1. The Work of "Physicists" 🍎
Slides 1-3Solving differential equations is the bread and butter of classical mechanics.
- Many equations in nature cannot be solved analytically.
- Numerical approximations are often good enough.
- Today: Euler, Runge-Kutta, and Animations!
🍏
"Let's get back to F=ma!"
2. Euler's Method
Slide 4-6Let's solve $\frac{dy}{dt} = y$ with $y(0)=1$. We know the answer is $e^t$.
Algorithm
$$ y_{n+1} \approx y_n + h \cdot f(y_n, t_n) $$
Precision: $O(h)$ (Not very good)
3. The Runge-Kutta Family 🚀
Slides 7-15RK2 (2nd Order)
Predictor-Corrector approach. Much better than Euler.
RK4 (4th Order)
The standard workhorse. Very precise ($O(h^5)$ local error).
RK45 (Adaptive Step Size)
Automatically adjusts step size $h$ to maintain precision.
4. Physics: The Pendulum 🕰️
Slides 16-25
$$ \frac{d^2\theta}{dt^2} = -\frac{g}{R}\sin(\theta) $$
1. $\dot{\theta} = \omega$
2. $\dot{\omega} = -\frac{g}{R}\sin(\theta)$
2. $\dot{\omega} = -\frac{g}{R}\sin(\theta)$
Euler fails to conserve energy. RK4 keeps it stable.
Method: RK4
Energy: 0.00
Energy: 0.00
Animation Code Logic (SciPy Version)
5. Using SciPy `solve_ivp`
Why write your own solver? Use industrial strength tools.
6. Springs & Chaos 🌀
Slides 30+
2D Spring
Energy: 0.00
Energy: 0.00
System Equations
Hooke's Law + Gravity in 2D.
Double Spring (Coupled System)
👐 Hands-On Session
Try simulating these scenarios:
- Gravity Shot: $a = 1/r^2$. Can you orbit a black hole?
- Damped Oscillator: Add friction $f = -bv$.