Kepler's laws
Polar basis vectors
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Kepler

Polar basis vectors.
Shortly we shall move on to Kepler's first law, which concerns itself with the shape of the orbit that a planet makes around the Sun. Before we do that, we must deal with polar basis vectors.

The polar coordinate system is an effective way of representing the positions of bodies with the angle they make with the origin, and the distance they are away from it. Polar coordinates are useful for dealing with motion around a central point -- just the case we have with planets moving around the Sun.

However, to continue with our use of vectors, we must define a few polar basis vectors. Our first defined vector will be the unit radial vector. It will be represented by r and will be defined as

r == cos theta i + sin theta j. equation 1
Note that this vector is a function of theta; in other words, the unit vector representing the direction in which the body is located from the Sun is naturally dependent on the angle.

This is a fortunate definition; according to it,

r = r r equation 2
something we were already using! Therefore we need make no change of notation. Our definition of r as a polar basis vector merely quantifies our work in the plane of the orbit.

Since we have two Cartesian basis vectors, i and j, we should also have two polar basis vectors. The second basis vector, which we shall call the unit transverse vector and represent with theta, is defined as the rate of change of r with respect to theta:

theta == dr/dtheta equation 3
theta = -sin theta i + cos theta j. equation 4
This definition means that theta always points orthogonally to the unit radial vector. This makes it easy to talk about the component of a vector along r, the radial direction, and the component along theta, the transerse direction.

Note that if we again take the derivative of theta with respect to theta we find that

dtheta/dtheta = -cos theta i - sin theta j equation 5
dtheta/dtheta = -(cos theta i + sin theta) j equation 6
dtheta/dtheta = -r. equation 7
We shall make use of this later.

Let us, for the sake of an example, see what our velocity vector v and our angular momentum vector l would look like in terms of this new polar system. (We shall require them later in the proof anyway.)

Velocity is the instantaneous rate of change of the position of the planet:

v = dr/dt equation 8
v = (d/dt) (r r) equation 9
v = dr/dt r + r dr/dt. equation 10
But this looks like something we've already dealt with! You may be tempted immediately to substitute theta in for dr/dt, but remember the definition: theta = dr/dtheta, something considerably different. We use the chain rule to expand dr/dt into a form which includes dr/dtheta:
v = dr/dt r + r dtheta/dt dr/dtheta. equation 11
Since dr/dtheta is theta, the unit transverse vector, and the angular speed, omega, is defined as
omega == dtheta/dt equation 12
we can obtain our final expression for velocity in polar coordinates as
v = dr/dt r + r omega dr/dtheta. equation 13
To find a similar expression for the angular momentum vector l in polar coordinates, we go back to the expression we found for angular momentum:
l = m (r cross v). equation 14
We can substitute r r for r and the expression we just found for v (I told you we'd need it), and get
l = m [(r r) cross (dr/dt r + r omega dr/dtheta)]. equation 15
We expand this expression to obtain
l = m [(r r) cross (dr/dt r + r omega dr/dtheta)] equation 16
l = m (r r) cross (dr/dt r) + m (r r) cross (r omega theta) equation 17
l = m r dr/dt (r cross r) + m r2 omega (r cross theta). equation 18
Since, again, a vector crossed with itself is the zero vector, the first term evaluates to zero and we find that
l = m r2 omega (r cross theta). equation 19
Since r cross theta = k, our final expression for the angular momentum vector is
l = m r2 omega k. equation 20
As we took the magnitude of this vector before, we shall do it again:
l = |l| equation 21
l = |m r2 omega k| equation 22
l = m r2 omega |k| equation 23
l = m r2 omega equation 24
as the magnitude of a unit vector is, by definition, unity.
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