In this lecture, the instructor posed the question to students “How would the system change if we had a large black box cover the central system and one pulley/weight set?”

In this case, the potential energy of the weight and the potential energy of the stretched central system would not be measurable, but it would be possible to measure the change in potential energy of the combination (assuming the force hidden in the box was constant). This led to expression of a quantity $V$, similar to a “free energy” in thermodynamics, expressed as $V=U-x_1F_1$. From this students find that $dV=F_2dx_2-x_1dF_1$, and if $F_1$ was held constant $\Delta V$ becomes an easy to measure quantity whereas $\Delta U$ is not.

A similar quantity $W$ was defined as $W=U-x_2F_2$, this is identical to the previous transform except now $F_2$ is the “special force” hidden inside the box.

The final Legendre Transform presented to students was to define a “potential energy-like” quantity for a system in which both weights were hidden inside the box. This is mathematically useful as it provides another Maxwell relation but is only physically useful if there are more than 2 sets of forces, allowing the system to be manipulated even with $F_1$ and $F_2$ hidden. This transform is given by the expression $Z=U-x_1F_1-x_2F_2$