Legendre transformation
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Legendre transformation (or Legendre transform), named after
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
, is an involutive
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...
on
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
s of one real variable. In physical problems, it is used to convert functions of one quantity (such as velocity, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
to derive the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
formalism out of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
formalism (or vice versa) and in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
to derive the
thermodynamic potentials A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermod ...
, as well as in the solution of
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
of several variables. For sufficiently smooth functions on the real line, the Legendre transform f^* of a function f can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as Df(\cdot) = \left( D f^* \right)^(\cdot)~, where (\phi)^(\cdot) means a function such that (\phi) ^(\phi(x))=x~, or, equivalently, as f'(f^(x^*)) = x^* and f^(f'(x)) = x in
Lagrange's notation In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with ...
. The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the
convex conjugate In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation ...
(also called the Legendre–Fenchel transformation), which can be used to construct a function's
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
.


Definition

Let I \sub \R be an interval, and f:I \to \R a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
; then its ''Legendre transform'' is the function f^*:I^* \to \R defined by f^*(x^*) = \sup_(x^*x-f(x)),\ \ \ \ x^*\in I^* where \sup denotes the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
, and the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
I^* is I^*= \left \ ~. The transform is always well-defined when f(x) is
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
. The generalization to convex functions f:X \to \R on a convex set X \sub \R^n is straightforward: f^*:X^* \to \R has domain X^*= \left \ and is defined by f^*(x^*) = \sup_(\langle x^*,x\rangle-f(x)),\quad x^*\in X^* ~, where \langle x^*,x \rangle denotes the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of x^* and x. The function f^* is called the
convex conjugate In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation ...
function of f. For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted p, instead of x^*. If the convex function f is defined on the whole line and is everywhere
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, then f^*(p)=\sup_(px - f(x)) = \left( p x - f(x) \right), _ can be interpreted as the negative of the y-intercept of the
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of f that has slope p. The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by f can be represented equally well as a set of (x,y) points, or as a set of tangent lines specified by their slope and intercept values.


Understanding the transform in terms of derivatives

For a differentiable convex function f on the real line with an invertible first derivative, the Legendre transform f^* can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. Explicitly, for a differentiable convex function f on the real line with a first derivative f' with inverse (f')^, the Legendre transform f^* (with derivative (f^*)' with inverse ((f^*)')^) can be specified, up to an additive constant, by the condition that f' and (f^*)' are inverse functions of each other, i.e., f' = ((f^*)')^ and (f^*)' = (f')^. To see this, first note that if f is differentiable and \overline is a critical point of the function of x \mapsto p \cdot x -f(x) , then the supremum is achieved at \overline (by convexity). Therefore, f^*(p)= p \cdot \overline - f(\overline). Suppose that f' is invertible and let g denote its inverse. Then for each p, the point g(p) is the unique critical point of x \mapsto px -f(x) . Indeed, f'(g(p))=p and so p-f'(g(p))=0 . Hence we have f^*(p) = p \cdot g(p) - f(g(p)) for each p. By differentiating with respect to p we find (f^*)'(p) = g(p)+ p \cdot g'(p) - f'(g(p)) \cdot g'(p). Since f'(g(p))=p this simplifies to (f^*)'(p) = g(p). In other words, (f^*)' and f' are inverses. In general, if h' is an inverse of f' , then h' = (f^*)' and so integration provides a constant c so that f^* = h +c . In practical terms, given f(x), the parametric plot of xf'(x)-f(x) versus f'(x) amounts to the graph of g(p) versus p. In some cases (e.g. thermodynamic potentials, below), a non-standard requirement is used, amounting to an alternative definition of with a ''minus sign'', f(x) - f^*(p) = xp.


