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

Questions › Category: Classical Mechanics › How to determine whether a given transformation is canonical or not?

Big Bang Boffin Moderator asked 3 years ago

Let’s say we are given a transformation of coordinates:
$Q_i=Q_i(q_i,p_,t)$
$P_i=P_i(q_i,p_i,t)$
Where $Q_i$ & $P_i$ are the new transformed coordinates and $q_i$ and $p_i$ are old(original) coordinates.
Then how do we determine whether this transformation is canonical or not?

Question Tags: Canonical Transformation, Classical Mechanics, Hamiltonian

1 Answers

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Big Bang Boffin Moderator answered 3 years ago

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,
$[Q_i,P_i]_{q_i,p_i}=[Q_i,P_i]_{Q_i,P_i}$
The Poisson Bracket of two variables u and v with respect to q and p is given by,
$[u,v]_{q,p}=\frac{\partial u}{\partial q_{i}}\frac{\partial v}{\partial p_{i}}-\frac{\partial v}{\partial q_{i}}\frac{\partial u}{\partial p_{i}}$
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.
$J=\begin{bmatrix} 0 & 1\\-1 & 0 \end{bmatrix}$
The third less practical way is to find a generating function for the given transformation.