Potentials - Dynamical Systems | Lecture 5

There is an alternative geometric perspective for flows on the line that comes from physics. In this lecture we present the geometric perspective of potentials. The terminology comes from potential energy in physics and allows one to think about a particle moving around on a potential energy surface. Here we describe the method and compare it with the previously analyzed phase lines.

This course is taught by Jason Bramburger for Concordia University.

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[ChaosBook]

V(x) = -x^2 + x^4 potential needs an overall - sign, I think. You want it to be bounded for large |x|🙂

[mrpop]

Thanks for uploading these videos Jason! Im trying to get a grasp of dynamical systems in general (im a 3rd year physics student) and these are really helpful. Unfortunately it seemswe don't have much courses on dynamical systems or then they are saved for our masters program, so these kind of free resources are a lifesaver. By the way, the last example should have a minus-sign, so V(x) = -(1/2*x^2 - 1/4*x^4).

[nahidsafari-yh]

Thank you for great explanation. I have a question about this approach. This is clear to me how it works, but I could not understand when this approach works better? What are the advantages/disadvantages of that?

[adrianramos]

The approach in Physics is that F = m*d2x/dt2 = -dV/dx. The potential you defined is related to the first derivative with time, not the the second. There is no problem with that but is not generally how is thought in Physics. That changes some things, mainly that minimums of the potential function are not ending points of the trajectory, but points where the particles oscillate around

[SG-cnik]

Interesting. Looks like cost-functions in neural networks, where you get a different (local) minima, depending on where you start. But I am not big at math, so this similarity just may be random.