ANGULAR MOMENTUM

Every translational quantity in physics has a rotational twin. Swap straight lines for circles and the whole vocabulary rotates with them.

Mass becomes moment of inertia — not just how much stuff there is, but how far out it sits from the axis. Force becomes torque — not just a push, but a push with leverage. And linear momentum, m·v, becomes angular momentum, the quantity this whole topic is about.

For a point mass, angular momentum is the rotational version of p = m·v:

Where r is the position vector from a chosen origin and p is the linear momentum. For a point going round a circle of radius r at tangential speed v, the magnitude simplifies to L = m·r·v. For an extended rigid body spinning about an axis, summing over every bit of mass gives the form you'll see most often:

I is the moment of inertia, ω the angular velocity. It is the rotational F = m·a's sister equation — p = m·v rewritten for things that turn.

Leonhard Euler laid out the rotational form of Newton's laws in the 1750s — torque plays the role of force, moment of inertia the role of mass, angular momentum the role of momentum. Half a century later, Siméon Denis Poisson gave L its modern vector definition and showed precisely how torques change it over time. The rotational ledger has been complete ever since.

Angular momentum is a cross product, so for one paragraph let's say what the cross product is.

Given two vectors r and v, the cross product r × v is a third vector, perpendicular to both, whose magnitude is |r|·|v|·sin θ — where θ is the angle between them. The direction is set by the right-hand rule: point your fingers from r toward v, curl them, and your thumb points along r × v.

Two consequences you should feel in your bones. If r and v are parallel, sin θ = 0 and the cross product vanishes — no rotation about the origin, no angular momentum. If r and v are perpendicular, sin θ = 1 and the cross product is as big as it can get — all of v contributes to turning.

This is why L "points out of the plane of motion" — because a cross product always does. It is also why torques and angular velocities are vectors that live perpendicular to the physical plane they describe. Spinning things live in one dimension more than they look like they do.

Newton's second law says F = dp/dt: force is the rate of change of momentum. The rotational twin is just as short:

Torque is the rate of change of angular momentum. The payoff is immediate: when the net external torque on a system is zero, L is constant. Fixed in magnitude. Fixed in direction. Forever.

This is the conservation law for this topic. Energy is conserved when the physics doesn't depend on time. Linear momentum is conserved when there is no net external force. Angular momentum is conserved when there is no net external torque. The three laws march in lockstep — and in the next topic, Noether's theorem, we'll see they are literally three faces of one idea.

Before we get there, the practical question is: when are torques zero? Surprisingly often. Any force that points straight at a fixed centre — a central force — exerts no torque about that centre, because r and F are parallel and r × F = 0. Gravity from the Sun on a planet. The Coulomb force on an electron. A string holding a whirling stone. All central. All torque-free about their centres. All conserve L.

And any system free of external influence entirely — an ice-skater on frictionless ice, a cat in free fall, a gymnast mid-flight — has zero torque acting on it. So its total L is locked. What it does with that locked L is the rest of this topic.

Stand a skater on the ice with her arms out, and set her spinning at a couple of rotations a second. Now she pulls her arms in to her chest. She speeds up — dramatically, visibly, the kind of acceleration a first-year student refuses to believe until they see it.

What happened? Her mass didn't change. No external torque acted; the ice is effectively frictionless for twists about a vertical axis. So L = I·ω is fixed. But pulling the arms in shrank I — mass that was sitting at r ≈ 70 cm now sits at r ≈ 20 cm, and each bit contributes m·r² to the moment of inertia. If the arm contribution drops by a factor of (70/20)² ≈ 12, the whole I can easily fall by a factor of two or three. Because I·ω is locked, ω has to climb by the same factor. Three rotations per second becomes eight or nine.

Slide the arms in. Watch ω rocket up. Watch L stay flat to four decimal places. Now notice the other readout: rotational kinetic energy, KE = ½·I·ω². It is not flat. Writing it in terms of L,

shrinking I at constant L makes KE grow. Where did that extra energy come from? The skater's muscles. Her arms felt a centrifugal pull outward; to draw them in, her shoulders had to do work against that pull. That work turned into rotational kinetic energy. Let her push her arms back out and her muscles have to do negative work — the spin feeds energy back into her body.

