Using energy conservation for equation of motion

Question

I am trying to get $\phi(t)$ out of this equation:

$$\frac12 J \dot\phi^2 + mgl ( 1- \cos\phi) = E_0.$$

I know it's a common method to say $\sin(\phi) \approx \phi $ and $\cos(\phi) \approx 1 $, but as you can see, this would destroy my whole equation.

I have considered writing $$\cos(\phi) = \sqrt{1-\sin^2(\phi) } = \sqrt{1-\phi^2},$$ but in that case I don't know how to solve the equation

you can expand cos for small theta and write cos(theta)=1-(tetha)^2 — P.A.M, May 31 '15 at 10:09
Separation of variables should work nicely. The integrals might not play nicely. — Sebastian Riese, May 31 '15 at 10:58
I also thought of that, but I don't know how to deal with the quadratic form — Christian, May 31 '15 at 11:06
just move everything else to the r.h.s. and take the sqare root. Then proceed as usual. — Sebastian Riese, May 31 '15 at 11:06

score 0 · Answer 1 · edited Apr 13 '17 at 12:39

Use this expansion: $$ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots $$ If you keep only the quadratic term you recover the simple harmonic oscillator.

(For the sake of completeness the corresponding expansion for the sine is $$ \sin x = x - \frac{x^3}{3!} + \cdots $$ which you can recover from the Euler identity, $e^{ix} = \sum_n \frac{x^n}{n!} = \cos x + i\sin x$.)

You were on the right track with with $\cos x \approx \sqrt{1-x^2}$; the step you are missing to make the integrals calculable is the binomial expansion, $(1+\epsilon)^n \approx 1 + n\epsilon$.

Using energy conservation for equation of motion

1 Answers1