Plotting Phase Space for Double Pendulum

[Robben]

Homework Statement

For a double pendulum, how do we plot the phase space for $\theta_2$ (the lower of the pendulum), i.e. the plot $\theta_2, \ \dot{\theta}_2?$

• $x$ = horizontal position of pendulum mass
• $y$ = vertical position of pendulum mass
• $\theta$ = angle of pendulum (0 = vertical downwards, counter-clockwise is positive)
• $L$ = length of rod (constant)

Homework Equations

F = ma
$x_1 = L_1\sin \theta_1$
$y_1 = L_1\cos \theta_1$

(the $_1$ subscript is the upper pendulum while $_2$ is the lower pendulum)

The Attempt at a Solution

I found the equation for $\theta''_2$ (which is pretty long to write in here, but I will write if you guys want me to) and I converted the second order equation into a first order, by substituting $\theta''_2$ to $\omega'_2$, i.e. $\omega'_2 = \theta''_2$, but I am wondering what exactly is $\theta_2$ suppose to be here?

Will our $\theta_2$ just be the $\theta_2$ in $x_2 = x_1 + L_2 \sin\theta_2$?

 

[BvU]

Can you clarify the question ? I mean, the answer "the same way as for $\theta_1$" is probably not what you are looking for...

Idem for "yes" to the question "will our $\theta_2$ just be the $\theta_2$ in $x_2 = x_1 + L_2\;\sin\theta_2$ ?"

What is it that we can do for you ? You are aware that there is no analytic solution for this seemingly simple device (see wiki) ?

 

[Robben]

Can you clarify the question ? I mean, the answer "the same way as for $\theta_1$" is probably not what you are looking for...

Idem for "yes" to the question "will our $\theta_2$ just be the $\theta_2$ in $x_2 = x_1 + L_2\;\sin\theta_2$ ?"

What is it that we can do for you ? You are aware that there is no analytic solution for this seemingly simple device (see wiki) ?

The link that you provided has a good picture. I want to plot a graph of the angular velocity and the $\theta_2$ that is given in the picture of the wiki link you provided. But how do I find $\theta_2$ and $\dot{\theta_2}$?

 

[BvU]

Same link says

$\theta_1 =$ (... complicated expression), $\theta_2 =$ (idem) ...(etc) are explicit formulae for the time evolution of the system given its current state. It is not possible to go further and integrate these equations analytically, to get formulae for θ₁ and θ₂ as functions of time. It is however possible to perform this integration numerically using the Runge Kutta method or similar techniques.

See also wolfram alpha

 

[Robben]

Same link says See also wolfram alpha

In the link, it provided $\dot{\theta_2}$ so in order to find $\theta_2$ I have to integrate $\dot{\theta_2}$ by using Runge Kutta?

 

[BvU]

The expression for $\dot\theta_2$ happens to contain all four time-dependent variables ( $\theta_1,\; \theta_2,\; p_{\theta_1}, \; p_{\theta_2}\;$), so you will have to provide initial values for all four of them and then integrate all four of them...

It's a bit of work, but then you'll get the nicest phase space plots corresponding to the spectacular animated pictures in the link !

 

[Robben]

Oh wow, that will be a lot of work. So that's the only option huh. What will be good initial conditions for them?

 

[BvU]

Check with the animated pics in the link ! But $\pi/2, \pi/2, 0,0$ seems to be one of them and it sure rocks !

PS plotting while stepping is much more attractive than calculating the whole thing first and then producing a still picture. It will be chaotic anyway, most of the time.

 

[Robben]

Thank you very much!