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How is the binomial theorem used here?

  1. Apr 30, 2015 #1
    The below image shows a portion of my current Analytical Mechanics textbook.

    My inquiry is how is the binomial theorem used to get from eq. 3.4.5a ⇒ 3.4.5b ?

    Thanks in advance


    image.jpg
     
  2. jcsd
  3. Apr 30, 2015 #2

    jedishrfu

    Staff: Mentor

    Here's a writeup on the binomial theorem:

    https://en.wikipedia.org/wiki/Binomial_theorem

    and you can see the (x+y)^2 = x^2 +2xy +y^2

    so what is x and what is y in your example?

    work it backwards and forwards and you'll see there's a step they didn't tell you with respect to gamma.
     
  4. May 1, 2015 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In fact, that's simply the factorisation of a quadratic equation. You have to be careful as Operators don't always commute, but in this case, as ##\gamma## and ##\omega_0## are constants, you get the same factorisation as if ##D## were a number.

    (I'm not sure I would call that the Binomial theorem. The Binomial Theorem does not apply for Operators, as they do not generally commute. I would call it the distributive law: which does apply for Operators as well as numbers.)

    As an exercise, you might like to compare:

    ##(x + y)(x - y)## (for numbers)

    and

    ##(X + Y)(X - Y)## (for operators).
     
    Last edited: May 1, 2015
  5. May 1, 2015 #4

    Thank you

    This clarified things
     
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