Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sign of Hamiltonians

  1. Mar 13, 2014 #1
    How do you decide on the sign of a Hamiltonian function?

    For example, I have the following system of differential equations:

    [itex]x'=y[/itex]
    [itex]y'=-\dfrac{3}{2}x^{2}-2x[/itex]

    With the following Hamiltonian:
    [tex]H^{\oplus}=\dfrac{1}{2}x^{3}+x^{2}+\dfrac{1}{2}y^{2}[/tex]

    because [itex]\dfrac{dH^{\oplus}}{dt}=0[/itex]. But if [itex]\dfrac{dH^{\oplus}}{dt}=0[/itex] then [itex]\dfrac{dH^{\ominus}}{dt}=0[/itex] is also true.

    We can write [itex]H=U\left(x\right)+T\left(v\right)[/itex] with [itex]v=y[/itex]. We can then use [itex]U\left(x\right)[/itex] to construct the phase portrait of the system of differential equations.

    With the use of Matlab I created the phase portrait of the system and it is obvious that the Hamiltonian with the positive sign leads to the correct plot. My question now is how do I know which Hamiltonian I should use?

    Plot of [itex]U^{\oplus}[/itex]:
    hpos.png

    Plot of phase portrait:
    untitled.png
     
    Last edited: Mar 13, 2014
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Sign of Hamiltonians
Loading...