angular momentum

Angular momentum is the rotational analogue of linear momentum, and the third great conserved quantity of classical mechanics. For a point particle at position r with momentum p, it is defined by L = r × p — a vector pointing perpendicular to the plane of r and p, with magnitude m·r·v·sin θ. For a rigid body rotating at angular velocity ω about an axis whose moment of inertia is I, L = I·ω, pointing along the axis of rotation.

Angular momentum is conserved in any system on which the net external torque is zero, in exact parallel to momentum conservation under zero external force. A central force (one pointing always toward a fixed centre, like gravity or a spring attached to a pivot) exerts zero torque about that centre, so angular momentum about the centre is conserved. This is why planetary orbits sweep out equal areas in equal times (Kepler's second law), why neutron stars spin hundreds of times per second (a collapsing stellar core preserves L while R shrinks enormously), and why a figure skater pulling in their arms speeds up (internal forces don't change L, but they do change I).

Noether showed in 1918 that angular-momentum conservation is the consequence of the fact that physics is invariant under rotations — a rotational symmetry generates a conserved rotational charge. It is a law built into the isotropy of space itself.