The Pendulum and Celestial Mechanics

One law, two stages: the pendulum and the planet

After Lotka-Volterra and the SIR model, you have seen how a system of two coupled rates can carry a whole story. The pendulum and the orbiting planet are the next great cast members, and they share a single script: Newton's second law, force equals mass times acceleration, which is always a second-order equation because acceleration is a second derivative. A pendulum of length L swinging through angle theta obeys theta'' = -(g/L) sin(theta); a planet of position r pulled by the Sun obeys r'' = -(G M / |r|^3) r. Different letters, same grammar — and the same toolbox you have been building.

Notice the crucial honesty in the pendulum equation: the right side is sin(theta), not theta. The familiar 'simple harmonic' pendulum you met earlier — theta'' = -(g/L) theta, with its tidy sines and cosines — is only the small-angle approximation, valid when theta is tiny enough that sin(theta) is close to theta. The true pendulum is nonlinear, and that one sin makes it impossible to solve in elementary functions. We are squarely in the territory the rung on qualitative methods prepared us for: most interesting ODEs have no closed-form solution, so we read the behaviour off the geometry instead.

Turning a second-order law into a phase portrait

The first move is one you have made many times: convert the second-order equation into a first-order system by naming the velocity. Let v = theta', so the pendulum becomes theta' = v and v' = -sin(theta). Now we have a two-dimensional autonomous system, exactly the kind whose phase portrait you learned to draw — the angle on the horizontal axis, the angular velocity on the vertical. Every state of the pendulum is one point in this plane, and its whole future is the curve threading out from that point.

Where are the equilibria? Setting both rates to zero needs v = 0 and sin(theta) = 0, so theta = 0, pi, 2pi, and so on. Two are physically distinct: theta = 0 is the pendulum hanging straight down, and theta = pi is it balanced perfectly upright. Linearizing near the bottom gives theta'' = -theta, a pure oscillation, so that point is a center — surrounded by closed orbits, the back-and-forth swings. Near the top, linearizing gives theta'' = +theta, with one growing and one decaying direction, the signature of a saddle.

Energy: the conserved quantity that organises everything

The deepest idea here is that the pendulum has a conserved quantity — its total energy never changes as it swings. For theta'' = -sin(theta), the energy is E = (1/2) v^2 + (1 - cos(theta)): a kinetic part from the speed plus a potential part from the height. You can check it directly: dE/dt = v v' + sin(theta) theta' = v(-sin(theta)) + sin(theta) v = 0. Because E stays fixed, every trajectory must live on a single level curve E = constant, and those level curves *are* the phase portrait. The conserved quantity hands us every orbit for free, without ever solving the equation.

Now the whole picture snaps into focus. Low energy means small closed loops around the bottom — gentle swings. As E grows the loops bulge until, at exactly E = 2, the curve stops closing and instead runs into the upright saddle: this special trajectory is the separatrix, the dividing line where the pendulum has *just* enough energy to creep toward standing on its head and never quite arrive. Above E = 2 the curves no longer turn back at all — the pendulum has so much energy it whirls over the top again and again, round and round. Three qualitatively different fates, all read straight off the energy contours.

energy level        what the pendulum does            phase-plane shape
----------------    -----------------------------     -------------------
E small (E < 2)     small back-and-forth swings       closed loops (center)
E = 2 exactly       creeps toward upright forever      separatrix (to saddle)
E large (E > 2)     whirls over the top, round         wavy open curves

The Hamiltonian view, and why it generalises

This energy-conserving structure is not a lucky accident of the pendulum; it is a whole class of system. When the equations can be written as theta' = dH/dv and v' = -dH/dtheta for a single function H — here H = E, the energy — we call it a Hamiltonian system. The clever skew pairing of those partial derivatives is exactly what forces dH/dt = 0, so a Hamiltonian system *always* conserves H. That is why its equilibria are centers and saddles and never spirals or nodes: there is no friction term to make energy leak away, so trajectories can never wind inward to a point.

Add real-world friction — air resistance, a stiff pivot — and you append a damping term: theta'' + b theta' + sin(theta) = 0. Now dE/dt = -b v^2, which is negative whenever the pendulum is moving, so energy steadily bleeds out. The conserved-quantity argument breaks, the center at the bottom becomes an attracting spiral, and every swing decays toward hanging still. This is the honest distinction between the idealised, frictionless model and the damped one you would actually build — and it is why a real grandfather clock needs a mainspring constantly feeding energy back in.

Kepler's orbits, from the same toolbox

Now lift the same machinery into the sky. The Kepler two-body problem — one planet around a much heavier Sun — is governed by r'' = -(G M / |r|^3) r, again Newton's law made into a second-order ODE. It looks far worse than the pendulum, but the same two ideas tame it. First, there are two conserved quantities, not one: the total energy E, and the angular momentum about the Sun. Each conservation law is a relation the motion must obey forever, and together they pin the orbit down almost completely.

Conservation of angular momentum is precisely Kepler's second law — the planet sweeps out equal areas in equal times, racing when close to the Sun and dawdling when far. Using it to eliminate the angle reduces the motion to a single equation for the radius in an *effective potential*, the genuine gravitational pull plus a 'centrifugal' term that conservation conjures up. That radial equation is once again organised by its energy: bound orbits sit in a potential well as closed orbits — ellipses — while just-unbound ones at the escape energy trace parabolas, the celestial cousins of the pendulum's separatrix.

When pendulums couple: normal modes

One last bridge, back toward the linear world you mastered earlier. Hang two pendulums and join them with a weak spring, keep the swings small so each sin(theta) collapses to theta, and you get a pair of coupled oscillators — a linear system of two second-order equations. Diagonalising it, just as you diagonalised systems in the eigenvalue rung, reveals special patterns of motion called normal modes: combinations in which both pendulums oscillate at a single shared frequency. Every motion, however tangled it looks, is a superposition of these modes.

1. Write Newton's law for the system as one or more second-order equations — for a pendulum, theta'' = -(g/L) sin(theta).
2. Convert to a first-order system by naming velocities (v = theta'), then locate the equilibria where all rates vanish.
3. Look for a conserved quantity — usually energy. If dE/dt = 0, the trajectories are its level curves and you get the whole phase portrait without solving.
4. Interpret each energy level: small swings, the critical separatrix, or whirling-over for the pendulum; ellipse, parabola, or hyperbola for the orbit.