I have one question regarding Canonical transformation and symplectic matrix. I have read some notions from the following note:
http://www.chim.unifi.it/orac/MAN/node6.html
For me it is not clear how it is obtained the following relation:
$$\dot y = MJM^t \frac{\partial H}{\partial y}.$$
Can you explain me, please.
Update1
What I have tried.
I'll consider the case when $n=2$.
We try to make a transformation $y \to y(x)$.
relation 1(a): $$\dot y = \frac{\partial y}{\partial x} \dot x $$ relation 2(a): $$\dot x = J \frac{\partial H}{\partial x}$$ BUT
relation 3(a): $$\frac{\partial H}{\partial x}=\frac{\partial H}{\partial y}\frac{\partial y}{\partial x}$$
So, we have:
$$\dot y= \frac{\partial y}{\partial x} \dot x = \frac{\partial y}{\partial x} J \frac{\partial H}{\partial x}=\frac{\partial y}{\partial x} J\frac{\partial H}{\partial y}\frac{\partial y}{\partial x}.$$
From here I get stuck.
Looks similar but I don't know how to continue to obtain same form.
I don't know why there appear the transpose of that matrix.
Can you help me to write in coordinates $\displaystyle \frac{\partial y}{\partial x}$.
UPDATE2
from relation 1(a):
$$\dot y = \frac{\partial y }{\partial x} \dot x $$ In matrix notation, for $n=2$, I'll have: relation (1b)
$$\begin{pmatrix} \dot y_{1}\\ \dot y_{2} \end{pmatrix}=\begin{pmatrix} \frac{\partial y_{1}}{\partial x_{1}}& \frac{\partial y_{1}}{\partial x_{2}}\\ \frac{\partial y_{2}}{\partial x_{1}}&\frac{\partial y_{2}}{\partial x_{2}} \end{pmatrix} \begin{pmatrix} \dot x_1\\\dot x_2 \end{pmatrix}.$$
Now, From relation 3(a) I want to obtain in a matrix form relation 3(b):
$$\begin{pmatrix} \frac{\partial H}{\partial x_1}\\ \frac{\partial H}{\partial x_2} \end{pmatrix}=\begin{pmatrix} \frac{\partial H}{\partial y_1}\\ \frac{\partial H}{\partial y_2} \end{pmatrix} \begin{pmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2}\\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} \end{pmatrix}.$$
So,the relation from UPDATE1
$$\dot y=\frac{\partial y}{\partial x} J\frac{\partial H}{\partial y}\frac{\partial y}{\partial x}.$$
will become: $$\begin{pmatrix} \dot y_{1}\\ \dot y_{2} \end{pmatrix}=\begin{pmatrix} \frac{\partial y_{1}}{\partial x_{1}}& \frac{\partial y_{1}}{\partial x_{2}}\\ \frac{\partial y_{2}}{\partial x_{1}}&\frac{\partial y_{2}}{\partial x_{2}} \end{pmatrix} \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}\begin{pmatrix} \frac{\partial H}{\partial y_1}\\ \frac{\partial H}{\partial y_2} \end{pmatrix} \begin{pmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2}\\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} \end{pmatrix} $$
And from here, again, it seems impossible for me... more and more, the matrix multiplication is not possible...
Thanks!