4Orbits

IA Dynamics and Relativity

4.3 Equation of the shape of the orbit
In general, we could determine r(t) by integrating the energy equation
E =
1
2
m ˙r
2
+ V
eff
(r)
t = ±
r
m
2
Z
dr
p
E V
eff
(r)
However, this is usually not practical, because we can’t do the integral. Instead,
it is usually much easier to find the shape r(θ) of the orbit.
Still, solving for
r
(
θ
) is also not easy. We will need a magic trick we
introduce the new variable
Notation.
u =
1
r
.
Then
˙r =
dr
dθ
˙
θ =
dr
dθ
h
r
2
= h
du
dθ
,
and
¨r =
d
dt
h
du
dθ
= h
d
2
u
dθ
2
˙
θ = h
d
2
u
dθ
2
h
r
2
= h
2
u
2
d
2
u
dθ
2
.
This doesn’t look very linear with
u
2
, but it will help linearizing the equation
when we put in other factors.
m¨r
mh
2
r
3
= F (r)
then becomes
Proposition (Binet’s equation).
mh
2
u
2
d
2
u
dθ
2
+ u
= F
1
u
.
This still looks rather complicated, but observe that for an inverse square
force, F (1/u) is proportional to u
2
, and then the equation is linear!
In general, given an arbitrary F (r), we aim to solve this second order ODE
for u(θ). If needed, we can then work out the time-dependence via
˙
θ = hu
2
.