Angular momentum

Angular momentum is a fundamental quantity in classical mechanics describing the rotational analogue of linear momentum. It plays a central role in the dynamics of rotating objects and systems, and its conservation underpins many familiar physical phenomena. Angular momentum possesses both magnitude and direction, the latter described by a pseudovector perpendicular to the plane of rotation. It is conserved in any closed system in which no external torque acts, making it a cornerstone of rotational dynamics.

Fundamental Concepts

Angular momentum describes how difficult it is to change the rotational motion of a body. For a point particle in three-dimensional Euclidean space, its angular momentum is defined as the cross product of its position vector relative to a chosen origin and its linear momentum. The choice of origin matters because position is measured from that point. The total angular momentum of a composite system is the sum of the angular momenta of its components, making it an extensive property, yet it also has intensive aspects due to its directional nature.
In rigid bodies and continua, angular momentum can be expressed as a volume integral of angular momentum density. This formulation is essential in fluid mechanics and astrophysical systems, where rotation is distributed across a continuous mass.
Angular momentum is closely related to torque, which is the rate of change of angular momentum. Internal torques always cancel out in a closed system, so only external torques can alter the total angular momentum. In the absence of external torque, angular momentum remains constant. This principle explains the stability of gyroscopes, the dynamics of bicycles and rifled projectiles, the spiralling form of hurricanes, and the rapid rotation of compact astronomical bodies such as neutron stars.

Orbital and Spin Angular Momentum

Angular momentum takes two primary forms. Orbital angular momentum refers to rotation about an external point. For example, the Earth possesses orbital angular momentum due to its revolution around the Sun. Spin angular momentum refers to rotation about an object’s own axis, such as the Earth’s daily rotation.
For a single particle, orbital angular momentum is always parallel to its orbital angular velocity, with proportionality depending on its mass and radial distance. For a rigid body, however, spin angular momentum is related to spin angular velocity through the moment of inertia tensor, reflecting how mass is distributed relative to the axis of rotation.
In two-dimensional motion, angular momentum can be treated as a pseudoscalar, with its sign indicating one of two possible directions perpendicular to the plane of motion. This simplification is often used in planar orbital mechanics.

Angular Momentum in Classical Mechanics

Angular momentum can be expressed in several ways depending on context. When the motion is circular, the quantity simplifies to the product of radius and linear momentum. For non-circular motion, one uses the component of velocity perpendicular to the radius vector. The perpendicular distance from the origin—also called the moment arm—is central to these definitions.
In Lagrangian mechanics, angular momentum is derived as the canonical momentum conjugate to a cyclic angular coordinate. For a particle constrained to move on a circle, the Lagrangian depends on the square of the angular velocity, and the resulting momentum corresponds directly to angular momentum. This approach demonstrates conservation of angular momentum when the Lagrangian is independent of the angular coordinate, implying rotational symmetry.

Example Calculations and Rotational Systems

The angular momentum of different rotating bodies can be modelled using simple formulae based on mass, frequency of rotation and radius. For a body in circular orbit, angular momentum depends on orbital radius and frequency. For the Earth, the orbital angular momentum relative to the Sun far exceeds its spin angular momentum.
A uniform rigid sphere rotating about its axis has angular momentum proportional to its moment of inertia and frequency. When the density is specified instead of mass, angular momentum can be expressed in terms of density and radius, illustrating how distribution of mass affects rotational behaviour. Spinning discs—such as those used in mechanical systems or recreational activities—follow similar relationships, with different coefficients depending on whether rotation occurs about the central axis or a diameter. These simplified models are essential for understanding how geometry influences rotational motion.

Three-dimensional Representation

In three dimensions, angular momentum remains a pseudovector quantity. The full vector form allows the description of complex rotational motion, including precession, nutation and rotation in non-planar systems. The generality of this vector representation enables the theory to address systems ranging from planetary orbits to rotating machinery and turbulent fluid flows. The vector form of angular momentum ensures that rotational dynamics can be described consistently regardless of the orientation or complexity of the motion.