Diagram illustrating the Legendre transformation of the function f(x) . The function is shown in red, and the tangent line at x₀  is shown in blue. The tangent line intersects the vertical axis at (0, −f^*) and f^*  is the value of the Legendre transform f^*(p)  where . Note that for any other point on the red curve, a line drawn through that point with the same slope as the blue line will have a y-intercept above the point (0, −f^*), showing that f^* is indeed a maximum.

In mathematics, it is often desirable to express a functional relationship as a different function, whose argument is the derivative of f , rather than x . If we let p = df/dx  be the argument of this new function, then this new function is written and is called the Legendre transform of the original function, after Adrien-Marie Legendre. Image File history File links Download high-resolution version (582x674, 30 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Legendre transformation ... Image File history File links Download high-resolution version (582x674, 30 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Legendre transformation ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Adrien-Marie Legendre (September 18, 1752 – January 10, 1833) was a French mathematician. ...

The Legendre transform of a function is defined as follows:

The notation max_x indicates the maximization of the expression with respect to the variable x while p is held constant. The Legendre transform is its own inverse. Like the familiar Fourier transform, the Legendre transform takes a function f(x) and produces a function of a different variable p. However, while the Fourier transform consists of an integration with a kernel, the Legendre transform uses maximization as the transformation procedure. The transform is well behaved only if f(x) is a convex function: In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ...

The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by f(x) 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. Duality in the projective plane refers to the interchangeability between points and lines which preserves incidence properties. ...

The Legendre transformation can be generalized to the Legendre-Fenchel transformation. It is commonly used in thermodynamics and in the hamiltonian formulation of classical mechanics. In mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: f and g are then said to be related by a Legendre transformation. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

Definitions

The definition of the Legendre transform can be made more explicit. To maximize f ^* with respect to x, we set its derivative equal to zero:

Thus, the expression is maximized when

.

This is a maximum because the second derivative is negative:

since f was assumed convex. Next we invert (2) to obtain x as a function of p and plug this into (1) , which gives the more useful form,

This definition gives the conventional procedure for calculating the Legendre transform of f(x): find , invert for x and substitute into the expression xp − f(x). This definition makes clear the following interpretation: the Legendre transform produces a new function, in which the independent variable x is replaced by , which is the derivative of the original function with respect to x.

Another definition

There is a third definition of the Legendre transform: and are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: In mathematics, a derivative is the rate of change of a quantity (e. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

We can see this by taking derivative of :

Combining this equation with the maximization condition results in the following pair of reciprocal equations:

We see that Df and are inverses, as promised. They are unique up to an additive constant which is fixed by the additional requirement that

Although in some cases (e.g. thermodynamic potentials) a non-standard requirement is used:

The standard constraint will be considered in this article unless otherwise noted. The Legendre transformation is its own inverse, and is related to integration by parts. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

Applications

Thermodynamics

The strategy behind the use of Legendre transforms is to shift the dependence of a function from one independent variable to another (the derivative of the original function with regard to this independent variable) by taking the difference between the original function and their product. They are used to transform among the various thermodynamic potentials. For example, while the internal energy is an explicit function of the extensive variables, entropy, volume (and chemical composition) In an experimental design, the independent variable (also known as predictor or regressor or manipulated variable) is the variable which is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ... In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, p = pressure, V = volume Differential definitions The following differential relations hold for the four potentials: If we write the above four equations generally as Then it is seen that yielding expressions for... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... see also: Entropy (disambiguation) Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Volume is how much space a thing has. ... The chemical composition of a substance refers to the elements of which the substance is composed. ...

the enthalpy, the (non standard) Legendre transform of U with respect to −PV In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as Δ or ΔH, or rarely as χ) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant conditions. ...

becomes a function of the entropy and the intensive quantity, pressure, as natural variables, and is useful when the (external) P is constant. The free energies (Helmholtz and Gibbs), are obtained through further Legendre transforms, by subtracting TS (from U and H respectively), shift dependence from the entropy S to its conjugate intensive variable temperature T, and are useful when it is constant. It has been suggested that this article or section be merged into intensive and extensive properties. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... It has been suggested that this article or section be merged into Helmholtz energy. ... In thermodynamics, the Gibbs energy or Gibbs energy function is the energy portion of a thermodynamic system available to do work. ... Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. ...

Hamilton-Lagrange mechanics

A Legendre transform is used in classical mechanics to derive the Hamiltonian formulation from the Lagrangian one, and conversely. While the Lagrangian is an explicit function of the positional coordinates q_j and generalized velocities dq_j /dt (and time), the Hamiltonian shifts the functional dependence to the positions and momenta,defined as Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... The velocity of an object is simply its speed in a particular direction. ... In physics, Hamiltonian has distinct but closely related meanings. ... In classical mechanics, momentum (pl. ...

Each of the two formulations has its own applicability, both in the theoretical foundations of the subject, and in practice, depending on the ease of calculation for a particular problem. The coordinates are not necessarily rectilinear, but can also be angles, etc. An optimum choice takes advantage of the actual physical symmetries. The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... A calculation is a deliberate process for transforming one or more inputs into one or more results. ... Rectilinear: Characterized by straight lines, as opposed to curvilinear which is characterized by curved lines. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... Sphere symmetry group o. ...

