In physics, mathematical transformations are often used to re-encode information, such as Lagrange transform/Fourier transform/Laplace transform. These changes provide people with a new perspective to understand the world, and also provide convenience for dealing with various physical problems.

This series will introduce the above three mathematical transformations respectively. This section will introduce the transformation of Lerang.

For a given function, the Lagrange transformation can provide better information expression if the following two conditions are met: (a) the function is strictly convex (that is, its second derivative is positive) and smooth enough; (b) Its first derivative can express physical concepts more intuitively or is easier to measure/control.

Used to represent its first derivative:

Construct a new function to

Then satisfy:

Form a pair of transformations.

The figure 1 provide a geometric explanation of equation (3). Represents the negative intercept tangent to the y axis.

Need to pay attention to the following points:

Zia, R.K., Redish, E.F. MacKay, S. R. (2009). Understand Legendre transformation. American Journal of Physics, 77(7), 6 14-622.