Equations of motion
We can write expressions for both the angular momentum and the total energy. If
p
_{
θ
}
is the magnitude of the momentum in the tangential direction, then
since this perpendicular to
,
L = rp
_{
θ
}
. But
p
_{
θ
} = mv
_{
θ
} = m
= mr
= mr
.
Hence
L = r(mr
) = mr
^{2}
. Hence:
L = mr
^{2}


We can also write an expression for the total energy as a sum of the radial
kinetic energy term, the angular kinetic energy term and the potential term:
Rearranging and dividing through the left side by
m
^{2}
r
^{4}
and the right by
L
^{2}
, and canceling factors
of
dt
^{2}
we find:
To find the equations of motion we want to find
r
in terms of
θ
. In principle we could take the square root of both sides of
the above equation, separate the variables, integrate to find
θ(r)
, and then invert to find
r(θ)
. This involves a lot of
messy algebra which is not very enlightening, so we will just state the
result:
= (1 + εcosθ)


where
ε
is the eccentricity that we saw in
Ellipses
and Foci and is now given by:
ε =


This equation determines the motion of all orbital systems in the universe.
We can also find the maximum and minimum values of
r
. The minimum occurs
where the expression for
1/r
is a maximum. This is when
cosθ = 1
and
the maximum is therefore
. Thus:
r
_{min} =


The maximum occurs for the minimum of
1/r
. There are two cases: first, when
ε < 1
, the minimum of
1/r
is
.
Thus:
r
_{max} =


When
ε≥1
, the expression for
1/r
can take on the value zero when
cosθ =  1/ε
. Hence the maximum value
r
can take on is infinite
in this case.
We can also take the equation and using
r
^{2} = x
^{2} + y
^{2}
, and
cosθ = x/r
, we can write:
Solving the orbits
The orbits are determined by the various values that
ε
can take.
Circular orbits
When
ε = 0
, the expression for
ε
tells us that
E = 
. The negative value of the energy just means that the
potential energy is more negative than the kinetic energy is positive. In this
case we have
r
_{min} = r
_{max} =
. The particle is trapped
at the very bottom of a potential well, and the radius does not change as it
goes around the orbit, hence forming a circle. Substituting this value for
r
into the energy we have
E = 
. Note that we could have derived
this directly by summing the potential energy we found for a circular orbit with
the kinetic energy (Gravitational Potential Energy).
E = 1/2mv
^{2} + U =  = 


Figure %: Potential well for a circular orbit.
In the case
ε = 0
we can see that this equation simplifies to
x
^{2} + y
^{2} =
. This describes a circle with
radius
.
Elliptical Orbits
Elliptical orbits occur when
0 < ε < 1
. This means that
 < E < 0
. Again the particle is trapped in a potential
well, oscillating now between
r
_{min}
and
r
_{max}
.
Figure %: Potential well for a elliptical orbit.
We can also solve the equation for
ε
in this range. The algebra
turns out to be complicated, but we find that:
+ = 1


where
a =
and
b =
. This is an ellipse with its center at
( L
^{2}
ε/GMm
^{2}(1  ε
^{2}), 0)
, and with semimajor and semiminor axis length
a
and
b
respectively. It can also be shown that one focus of this ellipse is at the
origin.