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How Hamilton's equations extremize the action

  1. Jan 18, 2009 #1
    A common exposition of Hamiltonian mechanics (e.g., Goldstein's Classical Mechanics) is to start with the action

    [tex]S=\int{L dt }[/tex]

    then show that the Euler-Lagrange equations extremize the action, then define the hamiltonian in terms of the lagrangian, then translate the Euler-Lagrange equations into the equivalent Hamilton equations.

    But what does it look like using the Hamiltonian from the beginning?

    [tex]S=\int{(\sum{p_j dq_j}-H(p_i,q_i)dt)}[/tex]

    And do we have to rewrite it as

    [tex]S=\int{(\sum{p_j \frac{dq_j}{dt}}-H(p_i,q_i))dt}[/tex]

    to accomplish it? I am eventually going to be relating it to some relativistic problems in which t becomes a coordinate.

    Any info and/or link is appreciated.
  2. jcsd
  3. Jan 18, 2009 #2


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    Just perform an independent variation of coordinates and momenta on the action as you have written it, with [itex] \delta q = 0 [/itex] at the end points, and equate the first order change to zero. It will lead you to Hamilton's equations.
  4. Jan 18, 2009 #3
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