Engineering physics, nonlinear dynamics, chaos, spacecraft dynamics, environmental fluid mechanics, disease spread in hundreds of lectures across several course series and the latest research by Dr. Shane Ross, a professor at Virginia Tech in engineering and mathematics. A world renowned teacher and researcher, he has a B.S. in physics and a Ph.D. in applied mathematics (control and dynamical systems), both from Caltech. He's worked at the University of Southern California, NASA/JPL, UC San Diego, Boeing, and Instituto de Ciencias Matemáticas, ICMAT, in Madrid.
Professor Ross' research group, the Ross Dynamics Lab, specializes in applications of nonlinear dynamics, performing mathematical modeling, simulation, visualization, and experiments with applications including orbital mechanics, and patterns of dispersal in oceanic and atmospheric flows. He's authored a book, more than a 100 scientific journal articles and has been funded by NSF, NASA, USDA, and the US Space Force.
I've made it to Topic 12, and plan on continuing to the end. I just wanted to say that these lectures are great, and I really (raised to the 4th power) appreciate you making these videos. Thank You !!!!!!!!!!!!
Great video! I really like the pacing. I noticed at 11:58 you mentioned that the noise was uniformly distributed, but I think you meant normally distributed with standard deviation of 4? randn.m is ~N(0,I) while rand.m is the uniform distribution for the range [0,1].
this is great! i just defended, and have time to explore areas i have always been curious about but weren't directly applicable to my work. i love the geometry here and am excited to dig into some of this myself <3
I am a very old man and this took me back 70 years ago, when I did both analogue and digital filters using the bilinear transformation to change the frequency domain to a circular one, for digital and switched filters. When I started all this, such along time ago, I used to recognise what a particular filter selects by plotting its Impulse function and then superimpose on it the signal that I want to investigate . Multiplying the two and integrating the product, the value of the Integral would be, how much of the signal is contained in the impulse function of the filter. For a low pass filter the Impulse response had to have a DC level plus an oscillation in it, decaying at a rate to decide the bandwidth. For a band pass filter the impulse response had to have a ringing at the frequency required to pass through and no DC.. For a high pass filter an impulse had to go right through, followed by an inverted version of a low pass filter whose area is equal to the initial impulse, and with a decay duration to decide the bandwidth For a band stop filter, I shall let the readers work out its impulse function! Many years ago I constructed three dimensional wooden models using toothpicks as impulses, to show the Laplace and the Convolution integrals of digital filters, The impulse looks like an exponential helix to chose the frequency and its exponential decay to chose the bandwidth. I still got them both. I should write a book showing how filters should be analysed in three dimensions and use three dimensional signals as V.e^( R+jw)t showing the real, imaginary and time axis. I always found it easier to start with analogue filters, as Butterworth and Chebyshev filter versions, then go to FFR then IIR , and then go to the Kalman filter. and other running filters. Congratulations for your video. Thank you for stirring and jolting my memories of my many years in UK, as a poor man, making ends meet with the little money I had to live on, but very rich in signal processing techniques, as used in communications and automatic state control systems, Thank you Oh, I had no computers to work with in those days and it was all hard ware, When about 50 years ago I had my first home computer I cried when, all I did with filters and signal processing, in both analogue and digital filters, for many years, I wrote in software in a 10 line program! . I also did N- path filters, switch8ing multiple low pass filters to obtain a band pass transfer function. Good old days, I do not think I would like to go back to those days, of so much dedication and concentration to this work, before I got married and had my own family. Sir, may I quote and modify what you said in one comment below, as it also applies to me, " I am just a very old man, who stood on the shoulders of giants and two great parents, and assisted by the company of a good wife for 60 years and children and grandchildren, and six great brothers and sisters and friends and a lot of luck in having good health.."
Dr. Ross, thank you for this great lecture series on an absolutely beautiful subject matter! I want to share one of my struggles, and resolution, for the benefit of fellow distance learners. Initially I thought that phi^dot, the rate of the final spin would be the spin component about the b_3 body frame axis. In which case it should be omega_s, the very premise at the beginning, which does not agree with phi_dot= (1-J/I_0)omega_s as calculated @38.51. But then I realized that after the first spin at rate psi_dot followed by the static tilt theta, there is already a spin component about the b_3 axis in the body frame, equal to psi_dot x cos(theta) = omega_0 x cot(theta) = (J/I_0) x omega_s. So, phi_dot is a (positive or negative) "correction" to bring it up/down to omega_s. But the net spin and precession are both always positive, for the prolate as well as oblate cases (?); i.e., the angular velocity as well as angular momentum are pointing towards the "northern hemisphere" for both cases. This can be seen in the video as well as the Wolfram demo in the next lecture. I will appreciate your advise if my assessment is incorrect. Thank you!
Bravo. I asked a question in a recent online math lecture; a mathematician in the chat pointed me to Limit Cycles. I appreciate Dr. Ross's examples of heart beats and walking. About 10 years ago, I stumbled across the interesting lateral rope technique: David Weck's "Dragon Roll" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-J25_41nPFLo.html . The Dragon Roll orbit is clearly a Limit Cycle; it follows Viviani's Curve over and around the body. When you change to orbit in the opposite direction, there are 3-4 cycles where you home in on the new stable orbit. The same thing happens with walking or running: a change in terrain or an obstacle will perturb the cycle, but you will rapidly find the nominal orbit. I abstractly understood the principle, but I couldn't get any further information until I got the name of the term and searched on it. Thanks for publishing this video, Dr. Ross.
