References
417
The action variable for a trajectory trapped in the well is given by
J =
1
2π
pdz =
√
2μF o
π
z −
0
dz
√
z
z 2 −
|E o |z
F o
+
κ 0 e 2
F o
=
2(z + )
3
2
√
2μF o
3π
[(1 + κ
2 )E(κ) − (1 − κ
2 )K(κ)],
(B.56)
where K(κ) and E(κ) are complete elliptic integrals of the first and second kinds,
respectively, and the modulus κ is defined as κ 2 =
z −
z +
(Byrd and Friedman 1971).
We cannot explicitly revert Eq. (B.56) to find E o as a function of J . However, we
can find the derivative of E o . Thus, the frequency, ˙
, is given by
˙
=
∂|E o |
∂J
=
π
√
μF o
√
2(z + )
1
2 (E(κ) − K(κ))
,
(B.57)
and the angle variable is given by
=
π
√
μF o t
√
2(z + )
1
2 (E(κ) − K(κ))
+ (0).
(B.58)
Using the relation p = m˙ z, we can write
2F o
m
dt = ±
√ zdz
(z − z − )(z − z + )
.
(B.59)
If we let
z(t) = z − sn
2 (u, κ),
(B.60)
then we find
u − E(u, κ) = ±
(E(κ) − K(κ)))
π
,
(B.61)
where E(u, κ) is the incomplete elliptic integral of the second kind.
References
Byrd PF, Friedman D (1971) Handbook of elliptic integrals for engineers and scientists. Springer,
Berlin
Goldstein H (1980) Classical mechanics. Addison-Wesley, Reading