Equations of motion of a double pendulum

[jamg.97]

Hello,

This is my first post on this forum, so please excuse me if I am not clear enough. I have recently been fascinated about chaos and decided to learn about the equations of motion in a double pendulum. I am in high school and have been so interested about chaos and its equations of motion that I have learned many mathematical concepts in order to do the equations. I have understood that chaotic objects can be graphed in a phase diagram and they make strange attractors. In a double pendulum, how can I make a phase diagram for a double pendulum. What would be the values of the x, y, and/or z axes to make the phase diagram. How could I make one in Mathematica? I also have some questions about the equations. This probably is a dumb question. Until how much can you solve the equations? For the kinetic energy, one of the variables is v for velocity. how can I solve for the velocity if I do not have any given time. I know that probably you are supposed to graph it with time being one axis and see how the graph evolves, but I have no idea on how to do this.

links I've used for the equations are
http://people.unt.edu/ctm0055/pendulum
http://scienceworld.wolfram.com/physics/DoublePendulum.html

PLEASE help me as quick as possible. These questions re killing me.

 

[Simon Bridge]

Welcome to PF;
Before dealing with the phase diagrams for chaotic systems - do you know how to draw a phase diagram for just a single simple pendulum?

 

[soothsayer]

In a double pendulum phase space, you typically graph angular momentum, ω vs. angular displacement, θ. This link might help:
http://www.phy.davidson.edu/stuhome/chgreene/chaos/Double Pendulum/phase_diagram.htm

I don't know how to use mathematica, so I can't help you there, but I'm sure someone here can.

In classical physics, I believe the double pendulum is technically completely deterministic for all time, in theory, but in practice, there is no way to predict the system indefinitely. Also, in real life, we can't set initial conditions to arbitrary precision due to quantum effects, so it's hard to say how long we can solve the system. Technically though, even you had enough computing power, you could solve the system for an arbitrary amount of time.

You shouldn't really have v in the kinetic energy equations, the coordinates are polar, so you should have ω, or dθ/dt, often written as \dot{\theta} in Lagrangian/Hamiltonian mechanics, to save space. In the double pendulum equations, you find the equation for \dot{\theta} in terms of the phase displacement, θ. There is no dependence on t, because that would mean the Langrangian, or energy, isn't conserved.