Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem,[1] is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function.
This article is about an involution transform commonly used in classical mechanics and thermodynamics. For the integral transform using Legendre polynomials as kernels, see Legendre transform (integral transform).
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 of a function 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
or equivalently, as and in Lagrange's notation.
The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called the Legendre–Fenchel transformation), which can be used to construct a function's convex hull.
Definition[edit]
Definition in [edit]
Let be an interval, and a convex function; then the Legendre transform of is the function defined by
The Legendre transform of a convex function, of which double derivative values are all positive, is also a convex function of which double derivative values are all positive.
Proof. Let us show this with a doubly differentiable function with all positive double derivative values and with a bijective (invertible) derivative. For a fixed , let maximize or make the function bounded over . Then the Legendre transformation of is , thus,by the maximizing or bounding condition . Note that depends on . (This can be visually shown in the 1st figure of this page above.) Thus where , meaning that is the inverse of that is the derivative of (so ). Note that is also differentiable with the ,Thus, the Legendre transformation is the composition of differentiable functions, hence it is differentiable. Applying the product rule and the chain rule with the found equality yieldsgivingso is convex with its double derivatives are all positive.
following derivative (Inverse function rule)
The Legendre transformation is an , i.e., .
Proof. By using the above identities as , , and its derivative ,Note that this derivation does not require the condition to have all positive values in double derivative of the original function .
![{\displaystyle {\begin{aligned}f^{**}(y)&{}=\left(y\cdot {\bar {p}}-f^{*}({\bar {p}})\right)|_{(f^{*})'({\bar {p}})=y}\\[5pt]&{}=g({\bar {p}})\cdot {\bar {p}}-f^{*}({\bar {p}})\\[5pt]&{}=g({\bar {p}})\cdot {\bar {p}}-({\bar {p}}g({\bar {p}})-f(g({\bar {p}})))\\[5pt]&{}=f(g({\bar {p}}))\\[5pt]&{}=f(y)~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e440cae9d7a3bce61655bfea18ef2fdb94db482)
involution
,
,
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As shown above, for a convex function , with maximizing or making bounded at each to define the Legendre transform and with , the following identities hold.
Applications[edit]
Analytical mechanics[edit]
A Legendre transform is used in classical mechanics to derive the Hamiltonian formulation from the Lagrangian formulation, and conversely. A typical Lagrangian has the form