Legendre Transform¶

This notebook allows you to create the same plots in the textbook for the function

$$ Y = X^2 + X + C. $$

Under “Functions,” the function $Y$ is defined, as well as $\psi$ (which you have to calculate yourself), the slope ($P$), intercept (which is to give a single point to denote where this happens on the $Y(X)$ plot), along with some functions to draw the tangent lines to the figure. In this case,

$$ P = \frac{dY}{dX} = 2X+ 1, $$

so

$$ \psi = Y – P X = X^2 – C = \left(\frac{P-1}{2}\right)^2 – C. $$

We have written $\psi$ in terms of $P$ and $X$. The former is more correct for the Legendre transform, but the latter is useful when drawing the tangent lines below.

You can skip down to the “Create Plots” section if you just wish to test the code for the given function. In that case, click “Run All” in the Cell menu above (if you are using CoLab, this is in the Runtime menu). After you have run the notebook once, you can then add lines below following the instructions and hit “shift-enter” to create your own plots.

You can go to the Functions section to change the function $Y$ to be used, but consider the comments in each function to be sure to change them all correctly. You might need to change how the plots are made (i.e., the $Y$ and $\psi$ ranges) if you do.

Functions¶

In this section, we have imported libraries that we need in order to simulate these outcomes. If you’re only interested in playing around with the results, just “run” the entire notebook and move to the next section. If you’re interested in coding and wish to modify this, have at it!

Create Plots¶

Here you can run the function

create_plots(X, C)

where X is the point you are interested in finding the value of $\psi$, along with the appropriate P (slope). C is a constant by which the function is shifted.

The function

create_plots2(X, C)

reads in a list of X values and the constant c (as the example shows), and plots several points to denote the appropriate point P on the $\psi$ plot.