Solving a Second-Order Differential Equation

[Saketh]

I've been trying to derive the period of a physical pendulum, and I ended up with a differential equation that boils down to this:

$$\frac{d^2 \theta}{dt^2} = C \sin{\theta}$$

where C is some constant.

With no experience in differential equations, I have no idea how to solve this. I can't find anything about how to solve this type of differential equation, so if someone could point me in the right direction, that would be helpful.

Thanks!

 

[StatusX]

It can't be solved nicely, which is why we usually assume the angle is small and make the approximation $\sin \theta \approx \theta$. If you want, there's a way to reduce it to finding the integral of something like [itex]1/\sqrt{a-\cos(\theta)}[/tex], but that's not a nice integral to do (ie, you need special functions).[/itex]

 

[HallsofIvy]

That is an extremely non-linear equation and cannot be solved in any simple form. As StatusX said, for smal values of the angle, you can make the approximation $sin(\theta)= \theta$ so the equation becomes $d^2\theta/dt^2= C\theta$.

Another technique is called "quadrature". Since t does not appear explictly in the equation, let $\omega= d\theta/dt$. Then apply the chain rule: $d^2\theta/dt^2= d\omega/dt= d\omega/d\theta d\theta/dt= \omega d\omega/dt= Csin \theta$. That's a separable first order differential equation which can be written as $\omega d\omega= Csin(\theta)d\theta$ and can be integrated directly:
$(1/2) \omega^2= -Ccos(\theta)+ D$. The rub is when you replace $\omega$ by $d/t\eta/dt$ and solve for $d\theta/dt$: $d\theta/dt= \sqrt{D -2C cos\theta}$ so
$\frac{d\theta}{\sqrt{D- 2C cos(\theta}}= dt$ and the left side cannot be integrated in terms of any elementary function. (That's called an "elliptic" integral. I've seen whole book cases of tables of values.)

 

[Saketh]

Does this mean that it is impossible to generate a theoretical formulation for pendulums that includes large-angle values?

I suspected that I would have to make a small-angle assumption. If I did have $d^2\theta/dt^2= C\theta$, how would I solve that? (Sorry for my lack of experience - once again, I just need to be pointed in the right diretcion.)

 

[HallsofIvy]

For small angles, yes. Of course, in order to get periodic solutions, C must be negative.

 

[daniel_i_l]

what function equals itself times a constant when differentiating twice?
(HINT: what's the derivative of e^x ?)

 

[robphy]

Does this mean that it is impossible to generate a theoretical formulation for pendulums that includes large-angle values?

It's [much] harder... but probably not impossible.

http://scienceworld.wolfram.com/physics/Pendulum.html
http://www.physics.orst.edu/~rubin/nacphy/JAVA_pend/
http://www.math.cornell.edu/~hubbard/pendulum.pdf

 

[arildno]

Does this mean that it is impossible to generate a theoretical formulation for pendulums that includes large-angle values?

I suspected that I would have to make a small-angle assumption. If I did have $d^2\theta/dt^2= C\theta$, how would I solve that? (Sorry for my lack of experience - once again, I just need to be pointed in the right diretcion.)

You can get an approximate solution to this, to whichever degree of accuracy you desire, by perturbation theory.

Alternatively, you may use numerics

 

[Saketh]

what function equals itself times a constant when differentiating twice?
(HINT: what's the derivative of e^x ?)

I don't understand - how does that help me solve the differential equation?

Thanks, everyone! http://scienceworld.wolfram.com/physics/Pendulum.html" answers all of my questions.

 

[HallsofIvy]

what function equals itself times a constant when differentiating twice?
(HINT: what's the derivative of e^x ?)

I can think of 4 such (independent) functions. That's why I said " Of course, in order to get periodic solutions, C must be negative."

 

[tim_lou]

just a curious question, has there ever been a series solution to the exact motion of undamped pendulum?