the charge is fixed in this configuration.

However, instead, suppose that the ''[[volt]]age'' between the plates {{math|''V''}} is maintained constant as the plate moves by connection to a [[battery (electricity)|battery]], which is a reservoir for electric charges at a constant potential difference. Then the amount of ''charges'' <math display="inline"> Q </math> ''is a variable'' instead of the voltage; <math display="inline"> Q </math> and <math display="inline"> V </math> are the Legendre conjugate to each other. To find the force, first compute the non-standard Legendre transform <math display="inline">U^*</math> with respect to <math display="inline"> Q </math> (also with using <math display="inline">C = \frac{Q}{V}</math>),

<math display="block">U^* = U - \left.\frac{\partial U}{\partial Q} \right|_\mathbf{x} \cdot Q =U - \frac{1}{2C(\mathbf{x})} \left. \frac{\partial Q^2}{\partial Q} \right|_\mathbf{x} \cdot Q = U - QV = \frac{1}{2} QV - QV = -\frac{1}{2} QV= - \frac{1}{2} V^2 C(\mathbf{x}).</math>

This transformation is possible because <math display="inline"> U </math> is now a linear function of <math display="inline"> Q </math> so is convex on it. The force now becomes the negative gradient of this Legendre transform, resulting in the same force obtained from the original function <math display="inline"> U </math>,
<math display="block"> \mathbf{F}(\mathbf{x}) = -\frac{dU^*}{d\mathbf{x}} = \frac{1}{2} \frac{dC(\mathbf{x})}{d\mathbf{x}}V^2 .</math>

The two conjugate energies <math display="inline"> U </math> and <math display="inline"> U^* </math> happen to stand opposite to each other (their signs are opposite), only because of the [[linear]]ity of the [[capacitance]]—except now {{math|''Q''}} is no longer a constant. They reflect the two different pathways of storing energy into the capacitor, resulting in, for instance, the same "pull" between a capacitor's plates.

===Probability theory===
In [[large deviations theory]], the ''rate function'' is defined as the Legendre transformation of the logarithm of the [[moment generating function]] of a random variable. An important application of the rate function is in the calculation of tail probabilities of sums of [[Independent and identically distributed random variables|i.i.d. random variables]], in particular in [[Cramér's theorem (large deviations)|Cramér's theorem]].

If <math>X_n</math> are i.i.d. random variables, let <math>S_n=X_1+\cdots+X_n</math> be the associated [[random walk]] and <math>M(\xi)</math> the moment generating function of <math>X_1</math>.  For <math>\xi\in\mathbb R</math>, <math>E[e^{\xi S_n}] = M(\xi)^n</math>.  Hence, by [[Markov's inequality]], one has for <math>\xi\ge 0</math> and <math>a\in\mathbb R</math>
<math display="block">P(S_n/n > a) \le e^{-n\xi a}M(\xi)^n=\exp[-n(\xi a - \Lambda(\xi))]</math>
where <math>\Lambda(\xi)=\log M(\xi)</math>.  Since the left-hand side is independent of <math>\xi</math>, we may take the infimum of the right-hand side, which leads one to consider the supremum of <math>\xi a - \Lambda(\xi)</math>, i.e., the Legendre transform of <math>\Lambda</math>, evaluated at <math>x=a</math>.

===Microeconomics===
Legendre transformation arises naturally in [[microeconomics]] in the process of finding the ''[[supply (economics)|supply]]'' {{math|''S''(''P'')}} of some product given a fixed price {{math|''P''}} on the market knowing the [[cost curve|cost function]] {{math|''C''(''Q'')}}, i.e. the cost for the producer to make/mine/etc. {{math|''Q''}} units of the given product.

A simple theory explains the shape of the supply curve based solely on the cost function. Let us suppose the market price for a one unit of our product is {{math|''P''}}. For a company selling this good, the best strategy is to adjust the production {{math|''Q''}} so that its profit is maximized. We can maximize the profit
<math display="block">\text{profit} = \text{revenue} - \text{costs} = PQ - C(Q)</math>
by differentiating with respect to {{math|''Q''}} and solving
<math display="block">P - C'(Q_\text{opt}) = 0.</math>

{{math|''Q''<sub>opt</sub>}} represents the optimal quantity {{math|''Q''}} of goods that the producer is willing to supply, which is indeed the supply itself:
<math display="block">S(P) = Q_\text{opt}(P) = (C')^{-1}(P).</math>

If we consider the maximal profit as a function of price, <math>\text{profit}_\text{max}(P)</math>, we see that it is the Legendre transform of the cost function <math>C(Q)</math>.