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Classical mechanics

  1. Nov 29, 2009 #1
    1. The problem statement, all variables and given/known data
    The problem is number 3d in the following file:
    http://phstudy.technion.ac.il/~wn114101/hw/wn2010_hw06.pdf [Broken]



    3. The attempt at a solution
    I think the difference comes from the using of the momentum p. In the Jaccobi function, we use only coordinate x and its derivatives.
    Is it correct?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Nov 29, 2009 #2
    And also number 4a ii.
    How Do I do that?
    thank you
     
  4. Nov 30, 2009 #3
    O.k
    I made some progress.
    In 4, i need only help with a)ii
    thanks
     
  5. Nov 30, 2009 #4
    The Euler-Lagrange equations of motion are second order differential equations while the canonical equations of motion are first order differential equations. So in the case of the 1D simple harmonic oscillator (with [itex]m=k=1[/itex]),

    [tex]
    H=\frac{p^2}{2}+\frac{x^2}{2}
    [/tex]

    the canonical equations come from:

    [tex]
    \dot{p}=-\frac{\partial H}{\partial x}=-x
    [/tex]

    [tex]
    \dot{x}=\frac{\partial H}{\partial p}=p
    [/tex]

    so how can you relate these to the Euler-Lagrange equations of motion:

    [tex]
    \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{x}}\right)-\frac{\partial L}{\partial x}=0
    [/tex]

    for the same problem?


    EDIT: Fixed the E-L eom so that its coordinates match the Hamiltonian & canonical equations of motion.
     
    Last edited: Nov 30, 2009
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