Simple Harmonic Motion

Simple Harmonic Motion

A mass on a spring oscillates. The restoring force is proportional to displacement:

$F = -k x quad Rightarrow quad m x'' = -k x$

Dividing by mass and letting $\omega^2 = k/m$:

$x'' = -\omega^2 x$

Reducing to a System

A second-order ODE like this is converted to a first-order system by introducing velocity $v = x'$:

$\frac{dx}{dt} = v$ $\frac{dv}{dt} = -\omega^2 x$

State vector: $[x, v]$. This is the standard trick — any nth-order ODE becomes a first-order system of $n$ equations.

Symplectic Euler

For oscillatory problems, the standard Euler method adds energy over time (unstable). The symplectic (semi-implicit) variant updates velocity first, then uses the updated velocity for position:

v = v + h * (-omega**2 * x)   # update v first
x = x + h * v                  # use new v for x

This conserves energy long-term, making it suitable for simulating oscillations.

Exact Solution

$x(t) = x_0 \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t)$

The motion is sinusoidal with period $T = 2\pi / \omega$.

Your Task

Implement harmonic_motion(omega, x0, v0, t_end, n) using the symplectic Euler update. Return (x, v).