Perturbations in Classical Mechanics
From the archive (2023). Corrected in October 2026; see the note at the end.
Both classical mechanics and quantum mechanics use the word perturbation for a small deviation from a reference state, but the two theories use the idea very differently. This post covers the classical side; the companion post covers perturbation theory in quantum mechanics.
Simple harmonic response
A small disturbance of a stable equilibrium typically produces (possibly damped) simple harmonic motion. Starting from Newton’s second law, \(F = m\ddot{x}\), a restoring force \(F = -\alpha x\) with \(\alpha > 0\) gives
\[m\ddot{x} = -\alpha x \quad\Longrightarrow\quad \ddot{x} + \frac{\alpha}{m}\,x = 0,\]whose solutions are sinusoidal with angular frequency \(\omega = \sqrt{\alpha/m}\).
Central forces and the radial equation
For a particle of mass \(m\) under a purely radial force \(F_r(r)\), the angular momentum per unit mass \(h = r^2\dot{\theta}\) is conserved (see Noether’s theorem), and the radial component of Newton’s second law becomes
\[\ddot{r} - \frac{h^2}{r^3} = \frac{F_r(r)}{m}.\]A circular orbit of radius \(r = a\) has \(\ddot r = 0\), so
\[-\frac{h^2}{a^3} = \frac{F_r(a)}{m} \quad\Longrightarrow\quad F_r(a) = -\frac{m\,h^2}{a^3}.\]Small radial perturbations of a circular orbit
Now displace the orbit slightly,
\[r(t) = a + \rho(t),\qquad |\rho| \ll a,\]keeping \(h\) fixed. Substituting into the radial equation,
\[\ddot{\rho} - \frac{h^2}{(a + \rho)^3} = \frac{F_r(a + \rho)}{m},\]and linearising both sides to first order in \(\rho\):
- Centrifugal term: \(\displaystyle \frac{1}{(a + \rho)^3} \approx \frac{1}{a^3}\left(1 - 3\frac{\rho}{a}\right).\)
- Radial force: \(F_r(a + \rho) \approx F_r(a) + F_r'(a)\,\rho.\)
The zeroth-order terms cancel by the circular-orbit condition, leaving
\[\ddot{\rho} + \left[\frac{3h^2}{a^4} - \frac{1}{m}F_r'(a)\right]\rho = 0.\]Defining
\[\omega^2 \equiv \frac{3h^2}{a^4} - \frac{1}{m}F_r'(a),\]this is \(\ddot{\rho} + \omega^2\rho = 0\), with general solution \(\rho(t) = A\cos(\omega t + \phi)\).
Stability condition. The perturbation stays bounded and oscillates provided \(\omega^2 > 0\), that is,
\[\frac{3h^2}{a^4} > \frac{1}{m}F_r'(a).\]For an inverse-square force \(F_r = -k/r^2\), using \(h^2 = ka/m\) gives \(\omega^2 = \frac{3k}{ma^3} - \frac{2k}{ma^3} = \frac{k}{ma^3} > 0\): circular Kepler orbits are stable, and the radial oscillation has the same frequency as the orbit itself, which is why the perturbed orbit closes into an ellipse.
Large kicks: when does the particle escape?
The linear analysis says nothing about large displacements, so for those we use the conserved energy
\[E = \tfrac12 m\left(\dot r^2 + r^2\dot{\theta}^2\right) + V(r), \qquad V(r) = -\frac{k}{r}.\]Displace the particle from the circular orbit of radius \(a\) to \(r_0\), with no radial velocity. What happens depends on what the kick preserves.
Same angular momentum. If \(h = \sqrt{ka/m}\) is unchanged, then
\[E = \frac{m h^2}{2 r_0^2} - \frac{k}{r_0} = \frac{k a}{2 r_0^2} - \frac{k}{r_0},\]which is negative for every \(r_0 > a/2\). A particle pushed outwards at fixed angular momentum therefore always stays bound. The figure shows \(r_0 = 1.8a\): the radius oscillates between about \(0.69a\) and \(1.8a\).
Same speed. If instead the particle keeps the circular-orbit speed \(v_0 = \sqrt{k/(ma)}\), then
\[E = \tfrac12 m v_0^2 - \frac{k}{r_0} = \frac{k}{2a} - \frac{k}{r_0},\]so \(E \ge 0\), and the particle escapes, exactly when \(r_0 \ge 2a\).
Summary
- Linear (local) stability. Small displacements, \(\lvert\rho\rvert \ll a\), produce bounded oscillations about \(r = a\) whenever \(\omega^2 > 0\).
- Global behaviour. Whether a large kick unbinds the particle is decided by the sign of the energy, and depends on how the kick changes the velocity as well as the position. Infinitesimal stability does not rule out escape after a large enough kick.
Credits
These notes are heavily inspired by Prof. Arjun Berera’s courses Introductory Dynamics (PHYS08052) and Principles of Quantum Mechanics (PHYS10094) at the University of Edinburgh.
Changes on 25 October 2026
What was wrong. The post said a particle escapes if it starts at \(r_0 \ge 2a\). That holds when the kick keeps the speed fixed, but the plot keeps the angular momentum fixed, and in that case a particle pushed outwards always stays bound.
What changed. Both cases are now worked out, each with its energy. Typos are fixed. Previously titled “Perturbations - Classical Mechanics”.