How is the binomial theorem used here?

[shanepitts]

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

 

[jedishrfu]

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.

 

[PeroK]

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

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).

 

[shanepitts]

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).

Thank you

This clarified things