8.4b ODE algorithm labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §8.4. Original drills with float-locked answers. Continues 8.4 ODE algorithms.

Explorer

Pure Python (math OK). No NumPy / VPython. Stubs fail until filled.

Shared oscillator. $m=1$, $k=(2\pi)^{2}$ so $\omega=2\pi$, $T=1$. IC $x=0$, $v=1$. Analytic amplitude $A=1/\omega$. Force $F=-kx$. Energy $E=\tfrac12 v^{2}+\tfrac12 k x^{2}$.

Formula refresh

Warm up: pick the matching equation among lookalikes (same walk as the theory lesson).

Labs

Activity: Euler one step

Take one Euler step with $h=0.01$ from $(x,v)=(0,1)$. Print x v with 10 decimals.

No input.

Write-ups

0/4 solved

8.4b.W1

Using “Euler”, “tangent”, and “local”, explain:

In one sentence, why does a single Euler step from (0,1) leave v unchanged for F=−kx?

Minimum 3 words. Your answer must include: “Euler”, “tangent”, “local”.

0 words/ 3 min

Meet the writing requirements, then you can open the model answer.

Explorers

# harmonic_rk_compare_sketch.py — original structure (no VPython)
import math

def rk2_step(x, v, h, k, m=1.0):
    def f(x, v):
        return v, -k * x / m
    f0, f1 = f(x, v)
    k1x, k1v = h * f0, h * f1
    f0, f1 = f(x + 0.5 * k1x, v + 0.5 * k1v)
    return x + h * f0, v + h * f1  # equivalent to y+k2

# Compare x(0.25) from Euler / RK2 / RK4 vs A*sin(ω*0.25)

Next: 8.5 Nonlinear oscillation solutions.