Properties

*The Legendre transform of a convex function is convex.Let us show this for the case of a doubly differentiable f with a non zero (and hence positive, due to convexity) double derivative and with a bijective derivative. For a fixed p, let x maximize px - f(x). Then f^*(p) = px - f(x), noting that x depends on p . Thus, f'(x) = p ~. The derivative of f is itself differentiable with a positive derivative and hence strictly monotonic and invertible. Thus x = g(p) where g \equiv (f')^, meaning that g is defined so that f'(g(p))= p. Note that g is also differentiable with the following derivative, \frac = \frac ~. Thus f^*(p) = pg(p) - f(g(p)) is the composition of differentiable functions, hence differentiable. Applying the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
and the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
yields \frac = g(p) + \left(p - f'(g(p))\right)\cdot \frac = g(p), giving \frac = \frac = \frac > 0, so f^* is convex. *It follows that the Legendre transformation is an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
, i.e., f^ = f ~: By using the above equalities for g(p), f^*(p) and its derivative, \begin f^(x) & = \left(x\cdot p_s - f^(p_s)\right), _ \\ pt& = g(p_s)\cdot p_s - f^(p_s) \\ pt& = g(p_s)\cdot p_s-(p_s g(p_s)-f(g(p_s)))\\ pt& = f(g(p_s)) \\ pt& = f(x)~. \end


Examples


Example 1

The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
f(x) = e^x has f^*(p) = p ( \ln p - 1 ) as a Legendre transform, since their respective first derivatives and are inverse functions of each other. This example illustrates that the respective
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
s of a function and its Legendre transform need not agree.


Example 2

Let defined on , where is a fixed constant. For fixed, the function of , has the first derivative and second derivative ; there is one stationary point at , which is always a maximum. Thus, and f^*(x^*)=\frac ~. The first derivatives of , 2, and of , , are inverse functions to each other. Clearly, furthermore, f^(x)=\fracx^2=cx^2~, namely .


Example 3

Let for . For fixed, is continuous on
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, hence it always takes a finite maximum on it; it follows that . The stationary point at is in the domain if and only if , otherwise the maximum is taken either at , or . It follows that f^*(x^*)=\begin 2x^*-4, & x^*<4\\ \frac, & 4\leq x^*\leq 6,\\ 3x^*-9, & x^*>6. \end


Example 4

The function is convex, for every (strict convexity is not required for the Legendre transformation to be well defined). Clearly is never bounded from above as a function of , unless . Hence is defined on and . One may check involutivity: of course is always bounded as a function of , hence . Then, for all one has \sup_(xx^*-f^*(x^*))=xc, and hence .


Example 5: several variables

Let f(x)=\langle x,Ax\rangle+c be defined on , where is a real, positive definite matrix. Then is convex, and \langle p,x\rangle-f(x)=\langle p,x \rangle-\langle x,Ax\rangle-c, has gradient and
Hessian A Hessian is an inhabitant of the German state of Hesse. Hessian may also refer to: Named from the toponym *Hessian (soldier), eighteenth-century German regiments in service with the British Empire **Hessian (boot), a style of boot **Hessian f ...
, which is negative; hence the stationary point is a maximum. We have , and f^*(p)=\frac\langle p,A^p\rangle-c.


Behavior of differentials under Legendre transforms

The Legendre transform is linked to
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
, . Let be a function of two independent variables and , with the differential df = \frac\,dx + \frac\,dy = p\,dx + v\,dy. Assume that it is convex in for all , so that one may perform the Legendre transform in , with the variable conjugate to . Since the new independent variable is , the differentials and devolve to and , i.e., we build another function with its differential expressed in terms of the new basis and . We thus consider the function so that dg = df - p\,dx - x\,dp = -x\,dp + v\,dy x = -\frac v = \frac. The function is the Legendre transform of , where only the independent variable has been supplanted by . This is widely used in thermodynamics, as illustrated below.


