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.