# Symplectic notation problem

1. Jan 15, 2008

### jarra

1. The problem statement, all variables and given/known data

My problem is: For all eigenvalues $$\omega_j$$ 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 $$K=i \sum_{j=1}^n \omega_j Q_j P_j$$

2. Relevant equations
$$H=\frac{1}{2}\vec{\varsigma}K\vec{\varsigma}$$ is given.

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

and $$\vec{\varsigma}$$ being a 2n-dimensional vector with entries: $$\vec{\varsigma}=[\vec q,\vec p]^T$$ with $$\vec q$$ and $$\vec p$$ 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:
$$$\left( \begin{array}{cc} 0 & 1 \\ -1 & 0\end{array} \right)$$$

The corresponding eigenvalues to the eigenvectors are $$\pm \omega_j$$ .

There should also be an ansatz putting $$\varsigma_j = \varsigma_0 e^{i\omega_j t}$$

3. The attempt at a solution
I get stuck at the relations $$\dot{\vec\eta}=M\dot{\vec\varsigma}=M\Omega \vec\varsigma$$

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