Environmental Engineering Reference
In-Depth Information
Putting
G M
λ
2
w
≡
u
−
and
M
2
λ
4
G
2
2
η
λ
2
c
2
≡
+
we have
d
w
√
c
2
θ
=−
−
w
2
cos
−
1
w
c
=
+
θ
0
,
(9.54)
where
θ
0
is a constant of integration which we are free to choose equal to zero
(because it corresponds only to a shift in what we call the zero on the polar angle
scale). Thus we have the solution that
w
=
c
cos
θ
and it is time to change back
to more familiar variables, i.e.
1
cos
θ.
G M
λ
2
G M
λ
2
1
r
−
2
ηλ
2
G
2
=
+
(9.55)
M
2
This is our final answer, for it tells us how
r
varies with
θ
. However, it is somewhat
cluttered with symbols and for that reason let us introduce two more quantities
λ
2
G
α
≡
(9.56)
M
and
1
.
2
ηλ
2
G
2
ε
≡
+
(9.57)
M
2
Notice that these are both constants of the motion. Thus the polar equation
describing the spatial trajectory
r(θ)
is simply
1
r
1
α
1
α
ε
cos
θ
−
=
(9.58)
which can be re-arranged to read
α
r
=
ε
cos
θ
.
(9.59)
1
+
