Proving the period of a pendulum

[GetSchwifty]

Homework Statement

I'm writing a small project as part of my degree, and I am stuck on proving one (probably simple!) thing towards the end. I've shown that the pendulum is periodic and has a total period of 4t₂.

Homework Equations

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The Attempt at a Solution

I think I need to rewrite the same ODE with starting point t=0 but I'm having a complete mindblank on how to proceed from here. Any help would be greatly appreciated!

 

[haruspex]

Not entirely sure about your notation, nor what your starting points are.
You seem to be drawing on some earlier equation $\ddot\theta=csin(\theta)$, but what is csin there? Do you just mean a constant c multiplied by sine? "csin" is a standard abbreviation for complex sine in some computer languages.
You also seem to know that $\theta(t_2)=\dot\theta(t_2)=\dot\theta(0)=0$.
Anything else?

 

[GetSchwifty]

n

Hi,

Sorry I should've given more information. The equation I'm drawing on as is as you said is $\ddot\theta = -csin\theta$ and this is just the regular sine function multiplied by a constant, c. No complex sine - sorry for the confusion.[PLAIN][PLAIN]http://blob:http://imgur.com/d719fbec-3e87-4f2c-a28f-955630eb054a [PLAIN][PLAIN]http://blob:http://imgur.com/d719fbec-3e87-4f2c-a28f-955630eb054a [PLAIN][PLAIN]http://blob:http://imgur.com/d719fbec-3e87-4f2c-a28f-955630eb054a [PLAIN][PLAIN]http://blob:http://imgur.com/d719fbec-3e87-4f2c-a28f-955630eb054a

Here is everything I know that's relevant to periodicity in my paper. Thanks for your reply!

 

[haruspex]

The latex is clearer if you use \sin rather than just sin.

As far as I can see, nowhere did you define the initial conditions (t=0). But in the second stage (post #1) you use $\dot\theta(0)=0$.

You have an equation $\theta(t)=-\theta(4t_2-t)$.
What would that imply at t=2t₂? Is the minus sign a mistake?

 

[GetSchwifty]

The latex is clearer if you use \sin rather than just sin.

As far as I can see, nowhere did you define the initial conditions (t=0). But in the second stage (post #1) you use $\dot\theta(0)=0$.

You have an equation $\theta(t)=-\theta(4t_2-t)$.
What would that imply at t=2t₂? Is the minus sign a mistake?

Noted for the future.

The minus sign is a typo, thanks for pointing that out!

At time t=2t₂, we have that it's equal to $\theta(-t)$ by symmetry, since it's the furthest "left" position of the mass (taking moving from left to right as being negative).

 

[GetSchwifty]

I think I need to prove that $\theta(4t_2) = \theta(0)$ but I'm not entirely sure... I think to do that I would need to basically run through a very similar argument again and then use cauchy-lipschitz to show uniqueness

 

[haruspex]

I think I need to prove that $\theta(4t_2) = \theta(0)$ but I'm not entirely sure... I think to do that I would need to basically run through a very similar argument again and then use cauchy-lipschitz to show uniqueness

You need to show that the initial conditions going forward from 4t₂ are the same as those going forward from t=0. So that's the values of $\theta$ and $\dot\theta$.

 

[GetSchwifty]

Thanks. It's very late here in the UK so I'll actually write this up tomorrow - the flu really isn't helping my thought process!

So in short I need to show that $\theta(0) = \theta(4t_2)$ and that $\dot\theta(0) = \dot\theta(4t_2)$? And then use a similar argument from before, from my second post with:

$$\begin{cases}

\ddot y& = -csiny(t)\\

y(t_2)&=-\theta(2t_2-t_2)=-\theta(t_2)=0 \\

\dot y(t_2)&=\dot\theta(2t_2-t_2)=\dot\theta(t_2)

\end{cases}$$

Or am I way off?

And then I would need to change this to reflect 0 and $4t_2$?

$$-\theta(2t_2-t)=\theta(t), t\in[t_2, 2t_2]$$

 

[haruspex]

I need to show that $\theta(0) = \theta(4t_2)$ and that $\dot\theta(0) = \dot\theta(4t_2)$?

Yes.

then use a similar argument

As you wish, but to me it would be suffice to say that the values of $\theta$ and $\dot\theta$ define the state of the system, so if the value pair recurs then the behaviour is cyclic. Likewise, for your antisymmetric result in post #3, I would just observe the effect of time reversal on the differential equation.
These are mechanistic arguments rather than purely mathematical ones.