Double Pendulum Chaos Light Writing (computer simulation) 1

[nickrohn93]

Well the math looks simple enough.

[LLit11]

is there a common theme in diffusion/mixture and chaotic behaviour?

[yuumeko]

everyone in the comments must be physicists

[Mathview]

Nice job on the double pendulum problem. Animation tells a fascinating story. Q1: What can you say about the onset of chaos and the island overlap criterion for this problem? Q2: What happens as you increase the number of oscillators, dimensions? Q3: Continuum limit? And a numeric question, if you reverse time in the final state, how accurately do you recover the initial state?

[mywickedchicken]

The Chaos Theory is fascinating. I just find it incredible how an orderly system that is completely balanced results in such disorder... and to think of how this influences the systems in our everyday lives!

[x89codered89x]

What software did you use to solve this?

Matlab/Simulink?

[khyar]

@x89codered89x I wrote everything from scratch in VB. NET. The equations were numerically integrated using the standard Runge-Kutta 4th order method.

[povidlo42]

@khyar RK4 is not a good idea for these chaotic simulations. It is much better to use some symplectic numerical scheme (e.g. Gauss-Legendre). Though you would not notice the difference in this animation, it would give you much more realistic Poincare sections...

[x89codered89x]

What software did you use to simulate this?

MATLAB/ SIMULINK?

[axllht]

can you simulate it in 3d? :)

[khyar]

@axllht Yes, this could be extended to 3D. Though the point of this video was to show how chaos can emerge in even the simplest and constrained non-linear systems. If the Universe was flat, it could still exhibit chaos!

[axllht]

@khyar this is looking so natural, like live's events

[overblast0000]

you say masses are always in pairs but shouldn't the lengths also be in pairs? in ratios is what i mean

[khyar]

@overblast0000 Looking at the equation for the angular velocities, the masses only appear in ratios (m1/m2) whereas the lengths appear both in ratios (L01/L12), (L12/L01) as well as (g/L01) and (g/L02). The rest of the terms in the equations are all related to the state variables (omega, theta, and omega_dot). This reveals how only three things control the dynamics of the system: the ratio of the pendulum bob masses, the ratio of the pendulum arm lengths, and the ratio of gravity to arm lengths.

[mariosoulbro1]

:O wow!!

[haj1111]

Interesting simulation with amazing results..

[MJsDisciple]

Haha, you make it sound so simple. But I do follow your methodology. I guess the idea is to not look at the big picture all at once but try to break down the problem in to chunks and go from there... I'll take that and use it. Thanks! Keep up with the vids.

[MJsDisciple]

Must say that this is very impressive. I'm an engineering student and find my mathematical abilities lack. How did you go from doing basic math to coming up with a formula that captures the movement of a double pendulum? I dunno, I can't imagine myself being able to do something of that calibur. Have you always been good at mathematics or did you just work at it from a young age. Thanks for the response, ur other vids are cool also.

[khyar]

@MJsDisciple Thanks a lot! Actually no, maths only "clicked" with me when I started university. I got a D grade in applied maths, didn't get the grades to get in to my first choice uni, only just got in to my second choice. That scared me a lot, so I told myself to sort it out or I will never get to do what I want. I was determined to grasp the subject and that happened during the 1st year of uni, the "maths barrier" broke!

[khyar]

[continued] Some fundamental pieces of the maths puzzle that had been missing for years were filled in during the 1st year lectures. Rather than learning to pass tests, I was learning to solve useful problems (i.e. really thinking about it because there was no answer in the back of the book).

Ultimately, mechanics boils down to the application of a set of rules: 1. Draw large, clear, labelled diagram (it is amazing how this step alone improves clarity of thought)

[khyar]

[continued 2]

2. Write down Newton's 2nd law for each body, F=ma (sum of external forces = mass * acceleration of body), for a pendulum the forces are rod tension and weight. Recall that force is a vector, so split it up into its x, y components.; 3. kinematic constraints - how are the bodies' positions, velocities and accelerations related to eachother; 4. solution of simultaneous equations to get accelerations 5. Integration of accelerations to get velocities and again to get positions.