41–50 of 55 posts
The first five chapters are about physical systems like this pendulum while the remaining chapters are mostly AI related.
The book use Processing but the concepts are language agnostic: http://natureofcode.com/book/
Interesting thing about the double pendulum - it is very difficult to get the numerical integration correct, even using very high order integrators like RK4. The simple Euler integrator used here has no hope of getting it right (not a criticism, just an observation!) The problem is that the double pendulum equations (in fact, any equations of motion derived from a Hamiltonian) have symplectic structure, i.e. they hav…
See this thesis: http://umu.diva-portal.org/smash/get/diva2:140361/FULLTEXT01...
the basis for this awesome physics simulator: http://www.algodoo.com/
Conservation of the hamiltonian with leapfrog integration is also the principle behind Hybrid Monte Carlo methods http://www.mcmchandbook.net/HandbookChapter5.pdf
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Me too. Lagrangian mechanics are almost the only thing about my physics undergrad where I still recall thinking "wow this is awesome!"
Where can I find a good introduction to Lagrangian mechanics?
Interesting thing about the double pendulum - it is very difficult to get the numerical integration correct, even using very high order integrators like RK4. The simple Euler integrator used here has no hope of getting it right (not a criticism, just an observation!) The problem is that the double pendulum equations (in fact, any equations of motion derived from a Hamiltonian) have symplectic structure, i.e. they hav…
Leapfrog integration will also get you conservation of energy. See this thesis: http://umu.diva-portal.org/smash/get/diva2:140361/FULLTEXT01... the basis for this awesome physics simulator: http://www.algodoo.com/ Conservation of the hamiltonian with leapfrog integration is also the principle behind Hybrid Monte Carlo methods http://www.mcmchandbook.net/HandbookChapter5.pdf
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I did basically this in my first year of college! See [1]. (Note: code was never intended to be visible to the public, and this is old and bad. But hey, it works!) http://kevingibbons.org/doublependulum.html
Your simulation has a "Daming factor"
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Me too. Lagrangian mechanics are almost the only thing about my physics undergrad where I still recall thinking "wow this is awesome!"
Where can I find a good introduction to Lagrangian mechanics?
I learned from a less established text by Taylor: http://www.amazon.com/Classical-Mechanics-John-R-Taylor/dp/1...
Looks unavailable, but a PDF is very clearly the second result on Google for "Classical Mechanics Taylor" so it may be a good option for the tight budget :)
Cool. Feature request: mount the pivot on a programmable cart. It'd be fun to write software to balance the double pendulum. e.g. https://www.youtube.com/watch?v=B6vr1x6KDaY and, balancing a triple pendulum: https://www.youtube.com/watch?v=cyN-CRNrb3E
I hacked together a WebGL version here:
https://dl.dropboxusercontent.com/u/1109/dp/index.html
Lots of fun - thank you.
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For me, the most interesting part of Chaos is that it is deterministic.
Meh, it's mathematically deterministic. In reality there is always error (modelling error, measurement error, simulation error) which means that chaotic systems are effectively random on long time scales, even though they are technically deterministic.
Another way of looking at real systems, is that the system is deterministic and the errors are random.