Linear Algebra for Normal Modes (IPhO Math Tools)

Normal modes are where linear algebra pays back instantly: a messy set of coupled equations becomes “find eigenvalues and eigenvectors”. In olympiad scripts, you often get full method credit by writing the matrix form and the determinant condition cleanly.

1. Definitions (Must Know)

A. Vectors and matrices (notation)

• A column vector of coordinates:
  x = (x₁; x₂; ⋮; x_N)
• A matrix mapping vectors to vectors: Ax.
• Transpose: A^T.
• Identity matrix: I.

B. Eigenvalues and eigenvectors

An eigenvector v ≠ 0 of A satisfies

Nontrivial solutions exist only if

C. Symmetric matrices

If A = A^T (real symmetric), then:

• eigenvalues are real,
• eigenvectors can be chosen orthogonal.

This is why normal-mode problems (built from energies) are usually clean.

D. Normal modes (physics meaning)

A normal mode is a motion where all coordinates oscillate sinusoidally at one frequency with fixed amplitude ratios:

The vector a is the mode shape.

2. Key Ideas (What Earns Marks)

• Write coupled oscillations as a matrix equation:
  Mx double dot + Kx = 0.
• Try a harmonic form x = ae^(iω t) to get
  (K-ω²M)a = 0.
• Nontrivial modes require:
  det(K-ω²M) = 0.
• Use symmetry to guess eigenvectors first, then find eigenvalues quickly.
• Check limiting cases (coupling spring constant to zero, one mass very large, and so on).

3. Detailed Explanations

A. Where M and K come from

For small oscillations in generalized coordinates x, the energy is typically quadratic:

Lagrange’s equations give:

Here:

• M is the (symmetric) mass matrix.
• K is the (symmetric) stiffness matrix.

B. Reducing to an eigenvalue problem

Assume a mode solution x(t) = ae^(iω t).

Then x double dot = -ω²x, so:

This is a generalized eigenvalue problem:

• eigenvalues are ω²,
• eigenvectors are mode shapes a.

C. The 2 by 2 shortcut (trace and determinant)

For a 2 by 2 matrix A, eigenvalues satisfy:

So:

If M = mI, then ω² are just eigenvalues of K/m.

D. Orthogonality (why modes decouple)

For symmetric M and K, different modes satisfy an M-weighted orthogonality:

This is why you can expand a general motion as a sum of modes and solve each one independently.

E. Limiting checks that catch algebra mistakes

• If a coupling constant goes to zero, modes should reduce to independent oscillators.
• If all masses scale by a factor, frequencies scale as ω ∝ 1/square root of m.
• If all spring constants scale by a factor, frequencies scale as ω ∝ square root of k.

4. Common Mistakes

• Writing the coupling forces with the wrong sign (a coupling spring exerts opposite forces on the two masses).
• Forgetting a “zero mode” when the system has translational symmetry (free system has a rigid-translation mode with ω = 0).
• Solving for ω and forgetting the square root (solving for ω² is often the clean step).
• Not checking the k_c → 0 or m₂ → ∞ limits.
• Mixing mode shapes with initial conditions (mode shapes are geometry; constants come later).

5. Exam Tips

• Draw the mode shapes first if symmetry is strong. For two identical masses, “in-phase” and “out-of-phase” are almost always eigenvectors.
• Write the determinant condition explicitly: det(K-ω²M) = 0.
• If you only need frequencies, you may not need normalized eigenvectors.
• Use energy language if it is quicker: V quadratic directly gives K.

6. Worked Examples

A. Two identical masses with a coupling spring

Two masses m are attached to walls by springs k and coupled by a spring k_c:

• left wall to mass 1: k
• between masses: k_c
• mass 2 to right wall: k

Find the normal-mode frequencies.

B. Two masses connected by one spring (free system)

Two masses m₁ and m₂ are connected by a spring of constant k in space (no walls). Find the mode frequencies.

7. Mind Stretchers

A. Three-mass chain between fixed walls

Three identical masses m are connected in a line between two fixed walls by four identical springs k:

wall, spring, mass, spring, mass, spring, mass, spring, wall.

Find the three normal-mode frequencies and a convenient set of mode shapes.

8. Practice