1. Limited time only! Sign up for a free 30min personal 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!

Poisson bracets - exam soon

  1. Nov 1, 2012 #1
    Okay there is a particular equation in my book, which I just can't seem to understand intuitively. I've been staring at it for an hour now without progress, so I hope some of you can explain it.
    Basically it's the one on the attached picture.
    Let me introduce the notation so you can help me:
    [itex]\varsigma[/itex] is a vector with the new set of canonical coordinates (Q1,...Qn,P1,...,Pn) which are viewed as function of the old coordinates [itex]\eta[/itex] = (q1,..,qn,p1,...,pn). The matrix poisson bracket [[itex]\varsigma[/itex],[itex]\varsigma[/itex]][itex]\eta[/itex] then comprise the matrix with the following poisson brackets as elements [[itex]\varsigma[/itex]l,[itex]\varsigma[/itex]k][itex]\eta[/itex].
    It should then be intuitive that this can be written as MJMT. Where M is the jacobian matrix with elements Mij = [itex]\partial[/itex][itex]\varsigma[/itex]i/[itex]\partial[/itex][itex]\eta[/itex]j
    How do I realize that?

    Attached Files:

  2. jcsd
  3. Nov 2, 2012 #2
    This is just the definition of Poisson brackets in symplectic notation.I dont think it follows from anywhere.
    I guess,you can explicitly write down the matrices explicitly for one or two independent co-ordinates,write down the matrix J explicitly(as defined in your textbook),and we will see the matrix multiplications grinding out the non symplectic familiar poisson bracket expressions.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook