Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 614

Alternatively, we can compute the angular momentum for an individual
point mass directly from its linear momentum by

Relationship between
linear and angular
momentum

L = rP sinφ, (12.28)

where P is the linear momentum. The purpose of the sinφ term is the
same as in the computation of torque: it isolates the tangential motion. If
the trajectory is known to be orbital, this term can be omitted since it will
always be unity. Notice that since Equation (12.28) contains a factor r,
it really is only for orbital angular momentum. Equation (12.27) can also
be used for orbital angular momentum, provided that J and ω are both
measured relative to the same pivot, but Equation (12.27) is probably more
appropriate for spin angular momentum of a rigid body, where J measures
the moment of inertia of the entire body. In any case, the spin angular
momentum of a rigid body can be computed by breaking the object into
mass elements and taking the sum of the orbital angular momenta of these
elements. Here are several ways in which the sum could be accomplished:

Spin angular momentum
of a rigid body

(r i 2 m i )ω L = ωJ = ω J i = J i ω =

i i i

= r i m i (r i ω) = r i m i v i = r i P i = L i .

i i i i

Here the subscripted variables refer to values for a particular particle. Note
that we dropped the sinφ term under the assumption that each particle is
in an orbital trajectory.
The second interpretation of linear momentum that we discussed is that
of momentum as the time integral of force. When we apply force over time,
linear momentum is accumulated. A similar relationship exists between
angular momentum and torque. As torque is applied over time, it builds
up angular momentum; equivalently, torque is equal to the rate of change
(derivative) of angular momentum. As with linear momentum, this can be
interpreted as a conservation law.

Torque and the Conservation of Angular Momentum

τ = dL L = τ dt,

dt .

In the absence of external torque, angular momentum is conserved.