Symplectic notation problem

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Homework Statement



My problem is: ``For all eigenvalues [tex]\omega_j[/tex] being distinct show that the normalization of the eigenvectors can be chosen in such a way that M has the properties of the Jacobian matrix.''

Another problem is to show that after this canonical transformation the new Hamiltonian, K, takes the form [tex]K=i \sum_{j=1}^n \omega_j Q_j P_j[/tex]



Homework Equations


[tex]H=\frac{1}{2}\vec{\varsigma}K\vec{\varsigma}[/tex] is given.

With K being a [tex] 2n \times 2n[/tex] matrix with the entries: [tex] \[ \left( \begin{array}{cc}
0 & \tau \\
\vartheta & 0\end{array} \right)\] [/tex]

and [tex]\vec{\varsigma}[/tex] being a 2n-dimensional vector with entries: [tex]\vec{\varsigma}=[\vec q,\vec p]^T[/tex] with [tex]\vec q[/tex] and [tex]\vec p[/tex] being n-dimensional consisting of the generalized coordinates and generalized momenta respectively.
To this there is a matrix M whose columns are eigenvectors of the matrix JK with J being the matrix:
[tex] \[ \left( \begin{array}{cc}
0 & 1 \\
-1 & 0\end{array} \right)\] [/tex]

The corresponding eigenvalues to the eigenvectors are [tex]\pm \omega_j[/tex] .


There should also be an ansatz putting [tex]\varsigma_j = \varsigma_0 e^{i\omega_j t}[/tex]


The Attempt at a Solution


I get stuck at the relations [tex]\dot{\vec\eta}=M\dot{\vec\varsigma}=M\Omega \vec\varsigma[/tex]

With [tex]\dot{\vec\eta}[/tex] being the new coordinates/momenta and [tex]\Omega=diag(i\omega_j)[/tex] is a [tex]2n \times 2n[/tex] matrix.
 

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