406
B Simple Models
Fig. B.1 The pendulum: x
versus V (x) = −g cos(x)
Fig. B.2 The pendulum:
phase space plot for m = 1
B.1.1 Libration—Trapped Orbits (E 0 < g)
Libration corresponds to motion trapped in the cosine potential well. The pendulum
bob never goes over the top. In this energy regime, the turning points of the orbit
(the points where p = 0) are given by
x ± = ±arccos
−
E o
g
.
(B.3)
The action for this case is defined as
J =
1
2π
pdx =
√
2m
π
x +
x −
dx
E o + g cos(x)
=
8
√ mg
π
[E(κ) − κ
K(κ)],
(B.4)
where K(κ) and E(κ) are the complete elliptic integrals of the first and second kinds,
respectively, and κ is the modulus, defined as κ 2 =
E o +g
2g (Byrd and Friedman
1971). We cannot explicitly write the energy, E o , as a function of J , but we can
obtain the derivative of the energy and therefore the angle variable, . We find
B Simple Models
Fig. B.1 The pendulum: x
versus V (x) = −g cos(x)
Fig. B.2 The pendulum:
phase space plot for m = 1
B.1.1 Libration—Trapped Orbits (E 0 < g)
Libration corresponds to motion trapped in the cosine potential well. The pendulum
bob never goes over the top. In this energy regime, the turning points of the orbit
(the points where p = 0) are given by
x ± = ±arccos
−
E o
g
.
(B.3)
The action for this case is defined as
J =
1
2π
pdx =
√
2m
π
x +
x −
dx
E o + g cos(x)
=
8
√ mg
π
[E(κ) − κ
K(κ)],
(B.4)
where K(κ) and E(κ) are the complete elliptic integrals of the first and second kinds,
respectively, and κ is the modulus, defined as κ 2 =
E o +g
2g (Byrd and Friedman
1971). We cannot explicitly write the energy, E o , as a function of J , but we can
obtain the derivative of the energy and therefore the angle variable, . We find