Legendre transformation

In physical problems, the Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively).

In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.

For sufficiently smooth functions on the real line, the Legendre transform

For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted

The Legendre transformation is an application of the duality relationship between points and lines.

In some cases (e.g. thermodynamic potentials, below), a non-standard requirement is used, amounting to an alternative definition of f * with a minus sign,

In analytical mechanics and thermodynamics, Legendre transformation is usually defined as follows: suppose

For x* fixed, x*x − f(x) is continuous on I compact, hence it always takes a finite maximum on it; it follows that the domain of the Legendre transform of

The Legendre transform is linked to integration by parts, p dx = d(px) − x dp.

Assume that the function f is convex in x for all y, so that one may perform the Legendre transform on f in x, with p the variable conjugate to x (for information, there is a relation

The function −g(p, y) is the Legendre transform of f(x, y), where only the independent variable x has been supplanted by p. This is widely used in thermodynamics, as illustrated below.

; the inner product used to define the Legendre transform is inherited from the pertinent canonical symplectic structure.

In this abstract setting, the Legendre transformation corresponds to the tautological one-form.

[further explanation needed] 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.

For example, the internal energy U is an explicit function of the extensive variables entropy S, volume V, and chemical composition Ni (e.g.,

(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 U with respect to volume V, the enthalpy H may be obtained as the following.

The non-standard Legendre transform here is obtained by negating the standard version, so

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

The force F between the plates due to the electric field created by the charge separation is then

If the capacitor is not connected to any electric circuit, then the 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 electrostatic potential energy as

An important application of the rate function is in the calculation of tail probabilities of sums of i.i.d.

A simple theory explains the shape of the supply curve based solely on the cost function.

For a differentiable real-valued function on an open convex 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

The multidimensional transform can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.

But the definition of Legendre transform via the maximization matches precisely that of the support function, that is,

If f is a real differentiable function over X, then its exterior derivative, df, is a section of the cotangent bundle T*X and as such, we can construct a map from X to Y.

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 convexity).

is strictly convex and bounded below by a positive definite quadratic form minus a constant, then the Legendre transform

A closed convex function f is symmetric with respect to a given set G of orthogonal linear transformations,