The Linearization of the Equations of Motion
75
So the complete solution of the inhomogeneous part has the form
x (s) =
1
ω 2 Λ (cos ωs − 1) + x i
cos ωs +
1
ω
1
ω
Λ sin ωs + a i
sin ωs
= x i cos φ +
1
ω
a i sin φ +
1
ω 2 Λ (1 − cos φ) ,
a (s) = −ω
1
ω 2 Λ (cos ωs − 1) + x i
sin ωs +
1
ω
Λ sin ωs + a i
cos ωs
= −ωx i sin φ + a i cos φ +
1
ω
Λ sin φ.
Thus we obtain
x f = x i cos φ + R 0 a i sin φ + R 0 (1 − cos φ)
1 + η 0
2 + η 0
δ i ,
a f = −
1
R 0
x i sin φ + a i cos φ + sin φ
1 + η 0
2 + η 0
δ i .
Finally we have to study the case of the time-of-flight part, which as we
said before can be obtained by mere integration. We have
l f =
L
0
−h
1 + η 0
2 + η 0
x +
1
(2 + η 0 )
2 δ
ds + l i
=
R0φ
0
−
1 + η 0
2 + η 0
1
R 0
x i cos
s
R 0
−
1 + η 0
2 + η 0
a i sin
s
R 0
−
1 − cos
s
R 0
1 + η 0
2 + η 0
2
δ i +
1
(2 + η 0 )
2 δ i
ds + l i
=
−
1 + η 0
2 + η 0
x i sin
s
R 0
+
1 + η 0
2 + η 0
R 0 a i cos
s
R 0
+
1 + η 0
2 + η 0
2
R 0
sin
s
R 0
δ i −
1 + η 0
2 + η 0
2
δ i s +
1
(2 + η 0 )
2 δ i s
R0φ
0
+ l i
= −
1 + η 0
2 + η 0
x i sin φ +
1 + η 0
2 + η 0
R 0 a i (cos φ − 1)
+
1 + η 0
2 + η 0
2
R 0 (sin φ) δ i −
η 0
2 + η 0
R 0 φδ i + l i .
As a result, we see that all the final coordinates indeed depend on all initial
coordinates in a linear fashion, and hence the relationship can be written in