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Generalized Velocity: Lagrangian

  1. Dec 4, 2017 #1
    1. The problem statement, all variables and given/known data
    Screen Shot 2017-12-04 at 9.58.27 PM.png

    In this example, I know that I can define the horizontal contribution of kinetic energy to the ball as ##\frac{1}{2}m(\dot{x} + \dot{X})^2##.

    In the following example,
    Screen Shot 2017-12-04 at 9.58.16 PM.png

    Mass ##M_{x1}##'s horizontal contribution to KE is defined as ##\frac{1}{2}m(\dot{X} - \dot{x_1})^2##. Why is this? I have a hunch that it is due to the "origin" (##X## line) ##x_1## and ##x_2## originate from, though I can't exactly put my finger on the exact reason.

    Assistance is greatly appreciated!

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Dec 4, 2017
  2. jcsd
  3. Dec 4, 2017 #2

    kuruman

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    It has to do with the way the generalized coordinates are defined. In the top drawing ##\dot{x}## increases to the right (although not clear from the double arrow) so the velocity relative to the ground is ##\dot{X}+\dot{x}##. In the second drawing, ##x_1## increases to the left while ##X## increases to the right, so the relative velocity would be ##\dot{X}-\dot{x_1}##.
     
  4. Dec 4, 2017 #3
    Thanks for the response,

    Would I be right to say that this is also the result of vector addition? Edit: With ##x_1## being defined as positive vector
     
  5. Dec 4, 2017 #4

    kuruman

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    You could say that considering that it involves defining cartesian coordinates relative to origin ##O_1## and then adding a cartesian coordinate to define origin ##O_1## relative to new origin ##O_2##.
     
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