Solving 2nd Order ODE: r\ddot\theta-g\sin\theta=0

[NeutronStar]

How would I go about finding a solution to this differential equation?

$$r\ddot\theta-g\sin\theta=0$$

Where r and g are constants.

 

[Dr Transport]

Numerically, use Runge-Kutta.

 

[NeutronStar]

Oh pooh,...

That's what I didn't want to hear!

 

[Tide]

Did you intend for the second term to have a plus sign?

IIRC, you should be able to reduce the solution to quadratures expressed in terms of elliptic integrals.

 

[NeutronStar]

Did you intend for the second term to have a plus sign?

IIRC, you should be able to reduce the solution to quadratures expressed in terms of elliptic integrals.

Yes, it should be plus. Sorry about that.

Hey, thanks for the tip about the elliptic integrals! That may be just what I'm looking for!

 

[Clausius2]

How would I go about finding a solution to this differential equation?

$$r\ddot\theta-g\sin\theta=0$$

Where r and g are constants.

If you want to solve the equations of your Lagrange Dynamics problems, you could also post it ¡n that thread you wrote. I didn't mention it to you, but the next step after writing the equations is solving them analytically. The usual assumptions made here by phsicists and engineers are to consider small displacements (i.e $$\theta\rightarrow 0$$). Then you could remove $$sen\theta$$ and $$cos\theta$$ of your equations and made it quasi-linears. Try to go about that, because it is the usual estrategy in Lagrange Dynamic courses.

 

[Galileo]

Such an equation usually appears for oscillating motions.
You can get a reasonably good approximation for small angles, where:

$$\sin \theta \approx \theta$$

and

$$\cos \theta \approx 1-\frac{1}{2}\theta^2$$

 

[dextercioby]

How would I go about finding a solution to this differential equation?

$$r\ddot\theta-g\sin\theta=0$$

Where r and g are constants.

The solution should be sinus amplitudinis,the Jacobi elliptic function.I'm sure of it.

 

[dextercioby]

I was curious as well to learn the solution of the simple pendulum ODE.The best approach i came across is the one in
Derek F.Lawden:"Elliptic Functions and Applications",Springer Verlag,1989,p.114 pp.117.
But the chapters 1 pp.3 (p.1 pp.94) are essential to understanding properly what he's doing when speaking of the simple pendulum.