Small Oscillations and Simple Harmonic Motion
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You should be comfortable with Taylor series expansions of functions and the basic calculus of single-variable functions, including the meaning of first and second derivatives at a point.
Why this matters
Simple harmonic motion is the universal local approximation for any stable physical system.
From the vibration of a guitar string to the oscillation of atoms in a crystal lattice, nearly all periodic phenomena in nature begin with the same mathematical structure. Understanding why the harmonic oscillator is not just a special case, but the inevitable first-order description of stability, provides the conceptual bridge between statics and dynamics. It allows us to predict the frequency of complex systems without solving their full, often intractable, equations of motion.
The Intuition of Stability
Imagine a ball resting at the bottom of a smooth, curved valley. If you nudge it slightly to the left, gravity pulls it back toward the center. If you nudge it to the right, gravity again pulls it back. This tendency to return to the original position is what we call stability. The key insight is that near the very bottom of the valley, the curve of the ground looks almost like a perfect parabola. No matter how complex the actual shape of the valley is further up the sides, the immediate neighborhood of the lowest point is dominated by its curvature. This geometric fact means that the force acting on the ball is proportional to its displacement from the center, but in the opposite direction. This linear restoring force is the defining characteristic of simple harmonic motion. We do not need to know the exact shape of the valley far away; we only need to know how sharply it curves at the bottom.
Mathematical Formulation
To quantify this, we consider a particle of mass \(m\) moving in one dimension under a conservative potential energy \(U(x)\). A stable equilibrium exists at a position \(x_0\) where the net force is zero. Since force is the negative gradient of potential energy, \(F = -dU/dx\), the condition for equilibrium is that the first derivative of the potential vanishes: \(U'(x_0) = 0\). For this equilibrium to be stable, the potential must have a local minimum at \(x_0\), which implies that the second derivative is positive: \(U''(x_0) > 0\).
We define the small displacement from equilibrium as \(q = x - x_0\). By expanding the potential energy in a Taylor series about \(x_0\), we get:
Since \(U'(x_0) = 0\), the linear term vanishes. The constant term \(U(x_0)\) does not affect the force. For sufficiently small oscillations, the cubic and higher-order terms are negligible compared to the quadratic term. Thus, the potential energy is approximated by a parabola:
The force is then \(F = -dU/dq = -U''(x_0)q\). Applying Newton's second law, \(m\ddot{q} = -U''(x_0)q\), we arrive at the standard harmonic oscillator equation.
The Frequency of Oscillation
The equation of motion derived above is \(\ddot{q} + \omega^2 q = 0\), where the angular frequency \(\omega\) is determined by the curvature of the potential well and the mass of the particle. The derivation cited in this lecture establishes that:
This result is profound because it shows that the frequency depends only on the local properties of the system at the equilibrium point. The "stiffness" of the system is given by \(k = U''(x_0)\). A steeper potential well (larger \(U''\)) results in a higher frequency, while a heavier particle (larger \(m\)) results in a lower frequency. The period of oscillation \(T\) is therefore:
This formula allows us to calculate the period of small oscillations for any system where we can determine the second derivative of the potential energy at equilibrium, without needing to solve the full differential equation for large amplitudes.
Generalization to Multiple Degrees of Freedom
While the derivation above focuses on one dimension, the principle extends naturally to systems with multiple coordinates. If a system has \(N\) degrees of freedom with generalized coordinates \(q_i\), and the potential energy \(V(q_1, \dots, q_N)\) has a minimum at \(q_i = 0\), we can expand \(V\) in a Taylor series. The linear terms vanish, and the quadratic terms form a matrix known as the Hessian matrix of second derivatives.
The equations of motion become a set of coupled linear differential equations. By diagonalizing the matrix of coefficients, we can decouple the motion into a set of independent normal modes. Each normal mode oscillates with its own characteristic frequency, determined by the eigenvalues of the system's stiffness and inertia matrices. This is the foundation of the study of vibrations in molecules, bridges, and buildings. The core idea remains the same: near a stable equilibrium, the dynamics are governed by the quadratic approximation of the potential energy.
