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Lagrangian dynamics

  1. Apr 19, 2017 #1
    1. The problem statement, all variables and given/known data
    New Bitmap Image (2).jpg

    2. Relevant equations
    The last part of this question is an example of this result:
    C.jpg

    3. The attempt at a solution
    Here is the solution
    a.jpg

    I think L' is missing a term: If we take the Earth as your frame of reference.(i.e. You are stationary, watching the movement of the railway carriage).Then there should be an extra term for the KE of pendulum,due to horizontal movement of the carriage. (see below, the y dot term corresponds to v of the carriage)
    b.jpg

    Why is it not involved in L' ?
     
  2. jcsd
  3. Apr 19, 2017 #2

    Orodruin

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    Your constraint equation is ##\dot y = v(t)##. This implies that the term you are referring to only depends on t and therefore is a total derivative (and hence irrelevant for the equations of motion).
     
  4. Apr 19, 2017 #3
    Sorry, could you explain it in more detail? I couldn't get it
     
  5. Apr 19, 2017 #4

    Orodruin

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    Which part do you have trouble with? There is only one degree of freedom for the pendulum, the angle ##\theta##. The final Lagrangian cannot depend on ##y## since the motion of ##y## is given by integrating ##v(t)##. The constraint gives you ##\dot y = v(t)## and so ##y(t)## is a primitive function of ##v(t)##. Inserting the constraint into your Lagrangian means your ##\dot y^2## term turns into
    $$
    \frac{\mu v(t)^2}{2}.
    $$
    This is a function of ##t## only and does not affect the equations of motion.
     
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