Legendre transformation

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Legendre transformations are used in thermodynamics, and to derive Hamiltonian mechanics from Lagrangian mechanics.

Let

Substituting

gives

A Legendre transformation is its own inverse, and is related to integration by parts.

Table of contents

1 Convex conjugates
2 Further properties
3 Geometric interpretation
4 References

Convex conjugates

More generally, for a function

one can define the Legendre-Fenchel transform or convex conjugate of f by

where

is the scalar product on Rⁿ.

This mapping has the following scaling properties:

(It follows that if a function f is homogeneous of degree r then the image of f under the Legendre transformation is a homogeneous function of degree s, where 1/r + 1/s = 1.)

Further properties

Further properties:

The convex conjugate of a closed convex function is again a closed convex function. For any proper convex function f and its convex conjugate f^* Fenchel's inequality holds:

For a differentiable function

we can use the derivative of f to find the Legendre-Fenchel transform explicitly. We write

where

If the supremum is finite it has to be a local maximum of h_p which we find by considering the zeroes of the first derivative of h_p with respect to x:

If we can solve the last equation for x (e.g. if f is a convex function and therefore has a strictly monotonic first derivative) we find

where x₀ is the point that maximizes or minimizes h_p. If x₀ in fact denotes a maximum of h_p we find for the Legendre-Fenchel transform of f at p the value

Geometric interpretation

For a convex function f the Legendre-transformation can be interpreted as the mapping between the graph of f and the family of tangents of the graph. (The tangents of f are well-defined almost everywhere since a convex function is differentiable at all but at most countably many points.)

Consider the equation of a line with slope m and y-intercept b:

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

f' is strictly monotone as the derivative of a convex function, and we can solve the last equation for x₀ to find the y-intercept b of the tangent as a function of its slope m:

References

Rockafellar, Ralph Tyrell, Convex Analysis, pp. 251, Princeton University Press (1996). ISBN 0691015864

Arnol'd, Vladimir Igorevich, Mathematical Methods of Classical Mechanics, second edition, Springer (1989). ISBN 0387968903