Law of Conservation of Angular Momentum
In an isolated system, total angular momentum remains constant over time: $$ \vec{L}_{\text{tot}} = \text{constant} $$
The conservation of angular momentum is one of the bedrock principles of physics. It explains how rotating systems behave when no external torques are acting on them.
This law stands alongside the conservation of energy and linear momentum as one of the cornerstones of mechanics. It applies universally-to rigid bodies, to individual particles, and on the grandest scales, to planets and stars.
What exactly is angular momentum? Angular momentum is a vector quantity that describes the rotational state of a body with respect to a fixed point or axis. For a particle of mass \$m\$, located at position \$\vec{r}\$ relative to a point \$O\$, moving with velocity \$\vec{v}\$, it is defined as: $$ \vec{L} = \vec{r} \times \vec{p} $$ where \$\vec{p} = m \vec{v}\$ is the particle’s linear momentum, and \$\times\$ denotes the cross product. The resulting angular momentum vector is perpendicular to the plane defined by \$\vec{r}\$ and \$\vec{p}\$.
Practical Examples
Example 1
A planet orbiting the Sun is governed by the Sun’s central gravitational pull.
Because there is no torque about the Sun, the planet’s angular momentum is conserved.
This is precisely what underlies Kepler’s second law: the radius vector sweeps out equal areas in equal intervals of time.
Example 2
A figure skater spinning on the ice can change their spin rate simply by pulling their arms inward toward the body.
The skater’s mass stays the same, but their moment of inertia $I$ decreases. To conserve angular momentum $L = I \omega$, the angular velocity $\omega$ must increase.
Theoretical Foundation: Noether’s Theorem
The conservation of angular momentum isn’t just an observed regularity-it rests on a profound theoretical basis.
Noether’s theorem states that every symmetry in the laws of physics corresponds to a conserved quantity.
In this case, rotational symmetry leads directly to the conservation of angular momentum.
If the laws of physics remain unchanged under rotations of the reference frame, then angular momentum must be conserved.
And so the principle holds.
