Symplectic Condition For Canonical Transformation

[CassiopeiaA]

I am reading Chapter 9 of Classical Mech by Goldstein.The symplectic condition for a transformation to be canonical is given as MJM' = J, where M' is transpose of M. I understood the derivation given in the book. But my question is : isn't this condition true for any matrix M? That is it doesn't matter whether we are talking about canonical transformation or not, any real matrix will satisfy the condition with the given J. And if so then how do I check if the given transformation is canonical without using the Poisson Bracket condition?

 

[vanhees71]

How do you come to that conclusion? Why don't you check the condition for the most simple case of a 2D phase space?

 

[CassiopeiaA]

How do you come to that conclusion? Why don't you check the condition for the most simple case of a 2D phase space?

I took 2X2, 3X3 and 4X4 matrices with unknown variables and did the operation on them. All of them satisfied the condition. I also did a small proof for NXN matrix, that the above condition is true for any matrix.

 

[vanhees71]

Then there must be some misunderstanding on your part. Let's check the 2D case. By definition
$$J=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$$
Now let $M \in \text{Sp}(2)$, i.e.,
$$M J M^T=J.$$
Set
$$M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$
Then the condition for symplecticity is
$$ad-bc=\det{M} \stackrel{!}{=}1.$$
This means for the 2D case the symplectic matrices are the $\mathrm{SL}(2,\mathbb{R})$ matrices, i.e., the matrices with determinant 1, and not all real $2 \times 2$ matrices fulfill this condition.

For more details see

https://en.wikipedia.org/wiki/Symplectic_group