Normal modes — coupled oscillators, normal coordinates: Foundations

Before you can read the parent note Normal Modes comfortably, every squiggle in it has to mean something to you. This page builds each symbol from nothing — plain words, then a picture, then the reason the topic can't live without it. Read top to bottom; each item leans on the one above.

1. Displacement $x$ — "how far from home"

Picture a mass sitting quietly where the springs are neither stretched nor squashed. Call that spot home (equilibrium). Now nudge it right by 3 cm: $x = + 3$ . Nudge it left instead: $x = - 3$ . The number and its sign together say exactly where it is relative to home.

Figure — Normal modes — coupled oscillators, normal coordinates

Why the topic needs it: everything — energy, forces, frequencies — is written in terms of how far from home each mass sits. If we measured from the wall instead, the equations would be cluttered with constants. Measuring from home makes "no force" happen exactly at $x = 0$ , which is the whole reason the maths stays clean.

The subscript tells you which mass: $x_{1}$ is mass 1, $x_{2}$ is mass 2. Two masses, so two numbers, so two subscripts.

2. The dot — velocity $\overset{x}{˙}$ and acceleration $\overset{x}{¨}$

Think of a movie of the moving mass.

$x$ = where it is in this frame.
$\overset{x}{˙}$ = how many centimetres its position jumps between frames — its speed and direction.
$\overset{x}{¨}$ = how much that speed itself is changing — is it speeding up or slowing down?

This is the same idea as a derivative: the dot is just physics shorthand for " $\frac{d}{d t}$ ", a slope-in-time. We use it because Newton's law is about acceleration ( $F = m \overset{x}{¨}$ ), so we need a compact way to talk about the second rate of change.

Why the topic needs it: the equations of motion say "acceleration of a mass = forces on it". No dots, no acceleration, no equation. The whole subject is a statement about $\overset{x}{¨}$ .

3. The spring constant $k$ — "how stiff"

Stretch a spring by amount $s$ and it pulls back with force $k \cdot s$ (Hooke's law). A floppy slinky has tiny $k$ ; a car suspension has huge $k$ .

In the parent problem there are two kinds:

$k$ = the outer springs joining each mass to a wall.
$k_{c}$ = the coupling spring in the middle, joining the two masses to each other. The little $c$ stands for "coupling".

Why the topic needs it: $k$ sets the restoring force, and restoring force sets the frequency. The famous answers $ω_{1}^{2} = k / m$ and $ω_{2}^{2} = (k + 2 k_{c}) / m$ are made of these stiffness numbers. The whole "in-phase is slow, anti-phase is fast" story is really a story about when $k_{c}$ does or doesn't get to push.

4. Stretch of a middle spring — why $(x_{2} - x_{1})$

The left end of the middle spring is glued to mass 1 (at $x_{1}$ ), the right end to mass 2 (at $x_{2}$ ). If both masses drift right by the same amount, the spring's length doesn't change at all — it's just carried along. Its stretch is the difference $x_{2} - x_{1}$ .

Check the cases:

Both move right equally ( $x_{1} = x_{2}$ ): stretch $= 0$ . Spring relaxed. This is the symmetric mode — the spring sleeps.
They move apart ( $x_{2}$ up, $x_{1}$ down): stretch is large. This is the anti-symmetric mode — the spring fights hardest.

Why the topic needs it: this single difference $(x_{2} - x_{1})$ is the coupling. Because it appears in both masses' equations (with opposite signs), the two masses can't move independently. Kill this term and the problem falls apart into two separate oscillators — which is exactly what normal coordinates achieve.

5. Energy — kinetic $T$ and potential $V$

You could write forces directly, but signs on springs are a minefield. Energy is a single number that's always positive-ish and never lies about direction. This is the whole reason the parent uses the Lagrangian $L = T - V$ : energy-first bookkeeping that manufactures the correct force signs automatically.

Why the topic needs it: the equations of motion in the parent are derived from $T$ and $V$ . If you don't know what these two letters store, section 2 is unreadable.

6. Angular frequency $ω$ and phase $ϕ$

A mass doing simple harmonic motion traces $x (t) = A cos (ω t + ϕ)$ .

