Coordinate conjugate to momentum.

[Petr Mugver]

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

[tex]L[x,\dot x][/tex]

I define the momentum

[tex]p=\frac{\partial L}{\partial\dot x}[/tex]

Now, if I make a change of coordinates

[tex]x\longmapsto y\qquad\qquad\qquad(1)[/tex]

I obtain a second lagrangian

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

and I can define a second momentum

[tex]q=\frac{\partial M}{\partial\dot y}[/tex]

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

[tex]p\longmapsto q\qquad\qquad\qquad(2)[/tex]

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.