Small Oscillations & Normal Modes
Small Oscillations & Normal Modes
Near a stable equilibrium, any conservative mechanical system behaves as a collection of independent harmonic oscillators. This is the physics behind crystal vibrations, molecular spectroscopy, and coupled pendula. The key insight is that to second order in the displacement from equilibrium, the kinetic energy is a positive-definite quadratic form in $\dot{q}$ and the potential energy is a positive-definite quadratic form in $q$: $$T=\frac{1}{2}\dot{\mathbf{q}}^\top M\dot{\mathbf{q}},\quad V=\frac{1}{2}\mathbf{q}^\top K\mathbf{q},$$ where $M$ is the mass matrix (symmetric positive-definite) and $K$ is the stiffness matrix (symmetric positive semi-definite).
The equations of motion in matrix form are $M\ddot{\mathbf{q}}+K\mathbf{q}=\mathbf{0}$. Seeking solutions $\mathbf{q}=\mathbf{a}e^{i\omega t}$ leads to the generalised eigenvalue problem $K\mathbf{a}=\omega^2 M\mathbf{a}$. The eigenvalues $\omega_r^2$ are the squared normal-mode frequencies; the eigenvectors $\mathbf{a}_r$ are the normal modes. Because $M$ and $K$ are both symmetric, $\omega_r^2$ are real; stability requires $\omega_r^2>0$. Zero eigenvalues correspond to rigid-body motions (neutral modes); negative eigenvalues signal instability.
The normal modes are orthogonal with respect to $M$: $\mathbf{a}_r^\top M\mathbf{a}_s=\delta_{rs}$ (after normalisation). In normal coordinates $\eta_r$ defined by $\mathbf{q}=\sum_r\eta_r\mathbf{a}_r$, the Lagrangian decouples into independent oscillators: $L=\sum_r\frac{1}{2}(\dot{\eta}_r^2-\omega_r^2\eta_r^2)$. Each $\eta_r$ evolves as $\eta_r=A_r\cos(\omega_r t+\phi_r)$, and the general motion is a superposition.
Definition — Mass and Stiffness Matrices
Near equilibrium $q_i=0$, expand: $T_{ij}=\frac{\partial^2 T}{\partial\dot{q}_i\partial\dot{q}_j}\big|_0$ and $K_{ij}=\frac{\partial^2 V}{\partial q_i\partial q_j}\big|_0$. The matrix equations of motion are $M\ddot{\mathbf{q}}+K\mathbf{q}=0$.
Theorem — Normal Mode Decomposition
The generalised eigenvalue problem $K\mathbf{a}=\omega^2 M\mathbf{a}$ has $n$ real eigenvalues $\omega_1^2\leq\cdots\leq\omega_n^2$ and $M$-orthonormal eigenvectors $\mathbf{a}_r$. In normal coordinates $\eta_r=\mathbf{a}_r^\top M\mathbf{q}$, the Lagrangian becomes $L=\sum_r\frac{1}{2}(\dot{\eta}_r^2-\omega_r^2\eta_r^2)$, a sum of independent oscillators.
Example 1: Two Coupled Pendula
Two identical pendula (mass $m$, length $l$) coupled by a spring (constant $k$) between the bobs. $V=\frac{1}{2}mgl(\theta_1^2+\theta_2^2)+\frac{1}{2}k l^2(\theta_1-\theta_2)^2$. Normal modes: (1) in-phase $\theta_1=\theta_2$, frequency $\omega_1=\sqrt{g/l}$ (spring unstretched); (2) anti-phase $\theta_1=-\theta_2$, frequency $\omega_2=\sqrt{g/l+2kl/m}$ (spring maximally stretched). Beats occur when the system is launched in one normal mode: energy shuttles between the two pendula at the beat frequency $\omega_2-\omega_1$.
Example 2: Linear Triatomic Molecule
Three masses in a line: $m$, $M$, $m$ connected by equal springs $k$. Three degrees of freedom, one zero mode (uniform translation). Two vibrational modes: symmetric stretch $\omega_1=\sqrt{k/m}$ (outer atoms move in phase, centre at rest); asymmetric stretch $\omega_2=\sqrt{k/m+2k/M}$ (outer atoms move together, centre moves oppositely). This is the model for $\mathrm{CO}_2$ stretching vibrations observed in IR spectroscopy.
Example 3: Stability Analysis via Normal Modes
The bead on a rotating hoop has equilibria at $\theta=0$ (top), $\theta=\pi$ (bottom), and $\cos\theta_0=g/(R\Omega^2)$. Linearising about $\theta=\pi$: $V_{\text{eff}}''=mR(g/R-0)>0$, so the bottom is stable. Linearising about $\theta_0$: stability requires $\omega^2=\Omega^2\sin^2\theta_0(1-\cos^2\theta_0\cdot 0)>0$, which holds for $\Omega^2>g/R$, confirming the bifurcation analysis.
