# What is a point transformation?

This problem comes from Goldstein.

What does $s=e^{\gamma t}q$ mean? Do I just put $q=e^{-\gamma t}s$ into the Lagrangian?

But I don't know what that means.

I think the point transformation may relate canonical transformation from wikipedia, but this is not enough to understand it. Could you give me an advice to learn about this?

In this context, it is a change of variables. The variable in the original Lagrangian is $q$, and Goldstein is asking you to use another variable $s$, which is related to the original $q$ via the "transformation": $$s = \exp(\gamma t) \ q$$ and later on, make sense of it (with the later questions). Point transformation in this context refers merely to this co-ordinate transformation. If you are seeking to understand from the point of view of canonical transformations, these point transformations are transformations of the adopted generalized co-ordinates. These would form a subset of canonical transformations, since this change of co-ordinates $q \rightarrow s$ would be accompanied by a corresponding change in the generalized momenta (via the Legendre transformation), and hence, will not affect the Hamilton's equations in this context.