Thank you very much for this interesting lecture... Homework (Excercise): In time 27:14: you should write down double pendulum equations of motion (from the Euler-Lagrange or Hamiltonian equations), and use the (classic) RK4 integration scheme for the problem to compute the trajectory of the double pendulum system. Utilizing the trajectory, calculate the system's energy and plot it versus time.
This is, as usual, a fantastic video. Especially the simulations/video clips help the understanding of this fascinating topic. :) However, I have one addition to make, the Lorenz attractor and other chaotic systems aren't necessarily non-deterministic. In fact the lorenz system is an example for a deterministic chaotic system. Simply due to the fact that if you'd knew the initial condition perfectly you could predict your future trajectory with 100% precision. This is a mathematical argument since physics tells us that we can't measure everything perfectly and computer science tells us that we we have rounding errors when storing and simulating data. Again, no offense I enjoy your videos very much! :) But this little detail bothered me a bit.
This was a great journey - I enjoyed this course and learned a lot! Thank you and looking forward to more!
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The double pendulum you showed was pretty nice, and made me think of this triple pendulum that's worth giving a look as well: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-meMWfva-Jio.html
Great lecture, professor! In addition to chaotic, turbulence can be also considered random. This makes me wonder whether there is any relation between chaos and randomness. Could you comment something about this?
Chaos is not random, we have a system of differential equations, everything is deterministic. But if you don't know exactly your initial conditions, then in this kind of systems you won't be able to determine the state of your system arbitraryly at long times
Fascinating video! 1) Interesting to see you mention Jerry Marsden. 2) Have you ever communicated with Jim Yorke? He is a co-author of the paper that introduced the term "chaotic differential equation" and gave a lecture at UWM involving chaos in the double-pendulum. 3) Settle a bet: have you ever learned the Lebesgue integral in your studies? (I'll explain the "bet" later if you like.)
Ok, so if you have a system with control inputs, like the inverted double pendulum, is it possible to apply symplectic integrators? It seems like whenever you have a motor or whatever, you're going to be adding or removing energy, so the fundamental assumption of those integrators is not met. Buuut, you know exactly where and how much energy is being added or removed (i.e. only through the actuators), so maybe it's possibly to exploit that? Like I don't know if this makes any sense, but if you pretended the motor in the double pendulum model had a battery, and added a term for the electric potential to the Hamiltonian?
I’m a high schooler taking calculus 3, and I love your videos. In college I am planning on combining computer science and physics, and your videos solidified this idea 😊
Fascinating lecture. I was one of the last people to walk across the Millennium Bridge before the police stopped people crossing. I did not see anyone all all fours but it was certainly alarmingly wobbly.
This was a great overview video. I was blown away by the metronomes synchronizing. Was scratching my head, on what would be the math, that can explain the eventual trajectory. Also, lots of other examples, relevant to space flight, such the (chaotic) dance amongst planetary bodies.
it might have a small error or i just didnt correctly understand it. About the simpletic integrator, why q = -dh/dp? q is supposed to be the position but take the partial derivative of momentum p.
About the pendulums: if both depends on the initial conditions and you started both with the same initial conditions and their behavior differ, then they don't depend on the initial conditions
The pendulums weren't started in the same conditions. The key here is to show that the smallest difference between initial conditions makes a big difference in the long run in these chaotic systems. Although through the human eye, the pendulums start in the same position and the same conditions, there are small variations (not only position, also drag variations, mechanic losses etc)
Identical twins begin as a single egg with identical DNA. Their lives often re-converge somewhat for a time, but they ultimately live very different lives. The difference in chemistry must be due to dietary and life choices. And many of the limitations in options are due to how finite resources get shared. One wife, for example, can marry only one of the twins. But differences in experiences lead to differences in attitudes and in making different choices even when given similar options.
Dear Professor Brunton, please try to make videos on how to code such question (possibly on MATLAB) and one thing I have notice you didn't talk about much on simulink .so please try to make videos on how to code them NOT THIS LECTURE BASED AS THERE ARE NUMEROUS LITERATURE AVAILABLE ONLINE thank you \
