Legendre Transformation - Legendre Transformation in One Dimension

Legendre Transformation in One Dimension

In one dimension, a Legendre transform to a function 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 to, making use of the fundamental theorem of calculus on the left hand side and substituting

on the right hand side to find

with . Using integration by parts the last integral simplifies to

$y_1 \, \dot{f}^\star(y_1) - y_0 \, \dot{f}^\star(y_0) - \int_{y_0}^{y_1} \dot{f}^\star(y) \, dy = y_1 \, x_1 - y_0 \, x_0 - f^\star(y_1) + f^\star(y_0).$

Therefore,

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

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

Read more about this topic: Legendre Transformation

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