Runge-Kutta Method for a double pendulum

[heycoa]

Hello, I am trying to program a double pendulum via the 4th order Runge-Kutta method and I cannot seem to be getting the right output. At first I used the Euler-Cromer method, but now I am aiming to make it more accurate.

Homework Statement

I have the equations of motion: d(omega)/dt and d(theta)/dt = omega

also, my step size is "h"

Homework Equations

The Runge-Kutta method can be found here: http://en.wikipedia.org/wiki/Runge–Kutta_methods

The Attempt at a Solution

for omega I tried the following:
k1=d(omega)/dt
k2=d(omega)/dt + 0.5*h*k1
k3=d(omega)/dt + 0.5*h*k2
k4=d(omega)/dt + 0.5*h*k3

omega= omega_initial + (1/6)*h*(k1 + 2*k2 + 2*k3 + k4)

and for theta:
k1=omega
k2=omega + 0.5*h*k1
k3=omega + 0.5*h*k2
k4=omega + 0.5*h*k3

theta = theta_initial + (1/6)*h*(k1 + 2*k2 + 2*k3 + k4)

 

[pasmith]

Hello, I am trying to program a double pendulum via the 4th order Runge-Kutta method and I cannot seem to be getting the right output. At first I used the Euler-Cromer method, but now I am aiming to make it more accurate.

Homework Statement

I have the equations of motion: d(omega)/dt and d(theta)/dt = omega

Homework Equations

The Runge-Kutta method can be found here: http://en.wikipedia.org/wiki/Runge–Kutta_methods

The Attempt at a Solution

for omega I tried the following:
k1=d(omega)/dt
k2=d(omega)/dt + 0.5*dt*k1
k3=d(omega)/dt + 0.5*dt*k2
k4=d(omega)/dt + 0.5*dt*k3

omega= omega_initial + (1/6)*dt*(k1 + 2*k2 + 2*k3 + k4)

and for theta:
k1=omega
k2=omega + 0.5*dt*k1
k3=omega + 0.5*dt*k2
k4=omega + 0.5*dt*k3

theta = theta_initial + (1/6)*dt*(k1 + 2*k2 + 2*k3 + k4)

Not quite; the $k_i$ are vectors with one component for each variable, and you need to compute each component of $k_1$ before computing any components of $k_2$ and so forth. Thus if you have $x = (x_1, \dots, x_N)$ and you wish to solve $\dot x = f(x)$ then you would need (in C):

double k1[N], k2[N], k3[N], k4[N], xTemp[N], xNew[N];

for(i = 0; i < N; i++)
{
   k1[i] = f(xOld)[i];
}

for(i = 0; i < N; i++)
{
   xTemp[i] = xOld[i] + 0.5*dt*k1[i];
}

for(i = 0; i < N; i++)
{
   k2[i] = f(xTemp)[i]; 
}
for(i = 0; i < N; i++)
{
   xTemp[i] = xOld[i] + 0.5*dt*k2[i];
}

with similar loops to calculate $k_3$ and $k_4$.

Of course if you're using a language which understands vector operations then you don't need to expressly loop through the components.

 

[heycoa]

Thank you very much for the response! But I'm not sure I understand. What is N here?

 

[heycoa]

I guess I just don't understand what K is. In my case, is K the slope of d(omega)/dt? So do I have to take the derivative of d(omega)/dt and then calculate the slope at each point?

 

[paranormal]

Hello, it looks like you are on the right track with using the 4th order Runge-Kutta method for your double pendulum problem. However, I noticed a few things that may be causing issues with your output.

First, in your equations for theta, you are using the same values for k1, k2, k3, and k4. This may be causing your results to be incorrect. Instead, for theta, you should use the equations:

k1 = d(theta)/dt
k2 = d(theta)/dt + 0.5*h*k1
k3 = d(theta)/dt + 0.5*h*k2
k4 = d(theta)/dt + h*k3

Then, for your final theta equation, you should use:

theta = theta_initial + (1/6)*h*(k1 + 2*k2 + 2*k3 + k4)

Second, it looks like you are using the same value for d(omega)/dt in each of your k equations for omega. This may also be causing issues with your results. Instead, you should use the equations:

k1 = d(omega)/dt
k2 = d(omega)/dt + 0.5*h*k1
k3 = d(omega)/dt + 0.5*h*k2
k4 = d(omega)/dt + h*k3

Then, for your final omega equation, you should use:

omega = omega_initial + (1/6)*h*(k1 + 2*k2 + 2*k3 + k4)

By using these updated equations, you should be able to get more accurate results for your double pendulum using the Runge-Kutta method. It's also important to double check your initial conditions and make sure they are accurate, as well as your step size, as these can also affect the accuracy of your results.

Good luck with your programming!