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To view to the latex in the Jupyter notebook you need to run the command init_vprinting Which I did in my import statements at the top of the program.

@@logandihel Yeah, I did that. I'm not sure what actual function you're using. I have another route that is working for me now, but it's a bit messy and requires a bit of setup. Your method seems cleaner and simpler unless you've abstracted way much of the details.

I'd recommend copying the code directly from my github page (link in description), and if that doesn't work you can submit an issue on my RUclips GitHub repository. It's also possible you're missing a required package, or the packages have changed versions since I posted the video

The energy of the system is constant only if friction is ignored. Set b to zero and you should see the energy is constant, perhaps with small changes due to numerical integration errors

@@logandihel Ahh it oscilates... Thanks anyway! I was searching a bit and I think it’s common in the way it solves the ODE right? I was doing the kepler problem potencial to see orbits and again the energy driftt. I guess now i’m reading a book about numerical method for python... I thought this day will never come!

Great question! I don't think that it really matters in Python whether you are doing a full derivative or a partial derivative because they both use diff(). I just make sure to explicitly declare that my derivatives are with respect to time

@@logandihel Thanks bro

* simply means to import everything (functions, classes, variables) from the sympy library into the current namespace. This way you don't need to use the module name (sympy) as a prefix: symbols() instead of sympy.symbols()

I used the matrix notation (state space model) because that is the gold standard in nonlinear system theory, and ode solvers in python/matlab expect that format

@@logandihel Thanks man

The velocity tends to zero when friction is added. Otherwise, no energy is lost and the velocity is sinusoidal

The solve_ivp() function allows you to input the simulation time (mine is set [0,10] meaning from 0 to 10 seconds) and also the evaluation time (mine is set from 0 to 10 seconds, evaluated 30 times per second). You can increase the 10s to a larger value like 25. You could also change the value of "b" (damping coefficient) to perhaps 5 to make the system lose energy faster

@@logandihel I changed the value but the duration of the GIF image remained the same. What can be done?

@olsea1229 which file are you editing? Is it SpringMass/animate.py ? If so, only lines 25 and 26 should need editing to change the .gif file time duration

@@logandihel I changed it like this: sol = solve_ivp(spring_mass_ODE, [0, 30], (x0, x_dot0), t_eval=np.linspace(0,10,10*30)) but the time of the gif animation has not changed

@@olsea1229 change t_eval to np.linspace(0,30,30*30)

Good catch! I haven't tried running in notebook collab before - I just use vscode with Jupter extension. Thanks for sharing

"Simple" in the sense that the pendulums are point masses and not rigid bodies with rotational inertia

Sorry, that is not the usual definition ​of a simple pendulum.

You're correct. A simple pendulum usually refers to a linear model of a single point mass pendulum. I just wanted to differentiate between point masses versus rigid bodies, and the phrase "point-mass double pendulum" was too long to fit on the screen!

@@logandihel Ok, no problem, thank you for your videos.

Not true, when he called scipy.integreate.solve_ivp() it numerically solves the "initial value problem" by running a time-step by time-step simulation. It does not use the analytical solution of the ODE. The description of the function says "This function numerically integrates a system of ordinary differential equations given an initial value". There is nothing more fitting of the name simulation than that.

Could you give an example of what sort of mathematical models you are looking for? Most of the content I've made so far is for modeling nonlinear dynamical systems.

@@logandihel Thank you so much for your kind reply. The modelling like SIER, Logistics growth, behaviour of functions with respect to different parameters, etc would be nice to know. If it is out of scope for your channel, I totally understand.

I think the topic of bifurcations with respect to varying parameters for 2-dimensional nonlinear systems would be a good fit for this channel. Thanks for the ideas!

Code for the animations are available on my github page, which is linked in the description. The motion picture was done using the manim library, which I may put out a video later.

Great question! It doesn't really matter how we define x, as long as we are consistent. If you want, simply replace x with (-y) in each equation, so that y positive is up. Fun fact: In aerospace literature, it is actually more common to define z to be down. This is just a convention

In the foundation of robotics course, the z-axis was always the axis of motion. Thanks for your video. I have been studying lagrangian for the past week and the available videos are long and complex. Although yours is short, it contains a lot of information and I keep watching so I enhance my understanding and then move into complex topics.

@mohamedeljahmi2454 good luck! I have some other videos for deriving equations of motion for the pendulum using Lagrangian mechanics, which is a great example of a nonlinear system

Well, I hope you like the animations at the end!

(in the introduction)

Yes, that would totally work! That would be a great exercise for you. In fact, you could generalized the spring mass system to oscillate back and forth about any angle!

@@logandihel Thanks for responding, I'll surely try it

Spring mass system for running? I'm not familiar with this system. Could you share a link?

@@logandihel Of course ! Here is the link : www.cs.cmu.edu/~hgeyer/Teaching/R16-899B/Papers/Blickhan89JBiom.pdf (thanks a lot, from France !)

​@@Arthur-e8x4n Same here bruh 👍 🇨🇵 (ça bosse le tipe)

Just saw this notification today! School is busy right now, but ill see what I can do

Yes, there's some math involved, but you can understand it! Check out the other videos in this playlist to learn where these equations come from 💪

my real question is... is this data science? data analytics? whats the main field for the videos --- im making animations with python and matplotlib animations but they are only random numbers using np.linspace... etc nothing more... thats why im asking brother. @@logandihel

You can convert any ordinary differential equation into a state space model. You just need to choose your state variables as theta, theta_dot, ell, and ell_dot

@@logandihel yeah doing that now, thanks for the info

hi logan, I have a doubt ,I am trying to solve a nonlinear differential equation, I am looking for closed solution as a function of time , I couldn't solve with dsolve in sympy, raising error->" cannot be solved by the factorable group method " using scipy integrate gives me the array of t and q(differential parameter) values, I want to simulate the model for t-> tends to infinity, how can I do that?

You can very rarely solve a nonlinear system. Most of the time it is actually impossible (like for the pendulum). You need to simulate starting with some initial conditions

Yes, but only because we defined the origin to be at the pivot, with the x axis to the right and the y axis vertical, and theta being counterclockwise from the downward position. So at resting position, x is zero and y is negative. At 0 < theta < pi/2, x >0 and y<0