Oscillations

Mass on a spring

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Intuition

Pull a block on a spring and let go. It overshoots the middle, slows, comes back, overshoots again: simple harmonic motion. The surprising fact is that the period does not depend on how far you pulled it. A bigger amplitude means a bigger speed through the middle, and the two cancel exactly. Watch kinetic and potential energy trade places in the energy graph while their sum stays flat.

The maths

$m \ddot{x} = -k x$ gives $x = A \cos(\omega t)$ with

$$\omega = \sqrt{k/m}, \qquad T = 2\pi \sqrt{m/k}.$$

Maximum speed $A \omega$, maximum force $k A$, energy $\tfrac{1}{2} k A^2$ throughout.

How the engine does it

The spring is a force with a potential, so its energy appears in the ledger. Velocity Verlet is symplectic: the energy oscillates slightly within a period but never drifts, and the measured period matches the known Verlet dispersion formula to 1e-8 and the textbook period to $(\omega \Delta t)^2 / 12$.

Simple pendulum

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Intuition

For small swings the pendulum keeps time regardless of amplitude, which is why clocks used them for three centuries. Pull it back far, say 120°, and the period grows noticeably: the restoring force is no longer proportional to the displacement. The playground shows both the small-angle formula and the exact period.

The maths

Small angles: $T_0 = 2\pi \sqrt{L/g}$. Any amplitude $\theta_0$:

$$T = 4 \sqrt{L/g}\, K\!\left(\sin \tfrac{\theta_0}{2}\right),$$

with $K$ the complete elliptic integral of the first kind, computed by the arithmetic–geometric mean.

How the engine does it

The string is a rod constraint solved with RATTLE, which keeps the length exact and the energy drift at 1e-10 over twenty periods. At 2 rad amplitude the period matches the elliptic integral to 1.7e-8 with second-order convergence.