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I don't understand conservative systems

  1. Feb 21, 2013 #1
    1. The problem statement, all variables and given/known data

    Look below.

    2. Relevant equations

    So my book says that you have a conservative system when E'(t) = [(1/2)m(x'(t))2 + V(x(t))]'(t) = 0. It gives an example of V(x) = -(1/2)x2 + (1/4)x4, whence setting y = x' gives E = (1/2)y2 -(1/2)x2 + (1/4)x4. According to the book, E = constant. Not sure how they got this since E' ≠ 0 as far as I can tell.

    3. The attempt at a solution


    Me no understand.
     
  2. jcsd
  3. Feb 21, 2013 #2
    I found this document with the same example. As soon as my headache leaves, I read it.
     
  4. Feb 21, 2013 #3

    vela

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    The system obeys Newton's second law, F=ma. It then follows that your expression for E is constant for any V(x). You should be able to show that E' = (ma + dV/dx)v(t), where v(t) is the velocity. Since the force F is given by -dV/dx, the factor in the parentheses vanishes, so E'=0.

    However, I'm guessing your book isn't really expecting you to show that E'=0. It's telling you because the system is conservative, E'=0 so that E is a constant.

    In the xy-plane, the possible behavior of the system is represented by the family of curves that satisfy ##E = \frac{1}{2}my^2 - \frac{1}{2}x^2 + \frac{1}{4}x^4##, where each curve can be associated with a specific value of E.

    Take the example of a free particle, where V(x)=0. Then you have ##y = \sqrt{2E/m} = \text{constant}##. The particle moves with constant velocity.
     
    Last edited: Feb 21, 2013
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