Coordinate conjugate to momentum.

[Petr Mugver]

Let's take a system, for simplicity with only one degree of freedom, described by a certain lagrangian

$$L[x,\dot x]$$

I define the momentum

$$p=\frac{\partial L}{\partial\dot x}$$

Now, if I make a change of coordinates

$$x\longmapsto y\qquad\qquad\qquad(1)$$

I obtain a second lagrangian

$$M[y,\dot y]=L[x(y(t)),\partial_t x(y(t))]$$

and I can define a second momentum

$$q=\frac{\partial M}{\partial\dot y}$$

My question is, if instead of the transformation (1) I want to consider the transformation of the momenta

$$p\longmapsto q\qquad\qquad\qquad(2)$$

How can I find the corresponding transformation (1)? In other words, given that I know how to do (1)-->(2), how can I do (2)-->(1)?

 

[lalbatros]

I think you need to read some material about canonical transformations.
Maybe wiki is a good start:

http://en.wikipedia.org/wiki/Canonical_transformation

Once you are on the starting block, read about generating functions.
The last section "Modern mathematical description" is like a summary.