Definition:Angular Momentum
Definition
Particle
Let $P$ be a particle of mass $m$ moving with velocity $\mathbf v$ relative to a point $O$.
The angular momentum of $P$ relative to (or about) $O$ is defined as:
- $\mathbf L = \mathbf r \times \mathbf p = m \paren {\mathbf r \times \mathbf v}$
where:
- $\mathbf p$ denotes the (linear) momentum of $P$
- $\mathbf r$ denotes the position vector of $P$ with respect to $O$
- $\times$ denotes the vector cross product.
Aggregation of Particles
Let $\PP = \set {P_i: i \in I}$ be an aggregation of particles, indexed by $I$, all in motion relative to a point $O$.
For all $i \in I$, let:
- the mass of particle $P_i$ be $m_i$
- the velocity of particle $P_i$ relative to $O$ be $\mathbf v_i$.
The angular momentum of $\PP$ relative to (or about) $O$ is defined as the sum of the angular momenta of each of the particles in $\PP$:
- $\ds \mathbf L = \sum_{i \mathop \in I} \mathbf r_i \times \mathbf p_i = m \paren {\mathbf r_i \times \mathbf v_i}$
where:
- $\mathbf p_i$ denotes the (linear) momentum of particle $P_i$ for $i \in I$
- $\mathbf r_i$ denotes the position vector of particle $P_i$ for $i \in I$ with respect to $O$
- $\times$ denotes the vector cross product.
Rigid Body
The angular momentum of a rigid body about a point $O$ is the vector cross product of its moment of inertia $\mathbf I$ about $O$ by its angular velocity $\omega$ about $O$:
- $\mathbf L = \mathbf I \times \omega$
Angular momentum is a vector quantity.
Dimension
The dimension of measurement of angular momentum is $\mathsf {M L}^2 \mathsf T^{-1}$.
Symbol
The usual symbol used to denote the angular momentum of a body is $\mathbf L$.
Also known as
Angular momentum is also (rarely) known as moment of momentum or rotational momentum.
Also see
- Definition:Orbital Angular Momentum
- Definition:Intrinsic Angular Momentum, or Definition:Spin Angular Momentum
- Results about angular momentum can be found here.
Linguistic Note
The plural of momentum is momenta.
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): angular momentum (moment of momentum)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): angular momentum (moment of momentum)