Let’s say we are given a transformation of coordinates:

Where & are the new transformed coordinates and and are old(original) coordinates.

Then how do we determine whether this transformation is canonical or not?

1 Answers

There are three ways to show that:

- Using
**Poisson Brackets**:

The fundamental**Poisson Brackets**of the transformed variables have the same value when evaluated with respect to any*canonical*coordinate set. In other words**the fundamental Poisson Brackets are invariant under canonical transformation.**

Therefore, if the original coordinates are canonical then you just need to prove that,

The Poisson Bracket of two variables u and v with respect to q and p is given by,

- The second method is known as the
**Symplectic Approach**. Where you need to show that the given transformation satisfies the*Symplectic Condition*given by,

where**M**is the Jacobian matrix of the transformation with matrix elements,

where ζ is the new transformed set of variables(basically a column matrix containing and ) and η is the original set of variables(a column matrix containing and ) such that

and**J**is the anti-symmetric matrix composed of four zero and unit matrices.

- The third less practical way is to find a generating function for the given transformation.

Note: The first and second condition imply the existence of a generating function. Also the first two methods are equivalent to each other.

For a detailed analysis and derivations of all these methods I would recommend going through the chapter 9 of Goldstein.

**References:** Goldstein Classical Mechanics 3rd Edition