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Hello I am trying to work through a little proof of the symplectic condition for Hamilton's equations for a classical mechanics course. I am trying to understand the meaning of the relation $\textbf{M}^T\textbf{J}\textbf{M}=\textbf{J}$.

This is what I have so far. \begin{equation} -\textbf{JM}=(\textbf{J})^T(\textbf{M}^{-1})^{-1} \end{equation} \begin{equation} \textbf{JM}=\textbf{J}(\textbf{M}^{T})^{-1} \end{equation} \begin{equation} \textbf{M}^T\textbf{JM}=\textbf{J} \end{equation}

I am essentially just playing around with the rules $-\Omega=\Omega ^{-1}=\Omega ^T$ for both $\textbf{M}$ and $\textbf{J}$ respectively. Is this correct that they both have these properties? Also as a bit of an aside if $\textbf{J}$ is a block matrix what is the significance of the matrix $\textbf{M}$, what is it called?

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In the context of a canonical transformation (CT) $$z^I~=~(q^i,p_i)~\longrightarrow ~(Q^j,P_j)~=~Z^J~=~f^J(z,t),$$ the matrix $$\textbf{M}^J{}_I~:=~\frac{\partial Z^J}{\partial z^I}$$ is the Jacobian matrix of the CT. Here the indices

$$i,j~\in~\{1,\ldots, n\} \quad\text{and}\quad I,J~\in~\{1,\ldots, 2n\}.$$

If the CCR reads

$$ \{z^I,z^J\}_{PB}~=~\textbf{J}^{IJ}~=~\{Z^I,Z^J\}_{PB}, $$

then it follows that

$$ \textbf{M}\textbf{J}\textbf{M}^T~=~\textbf{J}.$$

If $\Omega_{IJ}$ denotes the inverse matrix of $\textbf{J}^{IJ}$, then

$$ \textbf{M}^T\Omega\textbf{M}~=~\Omega.$$

References:

  1. H. Goldstein, Classical Machanics.
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