An example - variable capacitor

As another example from physics, consider a parallel-plate capacitor whose plates can approach or recede from one another, exchanging work with external mechanical forces which maintain the plate separation — analogous to a gas in a cylinder with a piston. We want the attractive force f between the plates as a function of the variable separation x. (The two vectors point in opposite directions.) If the charges on the plates remain constant as they move, the force is the negative gradient of the electrostatic energy The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... In physics, force is an influence that may cause a body to accelerate. ... A gas is one of the four major phases of matter (after solid and liquid, and followed by plasma, that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... A piston and cylinder from a steam engine A cylinder in an internal combustion engine is the space within which a piston travels. ... piston + connecting rod Components of a typical, four stroke cycle, DOHC piston engine. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... Electrostatics is the branch of physics that deals with the forces exerted by a static (i. ...

However, if the voltage between the plates V is maintained constant by connection to a battery, which is a reservoir for charge at constant potential difference, the force now becomes the negative gradient of the Legendre transform Josephson junction array chip developed by NIST as a standard volt. ... This does not cite its references or sources. ...

The two functions happen to be negatives only because of the linearity of the capacitance. Of course, for given charge, voltage and distance, the static force must be the same by either calculation since the plates cannot "know" what might be held constant as they move. The word linear comes from the Latin word linearis, which means created by lines. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ...

Examples

The exponential function e^x has  x ln x − x  as a Legendre transform since the respective first derivatives e^x and ln x are inverse to each other. This example shows that the respective domains of a function and its Legendre transform need not agree. The exponential function is one of the most important functions in mathematics. ... In mathematics, the domain of a function is the set of all input values to the function. ...

Similarly, the quadratic form In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

with A a symmetric invertible n-by-n-matrix has In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

as a Legendre transform.

Legendre transformation in one dimension

In one dimension, a Legendre transform to a function f : R → R with an invertible first derivative may be found using the formula

This can be seen by integrating both sides of the defining condition restricted to one-dimension

from x₀ to x₁, making use of the fundamental theorem of calculus on the left hand side and substituting The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

on the right hand side to find

with f*′(y₀) = x₀, f*′(y₁) = x₁. Using integration by parts the last integral simplifies to In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

Therefore,

Since the left hand side of this equation does only depend on x₁ and the right hand side only on x₀, they have to evaluate to the same constant.

Solving for f* and choosing C to be zero results in the above-mentioned formula.

Geometric interpretation

For a strictly convex function the Legendre-transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (For a function of one variable, the tangents are well-defined at all but at most countably many points since a convex function is differentiable at all but at most countably many points.) In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... In mathematics, a countable set is a set with the same cardinality (i. ... In mathematics, a derivative is the rate of change of a quantity (e. ...

The equation of a line with slope m and y-intercept b is given by Look up Slope in Wiktionary, the free dictionary. ... The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis of the coordinate system. ...

.

For this line to be tangent to the graph of a function f at the point (x₀, f(x₀)) requires

and

f' is strictly monotone as the derivative of a strictly convex function, and the second equation can be solved for x₀, allowing to eliminate x₀ from the first giving the y-intercept b of the tangent as a function of its slope m:

Here f* denotes the Legendre transform of f.

The family of tangents of the graph of f is therefore (parameterized by m) given by In mathematics, an index set is another name for a function domain. ...

or, written implicitly, by the solutions of the equation

The graph of the original function can be reconstructed from this family of lines as the envelope of this family by demanding The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. ...

Eliminating m from these two equations gives

Identifying y with f(x) and recognizing the right side of the preceding equation as the Legendre transform of f* we find

Legendre transformation in more than one dimension

For a differentiable real-valued function on an open subset U of Rⁿ the Legendre conjugate of the pair (U, f) is defined to be the pair (V, g), where V is the image of U under the gradient mapping Df, and g is the function on V given by the formula In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ...

where

is the scalar product on Rⁿ. In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...

Alternatively, if X is a real vector space and Y is its dual vector space, then for each point x of X and y of Y, there is a natural identification of the cotangent spaces T*X_x with Y and T*Y_y with X. If f is a real differentiable function over X, then ∇f is a section of the cotangent bundle T*X and as such, we can construct a map from X to Y. Similarly, if g is a real differentiable function over Y, ∇g defines a map from Y to X. If both maps happen to be inverses of each other, we say we have a Legendre transform. The fundamental concept in linear algebra is that of a vector space or linear space. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

Further properties

In the following the Legendre transform of a function f is denoted as f*.

Scaling properties

The Legendre transformation has the following scaling properties:

It follows that if a function is homogeneous of degree r then its image under the Legendre transformation is a homogeneous function of degree s, where 1/r + 1/s = 1. In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...

Behavior under translation

Behavior under inversion

Behavior under linear transformations

Let A be a linear transformation from Rⁿ to R^m. For any convex function f on Rⁿ, one has In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

where A* is the adjoint operator of A defined by In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

A closed convex function f is symmetric with respect to a given set G of orthogonal linear transformations, In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...

if and only if f* is symmetric with respect to G.

Infimal convolution

The infimal convolution of two functions f and g is defined as

Let f₁, …, f_m be proper convex functions on Rⁿ. Then

See also

• Projective duality

In the geometry of projective spaces, including the projective plane, duality concerns the interchangeability between points and lines which preserves incidence properties. ...

References

• Arnol'd, Vladimir Igorevich (1989). Mathematical Methods of Classical Mechanics (second edition). Springer. ISBN 0-387-96890-3.
• Rockafellar, Ralph Tyrell (1996). Convex Analysis. Princeton University Press. ISBN 0-691-01586-4.