How to determine whether a given transformation is canonical or not?

QuestionsCategory: Classical MechanicsHow to determine whether a given transformation is canonical or not?
1 Answers
Profile photo of Big Bang BoffinBig Bang Boffin Moderator answered 3 years ago

There are three ways to show that:

  1. 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,
    [Q_i,P_i]_{q_i,p_i}=[Q_i,P_i]_{Q_i,P_i}
    The Poisson Bracket of two variables u and v with respect to q and p is given by,
    [u,v]_{q,p}=\frac{\partial u}{\partial q_{i}}\frac{\partial v}{\partial p_{i}}-\frac{\partial v}{\partial q_{i}}\frac{\partial u}{\partial p_{i}}  
  2. The second method is known as the Symplectic Approach. Where you need to show that the given transformation satisfies the Symplectic Condition given by,
    MJ\tilde{M}=J
    where M is the Jacobian matrix of the transformation with matrix elements,
    M_{ij}=\frac{\partial \zeta_i}{\partial \eta_j}
    where ζ is the new transformed set of variables(basically a column matrix containing Q_i and P_i) and η is the original set of variables(a column matrix containing q_i and p_i) such that \zeta=\zeta(\eta)
    and J is the anti-symmetric 2nx2n matrix composed of four nxn  zero and unit matrices.
    J=\begin{bmatrix} 0 & 1\\-1 & 0 \end{bmatrix}
  3. 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