Applications


Analytical mechanics

A Legendre transform is used in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
to derive the Hamiltonian formulation from the Lagrangian formulation, and conversely. A typical Lagrangian has the form L(v,q)=\tfrac2\langle v,Mv\rangle-V(q), where (v,q) are coordinates on , is a positive real matrix, and \langle x,y\rangle = \sum_j x_j y_j. For every fixed, L(v, q) is a convex function of v, while V(q) plays the role of a constant. Hence the Legendre transform of L(v, q) as a function of v is the Hamiltonian function, H(p,q)=\tfrac \langle p,M^p\rangle+V(q). In a more general setting, (v, q) are local coordinates on the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
T\mathcal M of a manifold \mathcal M. For each , L(v, q) is a convex function of the tangent space . The Legendre transform gives the Hamiltonian H(p, q) as a function of the coordinates of the cotangent bundle T^*\mathcal M; the inner product used to define the Legendre transform is inherited from the pertinent canonical
symplectic structure Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ha ...
. In this abstract setting, the Legendre transformation corresponds to the
tautological one-form In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus p ...
.


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
extensive variable Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is on ...
to its conjugate intensive variable, which can usually be controlled more easily in a physical experiment. For example, the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
is an explicit function of the '' extensive variables''
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
, and
chemical composition A chemical composition specifies the identity, arrangement, and ratio of the elements making up a compound. Chemical formulas can be used to describe the relative amounts of elements present in a compound. For example, the chemical formula for ...
U = U \left (S,V,\ \right ), which has a total differential dU = T\,dS - P\,dV + \sum \mu_i \,dN _i. Stipulating some common reference state, by using the (non-standard) Legendre transform of the internal energy, , with respect to volume, , the
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
may be defined by writing H = U + PV = H (S,P,\), which is now explicitly function of the pressure , since dH(S,P,\) = T\,dS + V\,dP + \sum \mu_i \,dN _i. 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, , to the (often more convenient) intensive variable , resulting in the
Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
and Gibbs free energies. The Helmholtz free energy, , and Gibbs energy, , are obtained by performing Legendre transforms of the internal energy and enthalpy, respectively, A = U - TS ~, G = H - TS = U + PV - TS ~. 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.


An example – variable capacitor

As another example from
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, consider a parallel-plate
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
, 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 In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
acting on the plates. One may think of the electric charge as analogous to the "charge" of a
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
in a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
, with the resulting mechanical
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
exerted on a
piston A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tig ...
. Compute the force on the plates as a function of , 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 energy stored in a capacitor of
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
and charge is U (Q, \mathbf) = \frac QV = \frac \frac,~ where the dependence on the area of the plates, the dielectric constant of the material between the plates, and the separation are abstracted away as the
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
. (For a parallel plate capacitor, this is proportional to the area of the plates and inversely proportional to the separation.) The force between the plates due to the electric field is then \mathbf(\mathbf) = -\frac ~. If the capacitor is not connected to any circuit, then the ''
charges Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
'' on the plates remain constant as they move, and the force is the negative
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of the
electrostatic Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
energy \mathbf(\mathbf) = \frac \frac \frac. However, suppose, instead, that the ''
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defi ...
age'' between the plates is maintained constant by connection to a
battery Battery most often refers to: * Electric battery, a device that provides electrical power * Battery (crime), a crime involving unlawful physical contact Battery may also refer to: Energy source *Automotive battery, a device to provide power t ...
, which is a reservoir for charge at constant potential difference; now the ''charge is variable'' instead of the voltage, its Legendre conjugate. To find the force, first compute the non-standard Legendre transform, U^* = U - \left.\frac \_\mathbf \cdot Q =U - \frac \left. \frac \_\mathbf \cdot Q = U - QV = \frac QV - QV = -\frac QV= - \frac V^2 C(\mathbf). The force now becomes the negative gradient of this Legendre transform, still pointing in the same direction, \mathbf(\mathbf) = -\frac~. The two conjugate energies happen to stand opposite to each other, only because of the
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
ity of the
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
—except now 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 In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...
, the ''rate function'' is defined as the Legendre transformation of the logarithm of the
moment generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
of a random variable. An important application of the rate function is in the calculation of tail probabilities of sums of i.i.d. random variables.


