10.1: Point Transformations
- Page ID
- 29456
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It’s clear that Lagrange’s equations are correct for any reasonable choice of parameters labeling the system configuration. Let’s call our first choice \(\begin{equation}
q=\left(q_{1}, \ldots q_{n}\right)
\end{equation}\). Now transform to a new set, maybe even time dependent, \(\begin{equation}
Q_{i}=Q_{i}(q, t)
\end{equation}\). The derivation of Lagrange’s equations by minimizing the action still works, so Hamilton’s equations must still also be OK too. This is called a point transformation: we’ve just moved to a different coordinate system, we’re relabeling the points in configuration space (but possibly in a time-dependent way).