Mechanics · undergrad

Small Oscillations

Linearized motion near stable equilibrium and the normal-mode structure.

Published Apr 26, 2026 · 1 min

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Expansion Near Equilibrium

Near a stable equilibrium $q_0$ , expand the potential:

V(q)\simeq V(q_0)+\frac{1}{2}k(q-q_0)^2.

The motion becomes

m\ddot x+kx=0,

with angular frequency

\omega=\sqrt{\frac{k}{m}}.

For many coupled coordinates, diagonalizing the quadratic form reveals independent normal modes.

The differential equation obtained by making the action stationary.

Equations of motion from the principle of least action — the gateway to analytical mechanics.

The exactly solvable quantum system underlying vibrations and field modes.