Feynman, Orbits, and the Power of Doing It Step by Step

​By the time we reach Chapter 9–7 of The Feynman Lectures on Physics, the tone of the book has subtly shifted. Feynman is no longer primarily concerned with finding elegant analytical solutions. Instead, he is showing us how physicists actually wrestle complicated systems into submission. The subject is planetary motion, but the deeper lesson is about method, approximation, and the quiet power of Newton’s laws when they are allowed to operate incrementally.

Feynman begins by pointing out that techniques which work beautifully for simple systems, such as oscillating springs, do not transfer neatly to planets moving under gravity. The force law is different, and that difference matters. A planet does not experience a restoring force proportional to its displacement. Instead, the force depends on its distance from the Sun in a way that makes the mathematics stubbornly resistant to tidy solutions. Rather than fighting this, Feynman changes strategy.

The core idea is to stop thinking of an orbit as a curve to be discovered and start thinking of it as a history that unfolds in time. At any given moment the planet has a position and a velocity. From that position, gravity produces an acceleration directed toward the Sun. That acceleration changes the velocity, the changed velocity moves the planet to a new position, and the whole process repeats. Nothing mysterious is happening; it is simply Newton’s laws applied over and over again.

To make this workable, Feynman breaks everything into components. Motion is described separately in horizontal and vertical directions, even though the planet itself follows a curved path. The gravitational pull is always inward, but its influence on each direction depends on where the planet happens to be at that instant. What emerges is a slightly surprising result: when expressed component by component, gravity involves an inverse cube of the distance rather than the inverse square we usually quote. This is not a new force, just the familiar one seen through the lens of geometry.

At this point Feynman makes a deliberate simplification. By choosing units cleverly, the gravitational constant and the Sun’s mass are absorbed into the system, leaving behind a set of relations that express the essence of the motion without unnecessary clutter. This is characteristic of his style. He is not interested in carrying constants around unless they illuminate something important.

What follows is perhaps the most revealing part of the chapter. Feynman does not wave his hands and declare the orbit to be an ellipse. He actually performs the calculation. Time is divided into small steps. At each step, the planet’s acceleration is computed from its current position. That acceleration adjusts the velocity, and the updated velocity moves the planet onward. The process is tabulated carefully, one moment after another.

As the calculation unfolds, something deeply satisfying happens. When the successive positions are plotted, the orbit takes shape on its own. Near the Sun the planet moves quickly; farther away it slows. The curve bends naturally, not because we demanded it do so, but because Newton’s laws insist upon it. In a surprisingly small number of steps, the planet completes a large fraction of its journey around the Sun.

This moment carries an important message. We do not need a closed-form solution to understand planetary motion. We only need the laws of motion and the patience to apply them repeatedly. The orbit is not assumed; it is earned.

Feynman then widens the scope dramatically. If one planet can be handled this way, why not many? Why not let the Sun move as well? Each body in the system pulls on every other, and the acceleration of any one object becomes the result of all those interactions combined. The equations multiply rapidly, and the bookkeeping becomes formidable. But nothing fundamentally new is required. The same step-by-step logic still applies.

At this point Feynman introduces computation, not as a futuristic curiosity, but as a natural extension of Newtonian mechanics. A calculating machine can perform in seconds what would take a human an impractical amount of time. With sufficiently small time steps, the accuracy becomes astonishing. The motion of Jupiter, including the subtle perturbations from all the other planets, can be followed with extraordinary precision in a matter of minutes.

The chapter ends where it began, but with a profound change in perspective. What once seemed impossibly complex turns out to be completely calculable. Not because nature is simple, but because its laws are consistent. Armed with Newton’s principles and the ability to perform repeated calculations, we can follow the motion of planets, not approximately in a poetic sense, but quantitatively and in detail.

Chapter 9–7 is therefore not just about planetary motion. It is about learning to trust a method. When exact solutions fail, physics does not stop.