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

(Radial and angular velocity)

. ˙r

is the radial velocity, and

˙

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.