$A$ = amplitude, how far it swings.
$ω$ = the pace of swinging. Big $ω$ = fast buzz; small $ω$ = lazy sway.
$ϕ$ = where the cosine "starts" at $t = 0$ .
$ω^{2}$ shows up more than $ω$ because acceleration of a cosine brings down two factors: $\overset{x}{¨} = - ω^{2} x$ .

That last identity, $\overset{x}{¨} = - ω^{2} x$ , is the definition of an oscillator: acceleration always points home, proportional to displacement. It's the single equation every normal mode secretly obeys.

Why the topic needs it: a "normal mode" is precisely one shared value of $ω$ for all masses. The entire secular-equation machinery exists to hunt down the allowed $ω$ values.

7. Matrices, vectors, and the eigen-idea

Why bundle $A_{1}, A_{2}$ into one object $A$ ? Because the two algebraic equations in the parent are the same equation written for each row of a matrix. Stacking them lets us say the whole thing in one line: $(K - ω^{2} M) A = 0$ .

$K$ (stiffness matrix) packs together all the $k$ 's.
$M$ (mass matrix) packs the masses.
The special directions $A$ that survive this are the mode shapes — the "same way of moving together". These are the eigenvectors; the frequencies $ω^{2}$ are the eigenvalues.

Why the topic needs it: finding normal modes is solving an eigenvalue problem. Symmetric mode $(1 1)$ and anti-symmetric mode $(- 1 1)$ are the two eigenvectors. No eigen-idea, no modes.

8. The determinant $det$ — "when does zero have a way out?"

We're looking for real motion, so $A$ can't be all zeros. The equation $(K - ω^{2} M) A = 0$ then demands the matrix be collapsible — so we set its determinant to zero. That gives the secular equation, whose roots are the allowed $ω^{2}$ . This is the gate that lets only special frequencies through.

Why the topic needs it: $det (K - ω^{2} M) = 0$ is the master equation of the whole subject. Every mode frequency is a root of it.

9. Superposition and beats

Because the equations become independent, each mode runs on its own like a separate clock, and you add them to rebuild the true wobble. When two clocks have nearly equal $ω$ , their sum swells and fades — that slow throb is a beat, and it's why energy sloshes from one mass to the other.

Why the topic needs it: the "messy real motion = sum of clean modes" claim is superposition. And the beat example in the parent is superposition of two close frequencies.

Prerequisite map

Each box is one section above. Follow the arrows and you build the parent topic from the ground up. If any box feels shaky, reread that section before touching Small Oscillations or Phonons and Lattice Vibrations, which stack directly on this foundation.

Equipment checklist

Test yourself — cover the right side and answer aloud.

I can say what

x_{1}

and

x_{2}

mean and why we measure from equilibrium

They are each mass's signed distance from its resting spot; measuring from home makes zero force sit at

x = 0

I can read

\overset{x}{˙}

and

\overset{x}{¨}

Velocity (rate of change of position) and acceleration (rate of change of velocity) — the dot is

d / d t

I know what

k

and

k_{c}

physically are

Stiffness of the outer wall springs and of the middle coupling spring; bigger = stronger push per stretch.

I can explain why the middle spring's stretch is

x_{2} - x_{1}

A spring only feels the relative motion of its ends; equal drift changes nothing.

I can state

T

and

V

for one mass–spring

T = \frac{1}{2} m \overset{x}{˙}^{2}

(motion),

V = \frac{1}{2} k s^{2}

(stretch).

I know why we use energy / the Lagrangian

Energy bookkeeping produces correct force signs automatically, avoiding sign errors.

I can write SHM and recognise

\overset{x}{¨} = - ω^{2} x

x = A cos (ω t + ϕ)

; acceleration points home proportional to displacement.

I understand

ω

A

ϕ

Pace of wobble, size of swing, starting offset in the cycle.

I know what an eigenvector and eigenvalue are

A special vector a matrix only stretches; the stretch factor is the eigenvalue — here the mode shape and

ω^{2}

I can say why

det (K - ω^{2} M) = 0

Nonzero motion needs the matrix collapsed (singular); the determinant is zero exactly then.

I understand superposition and beats

Real motion is the moment-by-moment sum of modes; two close frequencies produce a slow throb.