Differential equations, a tourist's guide | DE1
Why Physics Stack Exchange recommends it
Physics Stack Exchange users link it when explaining the pendulum equation, why it is only approximately harmonic, and how phase space and representative points work.
Recommended by 2 people across 2 different months, 2019 to 2020.
What it is
The first episode of 3Blue1Brown's differential equations series, giving a visual overview of what ODEs are and why they matter. It uses the pendulum, phase space and the Romeo and Juliet love model as running examples, with animations by Grant Sanderson made in manim.
What people said
Recommends Dorothea · 2020-08 · Physics Stack Exchange · on “Integration of equation of motion in polar coordinates”
You cannot have a perfect solution for this. When you try to find the time period of pendulum (with oscillations of high $\theta$) the result you get is a second order differential equation which is unsolvable by current knowledge, although an approximation can be made. You can watch [this video] and the proceeding episode for an insight.
Recommends Himanshu · 2020-11 · Physics Stack Exchange · on “Concept of movement of the representative points in the phase space”
The best way to understand the idea is to take some simple example and then generalize it to prove the general result. Here I'll try to give the idea of representation point as you said: This idea of the motion of representative points is something that confuses me. The most simple example that every familiar with is of Simple pendulum but I'm involving some damping in it. The answer will make more sense if you go through the following visual [this video] . Now consider an ensemble of pendulums with different initial position and momentum. When you set the motion On. The pendulums will start to move, and the system will evolve in time under a given Hamiltonian equation. The following is a nice pi
Cites overseas · 2019-04 · Physics Stack Exchange · on “Why is a pendulum an example of harmonic motion?”
I've heard a lot of people say that a pendulum moving back and forth on a fixed-length spring is an example of harmonic motion. But when I derive the governing equation for a pendulum, I get $$\theta'' = \frac{-g}{L} \sin(\theta)$$ (see for example video here [this video] for a full derivation) and I thought the equation of a harmonic oscillator was $$x'' = -k x$$ like for springs. Why do people call a pendulum an example of harmonic motion if it doesn't satisfy the second equation? (I think it's just because they use the small angle approximation $\sin(\theta) \approx \theta$ and call it a day.)