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New to this idea and im tryna understand better: when you say that the lyapunov exponent is .9 for the Lorenz system, I'm a little confused. If you're closer to the trivial equilibrium point of the Lorenz system at <0, 0, 0>, shouldn't the lyapunov exponent be negative for some initial conditions?

Thanks Professor

It was great

As I understand it, Hamilton kinda got his big break as a mathematician/physicist with the Hamiltonian, but as an alternate formalism, it wasn't well known at the time. Quaternions came afterward - which became the precursor for vector analysis, after Oliver Heaviside more or less invented the notion of a vector, which was essentially a quaternion, but instead of the relationship between i,j,k, he took the two parts of the product of quaternions - the scalar part and _the rest,_ i.e. the vector part - and just defined two operations that would return each part when applied to a vector.

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Super content, but also tempo of presentation!

Sir, I have a doubt that has given me a headache. Sir, the non-linear dynamics and chaotic systems you talk about, these are higher level physics and maths concepts, right? I am in 11th grade; are we taught these concepts or not, or are we only taught about simple systems with some assumptions so that the system does not become chaotic, like assuming friction is not present or value of g is constant everywhere? Am I correct, sir?

The universe is probably a fractal on the largest scale with the number of dimensions equal to π.

great tutorial, enjoyed both 3 parts

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Hi great video! Question -- is there transient chaos in the windows of periodicity after around r = 28, or is it only in that first region where we have two fixed points?

Nice!

Unfortunately the link with the Matlab code is dead...😕

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Hello Dr. Ross, is the homework assignments you reference in the videos available?

this is utterly fucking great

Hi Prof. Ross, @38:00, is it implicitly assumed that the center of mass lies on the axis of rotation (omega)? If otherwise, the point G will itself be orbiting around the axis. Or, is the mathematical definition of angular momentum h_G applicable for any arbitrarily chosen body frame origin, whether it lies on the axis of rotation or not? I apologize if this had been discussed in an earlier lecture, in which case please advise - I had skipped a few. Thank you again for a great series.

We can predict outcomes in this case , by applying physics laws😅

I think you miss centripetal acceleration. Imagine the same dynamics, but you're swinging a dumbell on a string. Same stability applies. So the gg frequency should be faster. Gotta go study it a bit. Thoughts?

Hi Prof. at 13:31 what about trajectory that repeat it self after sometime but these trajectory are sensitive to initial conditions, would you call them periodic(because it does repeat) or chaotic (because of sensitivity to IC)? Thanks

Thank you ❤ From Nepal

I have watched/listened to only lectures 12 and 13 in his series yet, but what a sheer joy to go through this material❤❤ Thank you Professor Shane Ross!! I have a quick question. Is this course at the undergraduate or graduate level? My son will be doing his Masters in Aerospace (he did his undergrad in Mechanical Engineering), and I am planning to urge him to go through your lecture series in its entirety beforehand. So, I was wondering if this material would be comfortable for him assuming that he starts from Lecture 1 - my background is EE from many years ago, hence I am unable to make this judgment. Are there prerequisites? Thank you.

Perfect explanations. A great teacher explains why, not what.

Incredible work doctor , the accompanying documentation youve put into your videos and referencing them all together into a cohesive course is invaluable. Much appreciation from a fellow aerospace engineer who needs a refresher on attitude dynamics once in a while. Do you happen to have a book published that might accompany this course and serve as a reference manual?

This is like a GOD! Oh my God, Excellent!

Thank you for making this available!

you can see the distribution is skewed a bit

A wonderful set of lectures! Thank you so much for your work.

Thanks for great lecture, Professor. I need small clarification. In transport theorem, whether the velocity seen by B frame is expressed in B frame or E frame. If it is expressed B frame, how it is equal to velocity seen by E frame expressed in E frame

I am an aerospace masters student trying to specialise in control & simulation and I think these lectures are one of the best! Helped me understand the kinematics and rotation matrices very well! Thanks a lot for these free content!

In the far shots it doesn't look like the station is wobbling, but in the closer shots the station is actually wobbling from the view point of the smaller ship.

Hi! I'm really enjoying your videos! It's really helpful with my studies. I'm wondering what kind of software did you use to plot vector field and see the behavior? Thank you

Nice explanation! Also called EMA exponential moving average.

Great content, professor!

0:25 "people are walking around"? Eh? 😁

Could you recommend a book that has examples of non-canonical Hamiltonian systems? Also theory. Thank you

It was really helpful. Thank you so much

Oh my gosh!! This is wonderful

I'm not seeing any Chinese scientists being protested or going insane or any weird VR devices or infinitely expanded protons How is this related to 3 Body Problem?

gravity cause wobble .. in space i think no wobble

Thanks for the great video-series. I just have a question regarding the MATLAB Kalman filter using both data from the gyro and the accelerometer. If I understand it correctly, z is our measurement. Ignoring gravity, our measurement is omega_1, omega_2 and omega_3 given by the gyro. That is converted to quaternions to be able to write x_{k+1} = A * x_k where x is the state with the 4 quaternions. The kalman filter uses H as the matrix that maps states to measurements. With H = identity(4) means we're measuring directly the quaternions (I'm assuming because we can translate omega vector to quaternions. But when we incorporate the data from the accelerometer, I don't see how this fits into the Kalman filter. If I'm understanding the previous video correctly, we have two new measurements, meaning H should be a 6x6 matrix. Instead, the code seems to simply use roll and pitch obtained from the accelerometers as initial guesses rather than measurements. Could you clarify this? Thanks.

You are great, man!

That was great! I wasn't even looking for that content but I stayed because it was very well delivered and interesting! Thanks!

Really good and simple explanations of complicated stuff. Thanks.

You know what I love? Constant pitch noise. Can't get enough of it. Honestly, it's astonishing more music isn't just one or two tones for 2:30-4:00 minutes. 0/10 thumbs down

Why around 4:05 it was mentioned that Lagranigian is not always T-V? Is it hinting at non-mechanical systems? 2nd question: At around 38:28, it is said that the kinetic energy T can be a function of T(q_i, q_idot). Can you point to a system where T has explicit dependence on q_i. Moreover, at 46:13, we write {\partial L \over \partial q_i} = -{\partial H \over \partial q_i}. If the kinetic energy is a function of q_i in a general case, then {\partial H \over \partial q_i} = - {\partial L \over \partial q_i} + \sum_{i=1}^{n} \dot{q}_i {\partial p_i \over \partial q_i}, where {\partial p_i \over \partial q_i} may not turn out to be zero. So my question essentially is: how is kinetic energy a function of position vector explicitly? Thank you for your time

I am 18 years old and don't know much about these filters, and wanted to make a drone, almost everything was easy but I just couldn't grasp this concept, I think I watched more than 10 videos on kalman filter, nothing got through. But then I found this video, it was really easy to understand. thank u professor for your good work