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Classical Mechanics Goldstein 2.16

  1. Sep 14, 2014 #1
    1. The problem statement, all variables and given/known data

    In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion for the lagrangian ##L = e^{γt} (\frac{m\dot{q}^2}{2} - \frac{kq^2}{2})##. How would you describe the system? Are there any constants of motion? Suppose a point transformation is made of the form ##s = e^{γt/2}q##. What is the effective Lagrangian in terms of ##s##? Find the equation of motion for ##s##. What do these results say about the conserved quantities for the system.

    2. Relevant equations

    Lagrange-euler equation ##\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0##

    3. The attempt at a solution

    Using the lagrange-euler equation, I came up with one equation of motion ##m\ddot{q}+mγ\dot{q}+kq = 0##

    Is this correct? I'm not sure where to go from here. Can I solve this for ##q## like a normal second order differential equation? Can I treat ##γ## as a constant knowing nothing about it? If so, the factoring doesn't seem to come out right. (let ##y=e^{rt}##, ##my^2+mγy+k = 0##, then ?)

    Also, I'm not really sure what is meant by a point transformation. Does that simply mean replace that factor in the original lagrangian?
     
  2. jcsd
  3. Sep 14, 2014 #2

    BvU

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    Hello PM, and wecome to PF.

    I dug up your exercise (2.14) in the second edition and there it has ##s = e^{γt}q##. Would that have been changed ?

    The EL eqn gives you the equation of motion, which I think you found correctly.

    Note the exercise wants you to write out the effective Lagrangian for ##s## and then find the equation of motion for ##s##.

    Oh, and: yes ##\gamma## is a constant.
     
    Last edited: Sep 14, 2014
  4. Sep 14, 2014 #3
    Thanks BvU. I should have mentioned I have the third edition. The problem shows in this book exactly as I wrote it. I guess I'm stuck on the first part of the problem. I don't know how to interpret equation of motion from the lagrangian without solving the differential equation, and the equation I've come up with is unfactorable.
     
  5. Sep 15, 2014 #4

    vela

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    You should recognize this differential equation, particularly if ##\gamma=0##. It might also help to rewrite it as
    $$m\ddot{q} = -m\gamma \dot{q} - kq$$ or perhaps use ##x## instead of ##q## so it looks more familiar.
     
  6. Sep 15, 2014 #5

    BvU

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    Dear PM, what I meant to express is that you already found the equation of motion (there is only one coordinate, so there is only one equation in this exercise). It looks a lot like F = ma, which is not a coincidence. F consists of two terms, one (-kq) having to do with displacement from q = 0, the other something like -##\beta \dot q## (## \beta = \gamma ##). You are not asked to solve, but to describe the system. How would you describe the system if ##\gamma=0##? What could the other term represent ?

    And yes, Herbie wants you to rewrite L(q) as L(s). Work it out and show us...
     
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