6.6: Hamilton's Use of the Legendre Transform

We have the Lagrangian $\begin{equation} L\left(q_{i}, \dot{q}_{i}\right) \end{equation}$, and Hamilton's insight that these are not the best variables, we need to replace the Lagrangian with a closely related function (like going from the energy to the free energy), that is a function of the qi (that's not going to change) and, instead of the $\begin{equation} \dot{q}_{i} \text { 's, the } p_{i} \text { 's, with } p_{i}=\partial L\left(q_{i}, \dot{q}_{i}\right) / \partial \dot{q}_{i} \end{equation}$. This is exactly a Legendre transform like the one from $\begin{equation} f \rightarrow g \end{equation}$ discussed above.

The new function is

$$\begin{equation} H\left(q_{i}, p_{i}\right)=\sum_{i=1}^{n} p_{i} \dot{q}_{i}-L\left(q_{i}, \dot{q}_{i}\right) \end{equation}$$

from which

$$\begin{equation} d H\left(p_{i}, q_{i}\right)=-\sum_{i} \dot{p}_{i} d q_{i}+\sum_{i} \dot{q}_{i} d p_{i} \end{equation}$$

analogous to $\begin{equation} d F=-S d T-P d V \end{equation}$ This new function is of course the Hamiltonian.