Feynman's Lost Lecture (ft. 3Blue1Brown)

A lost Feynman lecture on planetary motion turns a familiar result—elliptical orbits—into a geometric inevitability. The core claim is that combining the inverse-square law of gravity with Kepler’s second law forces the planet’s velocity to behave in a way that, when translated back into position space, can only produce an ellipse. The payoff is less about heavy calculus and more about a chain of constructions: an ellipse appears from a rotated-line geometry trick, and the same tangency logic reappears when the lecture converts velocity information into orbital shape.

The argument begins with a precise definition of an ellipse using two foci: every point on the curve has a constant sum of distances to the foci. From there, it introduces a “mildly pleasing curiosity” that becomes central later. Start with a circle and pick an eccentric point inside it (not the center). Draw many chords from that eccentric point to points on the circle, then rotate each chord by 90 degrees about its midpoint. The rotated lines form tangents to a particular ellipse whose focal sum equals the circle’s radius. The proof hinges on similar triangles: along each rotated line (a perpendicular bisector of the original chord), the sum of distances from the two candidate foci stays constant at the intersection with the circle’s radius and increases elsewhere—meaning the line touches the ellipse at exactly one point. In short, rotating chords about their midpoints manufactures an ellipse because it manufactures its tangents.

With the geometry tool in hand, the lecture pivots to physics. Kepler’s second law says the area swept out per unit time is constant, a consequence of angular momentum conservation when the force always points toward the sun. For a small time step, the swept area is approximately ½·R·v_perp·Δt, and angular momentum conservation keeps R·v_perp fixed, so the area rate depends only on Δt. Crucially, this does not assume an ellipse; it only requires central (sunward) forces.

Next comes the inverse-square law. The sun’s gravitational force scales like 1/R^2, so the acceleration scales the same way. Over the time it takes the planet to traverse a small equal-angle slice, the lecture shows that the change in velocity is proportional to (Δt)/(R^2). But Kepler’s second law implies Δt itself scales like R^2 for equal angular slices, so the factors cancel: the velocity change across each slice has the same magnitude no matter where the planet is in its orbit. Because the force direction rotates by a constant angle from slice to slice, these equal-length velocity-change vectors line up as the sides of a regular polygon. As the slices get finer, that polygon approaches a circle.

The final step translates “velocity-space circle” into “position-space ellipse.” The lecture tracks how the planet’s velocity direction at a given orbital angle corresponds to tangency directions determined by points on the velocity circle. A clever 90-degree rotation—applied to the entire setup and then to each velocity direction—reorients those tangency constraints so they match the earlier ellipse construction: the rotated tangency lines correspond to the ellipse’s tangents, and the tangency points line up with the planet’s orbital positions. The result is a geometric QED: the orbit must be an ellipse, with the inverse-square law and Kepler’s second law jointly enforcing the necessary tangency structure.