Angular Velocity and Momentum - Kurt Nalty

1.2 Intrinsic Angular Velocity
I start by returning to the Frenet formulas. Curvature is the inverse of the
radius of the circle, whose curvature matches the curvature of the trajectory at
that tangent. When people are referring to a radius when discussing the angular
velocity of a particle, it is really the radius of curvature (⃗r = ⃗n/κ) to which they
are referring.
Given that the tangent (along ⃗v), normal (along ⃗r) and binormal (along
⃗ω) form an orthogonal triad, we can rewrite the particle velocity and angular
velocity equation as
⃗v = ⃗ω × ⃗r → ⃗ω = 1 d⃗u
⃗n × ⃗v = κ⃗n × ⃗v = × ⃗v (4)
r ds
There are a few interesting forms for this formula.
⃗ω = d⃗u × ⃗v
ds
(5)
dθ
⃗ = d⃗u
dt ds × d⃗r
dt
(6)
dθ ⃗ = d⃗u × d⃗r
ds
(7)
dθ
⃗ = d⃗u
ds ds × d⃗r
ds
(8)
d ⃗ θ
ds
Let’s write some expressions for ⃗ω.
= d⃗u × ⃗u (9)
ds
⃗ω = d⃗u × ⃗v (10)
ds
= κ⃗n × ⃗v (11)
[ ]
⃗v × (⃗a × ⃗v)
=
v 4 × ⃗v (12)
[ ⃗av 2 ]
− (⃗v · ⃗a)⃗v
=
v 4 × ⃗v (13)
=
⃗a × ⃗v
v 2 (14)
Now let’s check for ⃗ω × ⃗r = ⃗v, where ⃗r is the radius of curvature.
⃗ω × ⃗r =
( ) ⃗a × ⃗v
v 2 × v2 (⃗v × (⃗a × ⃗v))
|⃗a × ⃗v| 2 (15)
=
(⃗v × (⃗a × ⃗v))
(⃗a × ⃗v) ×
|⃗a × ⃗v| 2 (16)
=
⃗v (⃗a × ⃗v) · (⃗a × ⃗v) − (⃗a × ⃗v) (⃗v · (⃗a × ⃗v))
|⃗a × ⃗v| 2 (17)
=
⃗v (⃗a × ⃗v) · (⃗a × ⃗v)
|⃗a × ⃗v| 2 (18)
= ⃗v (19)
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