The derivation
Small Oscillations and Simple Harmonic Motion
Let a particle of mass m move in one dimension under a smooth conservative potential U(x) possessing a stable equilibrium at x_0, so that U'(x_0)=0 and U''(x_0)>0. Then for sufficiently small displacements q\equiv x-x_0 the motion obeys the harmonic-oscillator equation \(\ddot q+\omega^2 q=0\) with angular frequency \(\omega=\sqrt{U''(x_0)/m}\). Every stable equilibrium of a smooth potential is, to leading order, a simple harmonic oscillator whose frequency is fixed by the curvature of the well and the inertia.
It is a common misconception that the harmonic approximation is valid for all amplitudes. The validity of the approximation depends on the ratio of the third derivative to the second derivative. If the potential is highly asymmetric or if the displacement \(q\) is not small compared to the scale over which \(U''(x)\) changes, the cubic terms become significant, leading to anharmonic effects such as frequency-amplitude dependence and the generation of harmonics. Always check the smallness of the parameter \(q\) relative to the characteristic length scale of the potential well. C
Worked example
Problem. A particle of mass \(m = 2.0\ \mathrm{kg}\) moves along the \(x\)-axis under the influence of a potential energy function given by \(U(x) = 5x^4 - 2x^2 + 1\ \mathrm{J}\), where \(x\) is in meters. Determine the angular frequency of small oscillations about the stable equilibrium point near the origin.
- Find the equilibrium positions by setting the first derivative of the potential to zero: \(U'(x) = 20x^3 - 4x = 0\). This gives \(x(20x^2 - 4) = 0\), so \(x = 0\) or \(x = \pm \sqrt{4/20} = \pm \sqrt{0.2} \approx \pm 0.447\ \mathrm{m}\).
- Determine which equilibrium is stable by evaluating the second derivative: \(U''(x) = 60x^2 - 4\). At \(x = 0\), \(U''(0) = -4\ \mathrm{N/m}\). Since \(U''(0) < 0\), this is an unstable equilibrium (local maximum).
- At \(x = \pm \sqrt{0.2}\), \(U''(\pm \sqrt{0.2}) = 60(0.2) - 4 = 12 - 4 = 8\ \mathrm{N/m}\). Since \(U'' > 0\), these are stable equilibria.
- Calculate the angular frequency using \(\omega = \sqrt{U''(x_0)/m}\). Here, \(U''(x_0) = 8\ \mathrm{N/m}\) and \(m = 2.0\ \mathrm{kg}\). Thus, \(\omega = \sqrt{8/2.0} = \sqrt{4} = 2.0\ \mathrm{rad/s}\).
Units check. The second derivative of potential energy has units of \(\mathrm{J/m^2} = \mathrm{N/m}\). Dividing by mass (\(\mathrm{kg}\)) gives \(\mathrm{N/(m \cdot kg)} = (\mathrm{kg \cdot m/s^2}) / (\mathrm{m \cdot kg}) = 1/\mathrm{s^2}\). The square root gives \(1/\mathrm{s}\), which is the unit of angular frequency.
Three quick tests
- What condition must the second derivative of the potential energy satisfy at a stable equilibrium point?
Answer
It must be positive (\(U''(x_0) > 0\)), indicating a local minimum in the potential energy curve. - If the mass of an oscillator is quadrupled while the potential curvature remains unchanged, how does the period of oscillation change?
Answer
The period doubles, since \(T \propto \sqrt{m}\). - Why do we ignore the cubic and higher-order terms in the Taylor expansion for small oscillations?
Answer
For sufficiently small displacements \(q\), the term \(q^3\) is much smaller than \(q^2\), making the quadratic approximation dominant and accurate.
Problem set for PU-101 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.