Microeconomics

Legendre transformation arises naturally in
microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
in the process of finding the ''
supply Supply may refer to: *The amount of a resource that is available **Supply (economics), the amount of a product which is available to customers **Materiel, the goods and equipment for a military unit to fulfill its mission *Supply, as in confidenc ...
'' of some product given a fixed price on the market knowing the cost function , i.e. the cost for the producer to make/mine/etc. 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 . For a company selling this good, the best strategy is to adjust the production so that its profit is maximized. We can maximize the profit \text = \text - \text = PQ - C(Q) by differentiating with respect to and solving P - C'(Q_\text) = 0. represents the optimal quantity of goods that the producer is willing to supply, which is indeed the supply itself: S(P) = Q_\text(P) = (C')^(P). If we consider the maximal profit as a function of price, \text_\text(P), we see that it is the Legendre transform of the cost function C(Q).


Geometric interpretation

For a
strictly convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
, the Legendre transformation can be interpreted as a mapping between the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of the function and the family of
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s of the graph. (For a function of one variable, the tangents are well-defined at all but at most
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
points, since a convex function is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at all but at most countably many points.) The equation of a line with
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
p and y-intercept b is given by y = p x + b. For this line to be tangent to the graph of a function f at the point \left(x_0, f(x_0)\right) requires f(x_0) = p x_0 + b and p = f'(x_0). Being the derivative of a strictly convex function, the function f' is strictly monotone and thus
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. The second equation can be solved for x_0 = f^(p), allowing elimination of x_0 from the first, and solving for the y-intercept b of the tangent as a function of its slope p, b = f(x_0) - p x_0 = f\left(f^(p)\right) - p \cdot f^(p) = -f^\star(p) where f^ denotes the Legendre transform of f. The
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of tangent lines of the graph of f parameterized by the slope p is therefore given by y = p x - f^(p), or, written implicitly, by the solutions of the equation F(x,y,p) = y + f^(p) - p x = 0~. The graph of the original function can be reconstructed from this family of lines as the
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of this family by demanding \frac = f^(p) - x = 0. Eliminating p from these two equations gives y = x \cdot f^(x) - f^\left(f^(x)\right). Identifying y with f(x) and recognizing the right side of the preceding equation as the Legendre transform of f^, yields f(x) = f^(x) ~.


Legendre transformation in more than one dimension

For a differentiable real-valued function on an
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convex subset of the Legendre conjugate of the pair is defined to be the pair , where is the image of under the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
mapping , and is the function on given by the formula g(y) = \left\langle y, x \right\rangle - f(x), \qquad x = \left(Df\right)^(y) where \left\langle u,v\right\rangle = \sum_^n u_k \cdot v_k is the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
on . The multidimensional transform can be interpreted as an encoding of the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of the function's epigraph in terms of its
supporting hyperplane In geometry, a supporting hyperplane of a set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed half-spaces bounded by the hyperplane, * S has at leas ...
s. Alternatively, if is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
and is its
dual vector space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
, then for each point of and of , there is a natural identification of the
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
s with and with . If is a real differentiable function over , then its
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
, , is a section of the cotangent bundle and as such, we can construct a map from to . Similarly, if is a real differentiable function over , then defines a map from to . If both maps happen to be inverses of each other, we say we have a Legendre transform. The notion of the
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is commonly used in this setting. When the function is not differentiable, the Legendre transform can still be extended, and is known as the Legendre-Fenchel transformation. In this more general setting, a few properties are lost: for example, the Legendre transform is no longer its own inverse (unless there are extra assumptions, like
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).