Example 4: Rayleigh Quotient & Frequency Bounds
The Rayleigh quotient $R(\mathbf{x})=\mathbf{x}^\top K\mathbf{x}/(\mathbf{x}^\top M\mathbf{x})$ satisfies $\omega_1^2\leq R(\mathbf{x})\leq\omega_n^2$ for any nonzero $\mathbf{x}$. The lowest frequency is $\omega_1^2=\min_{\mathbf{x}}R(\mathbf{x})$ and the highest $\omega_n^2=\max_{\mathbf{x}}R(\mathbf{x})$. This variational characterisation is used to compute bounds on normal-mode frequencies without solving the full eigenvalue problem, and underlies the finite-element method in structural mechanics.
Practice Problems
- Find the normal-mode frequencies and eigenvectors for two masses $m$ connected by three springs of constant $k$ (mass–spring–mass–spring–wall on both ends) in a line.
- A particle of mass $m$ moves in the potential $V(x,y)=\frac{1}{2}(3x^2+xy+3y^2)$. Find the equilibrium, the mass matrix, the stiffness matrix, and the normal-mode frequencies.
- For two coupled pendula with coupling spring $k$, show explicitly that the normal coordinates $\eta_\pm=(\theta_1\pm\theta_2)/\sqrt{2}$ decouple the Lagrangian.
- A particle moves in $V=ax^4$ ($a>0$). Is $x=0$ a stable equilibrium? Can the small-oscillations formalism be applied? Explain.
- The general solution for two coupled pendula with initial conditions $\theta_1(0)=A$, $\theta_2(0)=0$, $\dot{\theta}_1(0)=\dot{\theta}_2(0)=0$ exhibits beats. Derive the beat frequency and the period of complete energy transfer between the two pendula.
Show Answer Key
1. Equations of motion: $m\ddot{x}_1 = -2kx_1 + kx_2$, $m\ddot{x}_2 = kx_1 - 2kx_2$. Trial $x_i = A_i e^{i\omega t}$: $\det(K-\omega^2 M)=0$ → $(2k/m-\omega^2)^2 - (k/m)^2 = 0$ → $\omega_1 = \sqrt{k/m}$ (symmetric mode, $A_1=A_2$), $\omega_2 = \sqrt{3k/m}$ (antisymmetric, $A_1=-A_2$).
2. Equilibrium: $\nabla V = 0$ at $(0,0)$. $V = \frac{1}{2}\mathbf{q}^T K \mathbf{q}$ with $K = \begin{pmatrix}3 & 1/2\\1/2 & 3\end{pmatrix}$, $M=mI$. Eigenvalues of $K/m$: $\omega^2 = (3\pm1/2)/m$, so $\omega_1 = \sqrt{5/(2m)}$ and $\omega_2 = \sqrt{7/(2m)}$.
3. $L = \frac{1}{2}ml^2(\dot{\theta}_1^2+\dot{\theta}_2^2) - \frac{1}{2}mgl(\theta_1^2+\theta_2^2) - \frac{1}{2}kl^2(\theta_1-\theta_2)^2$. Let $\eta_\pm = (\theta_1\pm\theta_2)/\sqrt{2}$. Then $L = \frac{1}{2}ml^2(\dot{\eta}_+^2+\dot{\eta}_-^2) - \frac{1}{2}mgl(\eta_+^2+\eta_-^2) - kl^2\eta_-^2$. The $\eta_+$ and $\eta_-$ equations are independent. ✓
4. $V = ax^4$: $V'(0)=0$ (equilibrium), $V''(0)=0$. The potential is flat to leading order — the small-oscillations formalism requires $V'' > 0$ for a restoring force proportional to displacement. The motion near $x=0$ is anharmonic ($\ddot{x} = -4ax^3/m$), not simple harmonic. The formalism cannot be applied.
5. General solution: $\theta_1(t) = \frac{A}{2}[\cos\omega_+t + \cos\omega_-t]$, $\theta_2(t) = \frac{A}{2}[\cos\omega_+t - \cos\omega_-t]$. Using sum-to-product: $\theta_1 = A\cos(\frac{\omega_--\omega_+}{2}t)\cos(\frac{\omega_-+\omega_+}{2}t)$. Beat frequency: $\omega_{\text{beat}} = |\omega_- - \omega_+|$. Period of complete energy transfer: $T_{\text{transfer}} = 2\pi/\omega_{\text{beat}}$.