4Orbits

IA Dynamics and Relativity

4.1 Polar coordinates in the plane
We choose our axes such that the orbital plane is
z
= 0. To describe the orbit,
we introduce polar coordinates (r, θ):
x = r cos θ, y = r sin θ.
Our object is to separate the motion of the particle into radial and angular
components. We do so by defining unit vectors in the directions of increasing
r
and increasing θ:
ˆ
r =
cos θ
sin θ
,
ˆ
θ =
sin θ
cos θ
.
x
y
r
ˆ
r
ˆ
θ
θ
These two unit vectors form an orthonormal basis. However, they are not basis
vectors in the normal sense. The directions of these basis vectors depend on
time. In particular, we have
Proposition.
d
ˆ
r
dθ
=
sin θ
cos θ
=
ˆ
θ
d
ˆ
θ
dθ
=
cos θ
sin θ
=
ˆ
r.
Often, we want the derivative with respect to time, instead of
θ
. By the
chain rule, we have
d
ˆ
r
dt
=
˙
θ
ˆ
θ,
d
ˆ
θ
dt
=
˙
θ
ˆ
r.
We can now express the position, velocity and acceleration in this new polar
basis. The position is given by
r = r
ˆ
r.
Taking the derivative gives the velocity as
˙
r = ˙r
ˆ
r + r
˙
θ
ˆ
θ.
The acceleration is then
¨
r = ¨r
ˆ
r + ˙r
˙
θ
ˆ
θ + ˙r
˙
θ
ˆ
θ + r
¨
θ
ˆ
θ r
˙
θ
2
ˆ
r
= (¨r r
˙
θ
2
)
ˆ
r + (r
¨
θ + 2 ˙r
˙
θ)
ˆ
θ.
Definition
. ˙r
˙
θ
is the
angular velocity.
Example
(Uniform motion in a circle)
.
If we are moving in a circle, then
˙r
= 0
and
˙
θ = ω = constant. So
˙
r = rω
ˆ
θ.
The speed is given by
v = |
˙
r| = r|ω| = const
and the acceleration is
¨
r = rω
2
ˆ
r.
Hence in order to make a particle of mass
m
move uniformly in a circle, we must
supply a centripetal force mv
2
/r towards the center.