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.