Small Oscillations and Normal Modes

At a stable equilibrium, $V$ has a local minimum. Expanding around equilibrium ($q_j = q_j^0 + \eta_j$):

$L \approx \frac{1}{2}\sum_{j,k} T_{jk}\dot{\eta}_j\dot{\eta}_k - \frac{1}{2}\sum_{j,k} V_{jk}\eta_j\eta_k$

Where $T_{jk} = \frac{\partial^2 T}{\partial \dot{q}_j \partial \dot{q}_k}\bigg|_0$ is the (constant) Mass matrix and $V_{jk} = \frac{\partial^2 V}{\partial q_j \partial q_k}\bigg|_0$ is the (constant) Stiffness matrix.

Both $T_{jk}$ and $V_{jk}$ are symmetric matrices. $T$ is positive definite (kinetic energy is always positive), and $V$ is positive definite at a stable equilibrium.

Writing the Lagrangian in matrix form:

$L = \frac{1}{2}\dot{\boldsymbol{\eta}}^T \mathbf{T}\, \dot{\boldsymbol{\eta}} - \frac{1}{2}\boldsymbol{\eta}^T \mathbf{V}\, \boldsymbol{\eta}$

The Euler-Lagrange equations become:

$\mathbf{T}\, \ddot{\boldsymbol{\eta}} + \mathbf{V}\, \boldsymbol{\eta} = \mathbf{0}$

Assuming solutions of the form $\eta_j = a_j e^{i\omega t}$The eigenvalue problem is:

$(\mathbf{V} - \omega^2 \mathbf{T})\mathbf{a} = \mathbf{0}$

The secular equation (characteristic equation) is:

$\det(\mathbf{V} - \omega^2 \mathbf{T}) = 0$

This is a polynomial of degree $n$ in $\omega^2$ whose roots are the squared normal mode frequencies $\omega_\alpha^2$.

Theorem 7.1. For a stable system, all normal mode frequencies are real and positive. The normal Modes are orthogonal with respect to both $\mathbf{T}$ and $\mathbf{V}$.

Proof. Since $\mathbf{T}$ is positive definite, we can write $\mathbf{T} = \mathbf{L}\mathbf{L}^T$ (Cholesky decomposition). Defining $\boldsymbol{\xi} = \mathbf{L}^T\mathbf{a}$ and $\mathbf{W} = \mathbf{L}^{-1}\mathbf{V}\mathbf{L}^{-T}$The eigenvalue problem becomes $\mathbf{W}\boldsymbol{\xi} = \omega^2\boldsymbol{\xi}$. Since $\mathbf{W}$ is symmetric and $\mathbf{V}$ is positive definite, all eigenvalues $\omega^2$ are real and positive. Orthogonality follows from the symmetry of $\mathbf{W}$. $\blacksquare$

Theorem 7.2. The normal mode vectors $\mathbf{a}^{(\alpha)}$ satisfy:

$\sum_{j,k} a_j^{(\alpha)} T_{jk} a_k^{(\beta)} = T_\alpha\, \delta_{\alpha\beta}, \quad \sum_{j,k} a_j^{(\alpha)} V_{jk} a_k^{(\beta)} = \omega_\alpha^2 T_\alpha\, \delta_{\alpha\beta}$

Where $T_\alpha$ is a normalisation constant.

This allows us to expand any motion as a superposition of normal modes.

Problem. Two identical simple pendulums of length $l$ and mass $m$ are coupled by a spring of constant $k$ connecting the bobs. Find the normal modes and frequencies.

For two equal masses $m$ on massless rods of length $l$:

The kinetic and potential energy matrices (to second order) give the eigenvalue problem with solutions $\omega_1 = \sqrt{g/l}(2 - \sqrt{2})^{1/2}$ and $\omega_2 = \sqrt{g/l}(2 + \sqrt{2})^{1/2}$.

The corresponding normal modes are:

• Mode 1: both pendulums swing in the same direction (in phase).
• Mode 2: the pendulums swing in opposite directions (out of phase).

• Assuming the mass matrix is always diagonal; in generalised coordinates it can have off-diagonal terms.
• Forgetting that both $T$ and $V$ must be positive definite for all eigenvalues to be real and positive.
• Confusing the secular equation $\det(\mathbf{V} - \omega^2\mathbf{T}) = 0$ with $\det(\mathbf{V} - \lambda\mathbf{I}) = 0$; the latter only applies when $T = I$.
• Neglecting to normalise eigenvectors with respect to the mass matrix, which leads to incorrect mode superpositions.
• Misidentifying degenerate modes: when $\omega_\alpha = \omega_\beta$, any linear combination of the corresponding eigenvectors is also a normal mode.
• Assuming that small oscillation analysis captures the full dynamics; it only describes motion near equilibrium and breaks down for large amplitudes.

flowchart TD
    A[7_Small Oscillations And Normal Modes] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Normal modes are like the natural hum of a system. When you pluck a guitar string, it vibrates at specific frequencies determined by its length and tension. Push a swing at just the right moment and it builds up; push at the wrong frequency and it barely moves. Small oscillation analysis decomposes any complex motion near equilibrium into these natural frequencies. Think of coupled pendulums: energy sloshes back and forth between them like water in connected tanks, creating the beat phenomenon. Each normal mode is a pattern where every part of the system moves in phase with the same frequency. The secular equation is like tuning a radio: it tells you exactly which frequencies the system can sustain. Any motion is just a superposition of these natural frequencies, like a chord on a piano is a superposition of individual notes.

• Lagrangian Mechanics: The Lagrangian formulation provides the foundation for analyzing small oscillations and normal modes.

• Hamiltonian Mechanics: The Hamiltonian formalism gives the total energy and phase space representation of normal mode oscillations.

• Rigid Body Dynamics: Rigid body rotations exhibit normal modes that describe small oscillations about equilibrium orientations.

• Calculus

• Linear Algebra

• Quantum Computing

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

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