How do you find the Legendre transform of a function?

The Legendre transform is linked to integration by parts, pdx = d(px) − xdp. Assume that it is convex in x for all y, so that one may perform the Legendre transform in x, with p the variable conjugate to x.

Why is Legendre transform convex?

Hence, the Legendre transform of a convex function can be viewed as the ordinate value of the tangent to f(x) of slope p: f⋆(p) = tx0(p)(0). For the same reason, a Legendre transform using the “min” procedure in Eqn. (6) leads to a concave function, even if the function to start with is not concave.

What is Legendre transformation in classical mechanics?

The Legendre transformation connects two ways of specifying the same physics, via functions of two related (“conjugate”) variables. mechanics, the Lagrangian L and Hamiltonian H are Legendre transforms of each other, depending on conjugate variables ˙x (velocity) and p (momentum) respectively.

What is the meaning of Legendre transformation?

A Legendre transform converts from a function of one set of variables to another function of a conjugate set of variables. Both functions will have the same units. The two conjugate pairs of variables are pressure P and volume V, and temperature T and entropy S.

Why is Legendre transform useful?

The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled.

What is the importance of Legendre transformation?

The Legendre transformation is a mathematical concept of great significance to physics. In mechanics and field theory, it provides the transition between Hamiltonian and Lagrangian descriptions, and in thermodynamics it relates the different thermodynamic potentials.

Which is the Legendre transform of a convex function?

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. 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 . {\\displaystyle f^ {\\prime } (x)=p~.}

How is the functional relationship specified in Legendre transformation?

The functional relationship specified by points, or as a set of tangent lines specified by their slope and intercept values. 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 . (by convexity). Therefore, .

Which is the formula for the Legendre transform?

The Legendre transform produces a formula, in terms of p, for a new function g(). The transform is invertible, so knowing g(p) tells you everything about f(x). 2Geometric interpretation of the Legendre transform. Plot f(x). At each point, imagine a line tangent to the plot.

How is the Legendre transformation used in LaGrange notation?

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.