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Why is the Harmonic Oscillator so common in physics?

  1. Jul 13, 2012 #1
    I've heard before that it's because when you expand around a minimum point in the potential energy you get a quadratic function, but I can't recall where I read this. Can anyone point me in the right direction, or give their own explanation?

    I only ask because I just solved a problem in my research by assuming that the resistive force to something is F=-kx which of course leads to harmonic oscillation - and this was completely unexpected - nobody knew that the resistive force should be, it just happens that it fits the data bang on.
     
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  3. Jul 13, 2012 #2

    AlephZero

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    When you expand any "smooth" function as a Taylor series about ##x = a## you have
    ##f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2## plus higher powers of ##(x-a)##

    At a minimum (or maximum) the first derivative ##f'(a) = 0## so ##f(x)## is approximately a quadratic.

    Any book on optimisation theory or multi-variable calculus should give the corresponding results for functions of more than one variable.
     
  4. Jul 13, 2012 #3

    haruspex

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    To expand a little on what AlephZero wrote...
    If the potential approximates f(a) + f''(a)x2 then the restorative force, the derivative of the potential, is linear. At an energy minimum, the coefficient is negative, producing SHM.
     
  5. Jul 18, 2012 #4
    That's really helpful guys, cheers.
     
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