To be even more difficult about the topology (I know no topology) not every family of (n-1) spheres gives us an n sphere. I can think of conic sections, parabolas and a Gaussian cloud as some examples of families of (n-1) spheres.
I take it the seperatrix for the pendulum is just the inverted position? It would take an infinite amount of time not just to return to its original state, but also any state, no?
I've been trying to learn Kalman Filters for a while with no luck.. After watching these videos, things are coming together. Thank you very much for these excellent videos.
If the GF is the action, and the path taken is the one s.t. dS/dt = 0, then according to HJ: S = W - tE => E = W/t - S/t, for the interval t of interest. Taking differential intervals then, wouldn't: => E = dW/dt - dS/dt = dW/dt Meaning that W would essentially be a sum of total energies kind of term? EDIT: in the time-independent case ofc.
I think it makes sense that the Hamilton-Jacobi theorem is nonunique, given the dependence of the generating function of the Lagrangian, which is also nonunique. Several Lagrangians would imply several generating functions. It's probably also worth keeping in mind that all of this business around the constraints for generating functions being canonical to begin with is built on a specific example of a nonunique Lagrangian where the function is added to the orginal Lagrangian and is of the form dF/dt. Maybe there are other functions that could work as canonical, which will have differing constraints, and then perhaps the constraints would then be different, and then the generating function that gives us back equillibrium points along all the points in ${p} \times {q}$ could be something other than the action...
Thank You Professor Shane Ross for this video. Currently I'm working on Research in Unmanned Surface Vehicle (USV) and this video help a lot for me to have more depth understanding Kinematics Equation of USV. I Have a question related to the Convention. will the Yaw-Pitch-Roll (3-2-1 Convention) sequence yield the same results or conditions if the order of "Pure Rotation" is changed to Roll-Pitch-Yaw or Pitch-Roll-Yaw?
If they are near the spin axis, even at 68 rpm, they are experiencing very small G force. Telescoping and articulatable ports, capable of articulation at a rate barely above 1hz. Automated to align itself exactly to the constant changing procession. Then I would believe it possible. Then you could spin your ship up and extend the docking probe along that axis.
@ProfessorRoss Thanks a lot for your very interesting videos. I am rediscovering a love for dynamical systems. >>>> Is there a minor error on your definition of the Baker's Map? x_n=1/2 seems to have two different definitions, I think because of a spurious <= rather than <.
Did 5yo discover this in a swing by rhythmically shifting their weight with no external force, increasing the amplitude of the swinging on their own? And nobody noticed anything strange about it?
You're completely right. From the scene around 6:45 you can count the time it takes for a revolution, and it's closer to once every 3 seconds, so a rotation of about 20 rpm. But I'm working within the "world" of the movie, so I'm trusting the robot's assessment of 67-68 rpm. If we can't trust our AI robots in a clutch situation like this, who can we trust? 😏 I don't think the actual number matters to Cooper, but it makes a difference in terms of the energy of the rotation. The difference between ω=20 rpm and ω=68 rpm is about (68/20)^2 ≈ 10, so a factor of 10 difference in rotational energy.
Forgive me, since I'm going through these one by one, if you cover it later, but I expanded the condition for a coordinate transform. Isn't this the same as anticommutivity? As in like the Poisson bracket {Q, P} = 1?
Wrt. the diagram of the stream function, aren't crossings disallowed, since the EoM would be underdetermined at those points? *unless it's a projection ofc.
Great Lecture sir. Just a small question in the first section,estimating velocity through position, may be I'm new, X = Ax x = [pos,vel] A = [1 dt 0 1] This is the prediction model A for kalman filter. We are using for velocity estimate with just providing the measurement of position alone. How Kalman filter estimates the velocity, because no prediction is there for velocity also measurement has no velocity input considered as H neglects the same. How its estimating velocity from position? Curious!
I literally loved your Kalman Filter course , where Im an Autonomous Vehicle Engineer, Now this pulled me to this course, and Im so much interested to go all through this. Thank you so much for the work!!
Excellent discussion Prof. Ross. I was just wondering if we can show rigorously (what we assume here) that the Poisson bracket properties are true for canonical as well as non canonical Hamiltonian systems, Thanks.
@@ProfessorRoss I do Prof, thankyou for the amazing lecture. I have masters in electrical and 4 years of experience in robotics and control. I want to do PhD in ADCS control and start a company afterwards. Can you please share your email/LinkedIn to connect?
That visualization is of the restricted three-body problem for a particle (or asteroid) in the Sun-Jupiter system. Each frame is a Poincaré section at a different value of the Hamiltonian energy.
Yes, but a spinning top is a different situation than an object spinning in space, because of the effect of the ground "pushing" on the top, the friction at the contact point, and the effect of gravity. In particular, the effect of friction is to make the top stabilize so it wobbles less (angle between spin axis and gravity). An object spinning in space has no ground to push off of, no friction, and no gravity (or very little--the effect of the gravity gradient is negligible over the timescale shown; ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-gz9uCWctN9I.html )
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?
Great question. The Lyapunov exponent is independent of initial location; it's determined from following any initial condition for long enough. Even when you're near the equilibrium point at <0, 0, 0>, the dynamics will still take the state away from <0, 0, 0> zipping around, and eventually going onto the strange attractor, where it will wander chaotically forever. (I'm assuming we're talking about a parameter r in the regime where the strange attractor exists).
