1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Symplectic notation problem

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data

    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]

    2. Relevant 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]

    3. 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.
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?

Similar Discussions: Symplectic notation problem
  1. Problem is. (Replies: 0)

  2. Matrix problem (Replies: 0)

  3. Lagrangian problem (Replies: 0)

  4. MCQ problem (Replies: 0)

  5. Scattering problem (Replies: 0)