Legendre transformation on manifolds

Let M be a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, let E and \pi : E\to M respectively be a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
on M and its associated bundle projection, and let L : E\to \R be a smooth function. We think of L as a
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by analogy with the classical case where M = \R, E = TM = \Reals \times \Reals and L(x,v) = \frac 1 2 m v^2 - V(x) for some positive number m\in \Reals and function V : M \to \Reals. As usual, we denote by E^* the dual of E, by E_x the fiber of \pi over x\in M, and by L, _ : E_x\to \R the restriction of L to E_x. The ''Legendre transformation'' of L is the smooth morphism \mathbf F L : E \to E^* defined by \mathbf FL(v) = d(L, _)(v) \in E_x^*, where x = \pi(v). In other words, \mathbf FL(v)\in E_x^* is the covector that sends w\in E_x to the directional derivative \left.\frac d \_ L(v + tw)\in \R. To describe the Legendre transformation locally, let U\subseteq M be a coordinate chart over which E is trivial. Picking a trivialization of E over U, we obtain charts E_U \cong U \times \R^r and E_U^* \cong U \times \R^r. In terms of these charts, we have \mathbf FL(x; v_1, \dotsc, v_r) = (x; p_1,\dotsc, p_r), where p_i = \frac (x; v_1, \dotsc, v_r) for all i = 1, \dots, r. If, as in the classical case, the restriction of L : E\to \mathbb R to each fiber E_x is strictly convex and bounded below by a positive definite quadratic form minus a constant, then the Legendre transform \mathbf FL : E\to E^* is a diffeomorphism.Ana Cannas da Silva. ''Lectures on Symplectic Geometry'', Corrected 2nd printing. Springer-Verlag, 2008. pp. 147-148. . Suppose that \mathbf FL is a diffeomorphism and let H : E^* \to \R be the “
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” function defined by H(p) = p \cdot v - L(v), where v = (\mathbf FL)^(p). Using the natural isomorphism E\cong E^, we may view the Legendre transformation of H as a map \mathbf FH : E^* \to E. Then we have (\mathbf FL)^ = \mathbf FH.


Further properties


Scaling properties

The Legendre transformation has the following scaling properties: For , f(x) = a \cdot g(x) \Rightarrow f^\star(p) = a \cdot g^\star\left(\frac\right) f(x) = g(a \cdot x) \Rightarrow f^\star(p) = g^\star\left(\frac\right). It follows that if a function is homogeneous of degree then its image under the Legendre transformation is a homogeneous function of degree , where . (Since , with , implies .) Thus, the only monomial whose degree is invariant under Legendre transform is the quadratic.


Behavior under translation

f(x) = g(x) + b \Rightarrow f^\star(p) = g^\star(p) - b f(x) = g(x + y) \Rightarrow f^\star(p) = g^\star(p) - p \cdot y


Behavior under inversion

f(x) = g^(x) \Rightarrow f^\star(p) = - p \cdot g^\star\left(\frac \right)


Behavior under linear transformations

Let be a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
. For any convex function on , one has (A f)^\star = f^\star A^\star where is the
adjoint operator In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
of defined by \left \langle Ax, y^\star \right \rangle = \left \langle x, A^\star y^\star \right \rangle, and is the ''push-forward'' of along (A f)(y) = \inf\. A closed convex function is symmetric with respect to a given set of orthogonal linear transformations, f(A x) = f(x), \; \forall x, \; \forall A \in G
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
is symmetric with respect to .


Infimal convolution

The infimal convolution of two functions and is defined as \left(f \star_\inf g\right)(x) = \inf \left \. Let be proper convex functions on . Then \left( f_1 \star_\inf \cdots \star_\inf f_m \right)^\star = f_1^\star + \cdots + f_m^\star.


Fenchel's inequality

For any function and its convex conjugate ''Fenchel's inequality'' (also known as the ''Fenchel–Young inequality'') holds for every and , i.e., ''independent'' pairs, \left\langle p,x \right\rangle \le f(x) + f^\star(p).


See also

*
Dual curve In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. I ...
*
Projective duality In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of du ...
*
Young's inequality for products In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality f ...
*
Convex conjugate In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation ...
*
Moreau's theorem In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approxim ...
*
Integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
*
Fenchel's duality theorem In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel. Let ''ƒ'' be a proper convex function on R''n'' and let ''g'' be a proper concave function on R''n''. Then, if regularity condi ...


References

* * * Fenchel, W. (1949). "On conjugate convex functions", ''Can. J. Math'' 1: 73-77. * *


Further reading

* * *


External links

{{Commons category, Legendre transformation
Legendre transform with figures
at maze5.net
Legendre and Legendre-Fenchel transforms in a step-by-step explanation
at onmyphd.com Transforms Duality theories Concepts in physics Convex analysis Mathematical physics