Canonical Transformations

QuantumDefect

Hello,
Ive looked through a couple books on this subject and found the basic theory but none actually apply it to a problem. I was wondering if someone would be so kind as to maybe do a practice problem for me? The reason I say this is because I have a homework problem and have solved for the hamiltonian and the canonical equations however, I would like to find a new set of co-ordinates in which the momenta might be constants. In my problem I have a co-ordinate velocity dependant potential and would like to find the transformations in which the hamiltonian is a constant. Is this possible? I have just gotten into the Hamiltonian formalism and am extremely excited, more so when I first learned about lagrangian mechanics. Thank you so much.

Physics Monkey

Hi QuantumDefect,

These methods are really cool, aren't they? I could do a problem for you, something simple like the harmonic oscillator. Such a problem is almost certainly worked in greater detail in your book, but if you're actually interested I could step you through it.

Unfortunately, there is no good way to just guess or figure out a canonical transformation that leaves the Hamiltonian independent of the coordinates. Is it possible? Sure. Is it easy to find? Almost never. A general approach starts from the Hamilton-Jacobi equation. The general approach says that the solution to the HJ equation (a non-linear differential equation) is the desired generator of the transformation. This method has the advantage of casting the problem in a form (differential equation) familiar to most physicists, but the problem is equally insoluable from an analytic point of view.

Without some details, I probably couldn't say much more. Feel free to post your problem and we can talk about it.

QuantumDefect

Thanks Physics Monkey!
Thats why none of the books took it into greater depth! I'll read up more on the Hamilton-Jacobi equation and if I have any questions, I'll come back. However, you answered my question and I am extremely grateful.


Many thanks,

~QuantumDefect