L is conserved. KE is not. They are separate ledgers, and the skater's muscles are the bookkeeper running between them.

A cat dropped upside-down, back held rigid, lands on its feet. Everyone has watched this. Nineteenth-century physicists, watching it too, thought it was impossible.

The problem is simple to state. The cat starts at rest in the air. No external torque acts on it while it falls. Therefore its total angular momentum must stay exactly zero. But when it hits the ground it has visibly rotated by 180°. How did a rigid body turn itself over while keeping L = 0?

The answer, which took nearly a century to nail down, is that the cat is not rigid. In 1894 the French physiologist Étienne-Jules Marey photographed a falling cat with a chronophotograph firing 60 frames per second, and revealed what the naked eye missed: the cat bends at the waist. Its front half rotates one way, its back half rotates the other. Each half carries its own angular momentum. The two are equal and opposite — so they sum to zero, exactly as conservation demands.

But here is the subtle bit. If the front and back had the same moment of inertia, their rotations would cancel and the cat would end up exactly as it started. The cat's trick is to change its moments of inertia as it twists. It tucks its front legs in (small I) while extending its back legs (large I), then rotates its torso. For the total L to stay zero, the small-I front has to swing through a large angle, while the large-I back swings through a small angle in the opposite direction. Then it swaps — extends the front, tucks the back, and rotates the other way. Now the small-I back swings through a large angle in the same direction the front went the first time. Two half-twists, each of which respects L = 0, combine into a net flip.

This was not properly formalised until 1969, when T. R. Kane and M. P. Scher at Stanford wrote down the two-link "cat model" in the International Journal of Solids and Structures and showed that the motion is a geometric phase — the cat's shape traces a closed loop in its configuration space, and the orientation of its body is the holonomy of that loop. The cat does not disobey angular-momentum conservation. It exploits a subtlety of it that took a hundred years, from Marey's photographs to Kane–Scher's equations, for physics to fully articulate.

In 1609 Kepler announced three laws about the planets. The first said they move on ellipses. The third related period squared to semi-major axis cubed. They get most of the attention. But it is the second — a line from the Sun to a planet sweeps out equal areas in equal times — that was the real computational breakthrough, and with what we now know it takes three lines to prove.

In polar coordinates around the Sun, a small patch of area is dA = ½·r²·dθ. Differentiate:

where v_tan = r · dθ/dt is the tangential velocity. Multiply through by m/m:

with L = m·r·v_tan the angular momentum about the Sun. So "equal areas in equal times" is exactly the statement that L is constant. Which it has to be — gravity on the planet points straight at the Sun, and a central force exerts zero torque about its centre.

Kepler himself did not have this derivation. He got his law by staring at Tycho Brahe's tables of Mars for a decade and noticing that Mars moved faster near perihelion and slower near aphelion in the exact ratio "equal areas in equal times" would require. It was an empirical regularity wrung out of a mountain of numbers. A century later, Newton showed it was a one-line consequence of central-force dynamics — the same conservation law the figure skater obeys, playing out on a scale of hundreds of millions of kilometres.

Everything in this topic has leaned on one condition: no net external torque. Under that condition L is locked and every beautiful effect follows — the skater's spin-up, the cat's flip, the planet's sweep, the pulsar's metronome.

But of course real systems have torques all the time. A wrench on a bolt. Gravity on a leaning tower. The wind on a windmill. When τ ≠ 0, L changes at rate τ = dL/dt — and because L and τ are both vectors, the direction of L can change even when its magnitude doesn't. That is how a spinning top, tilted under gravity, precesses instead of falling over. It is how a gyroscope stabilises a ship. It is how the Earth's axis slowly traces a circle against the stars over 26,000 years.

The next chapter of classical mechanics is rotation itself, taken seriously. Torques, moments of inertia as full tensors, rigid-body dynamics, precession, nutation, Euler's equations. Most of it was first written down by Euler in the 1750s, and it is still the language every engineer who works with spinning things speaks today.

Before we get there, though, one more step in the conservation-law ladder. You have now seen three conserved quantities — energy, momentum, angular momentum — each following from a different condition on the dynamics. In 1918 Emmy Noether proved that all three are the same theorem in disguise, each tied to a symmetry of space or time. Her result is the deepest thing in classical mechanics, and it is where we go next.