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Hamilton equation for a block on an inclined plane

  1. Nov 30, 2016 #1
    1. The problem statement, all variables and given/known data
    I am asked to find the Hamilton equations for a block on an inclined plane (no friction)

    2. Relevant equations


    3. The attempt at a solution
    Please ignore the fact that my steps are written in French (sorry!)
    upload_2016-11-30_16-50-45.png
    I am no longer sure of what i'm doing when it comes to finding the momentum. I here chose y' and just ignored the x', because that seemed logical to me at the time. How can I correct my solution if it isn't right?
     
  2. jcsd
  3. Nov 30, 2016 #2

    TSny

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    You should make clear the orientation of the x and y-axes.

    Treat the ##\dot{x}## term similarly to how you treated the ##\dot{y}## term.
     
  4. Nov 30, 2016 #3
    Will I have two different equations for momentum then? And then add them together for a total momentum?
     
  5. Nov 30, 2016 #4

    TSny

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    You will have a differential equation for the x-component of momentum and a separate differential equation for the y-component of momentum.
    Since you didn't state the orientation of your axes, I am having to guess the orientations based on your equations.
     
  6. Nov 30, 2016 #5
    x measured horizontally across the slope, and y measured down the slope
     
  7. Nov 30, 2016 #6

    TSny

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    OK. Thank you.
     
  8. Nov 30, 2016 #7
    I think I understand, I will have to use the definition:
    upload_2016-11-30_18-1-41.png
     
  9. Nov 30, 2016 #8

    TSny

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    Yes. But ##p_i## should not be "dotted" (time derivative) in ## H = \sum p_i \dot{q}_i - L ##.
     
  10. Nov 30, 2016 #9
    This is my final answer:
    upload_2016-11-30_19-34-15.png
     
  11. Nov 30, 2016 #10

    TSny

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    In your second equation for ##L##, there is a sign error in the kinetic energy part. But you corrected it later.

    In your expressions for ##H## you've dropped ##y## from the potential energy part.

    Otherwise, looks OK.

    Are you also supposed to write out the equations of motion?
     
  12. Dec 1, 2016 #11
    Yes. How do we go about doing this?
     
  13. Dec 1, 2016 #12

    TSny

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    You showed at the end of your first post how to get the equations of motion by taking partial derivatives of H. But you have two components of p: px and py.
     
  14. Dec 1, 2016 #13
    So will I derive 3 times, once for px, once for py, and then for theta?
     
  15. Dec 1, 2016 #14

    TSny

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    No, just twice. Theta is not a variable. You could call theta a parameter. Theta is assumed to have a fixed value as the block slides on the incline.
     
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