4.3 Scattering Chaos in a Magnetic Dipole
111
H =
1
2m
p −
e
c
A(r)
2
,
(4.11)
where r and p are the position and momentum, respectively, of the particle, A(r) =
A 0
−
y
r 3 ˆ
x +
x
r 3 ˆ
y
is the magnetic vector potential, and c is the speed of light (this
system was first studied extensively in Stomer 1907 and later in Contopoulis and
Vlahos 1975 and Dragt and Finn 1976). We can write the Hamiltonian in terms of
cylindrical coordinates, (ρ, φ, z), where x = ρcos(φ) and y = ρsin(φ). Then, the
Hamiltonian takes the form
H =
1
2m
p
2
ρ + p
2
z +
p φ
ρ
−
αρ
r 3
2
,
(4.12)
where α = eA 0 /c and r =
ρ 2 + z 2 . It is easy to see that the angular momentum
p φ is a constant of the motion, reflecting the fact that the system has azimuthal
symmetry about the z-axis. All the interesting motion occurs in the (ρ, z) plane.
We can write the Hamiltonian in terms of dimensionless momenta and coordinates (Dragt and Finn 1976). We shall assume that p φ > 0. Let ρ = ρ o α/p φ ,
z = z o α/p φ , p ρ = p o
ρ p 2
φ /α, p z = p o
z p 2
φ /α, and H = H o p 4
φ /mα 2 , where
(p o
ρ , p o
z , ρ o , z o , H o ) are dimensionless. Then let p o
ρ →p ρ , etc. The Hamiltonian, in
terms of dimensionless coordinates and momenta, takes the form
H =
1
2
(p
2
ρ + p
2
z ) + V (ρ, z) = E,
(4.13)
where E is the total energy, V (ρ, z) is the effective potential,
V (ρ, z) =
1
2
1
ρ
−
ρ
r 3
2
,
(4.14)
and all quantities are now dimensionless. The only controllable parameter is the
total energy E.
It is useful to look more carefully at the potential energy V (ρ, z). A contour plot
showing lines of constant V (ρ, z) is given in Fig. 4.9. There is an oddly shaped
potential energy bowl that lies to the left of the point (ρ = 2, z = 0). The lowest
value of the potential energy, V (ρ, z) = 0, is in the bowl (and at ρ = 0) and occurs
along a continuous line, ρ 2 − (ρ 2 + z 2 ) 3/2 = 0, called the thalweg, which intersects
the z-axis at ρ = 0 and ρ = 1. A saddle point occurs at point P S = (ρ = 2, z = 0).
The value of the potential energy at the saddle point is V (ρ, z) =
1
32 = 0.03125.
The saddle point provides a passageway from the outer region, ρ > 2, to the interior
of the bowl. As ρ→0, the potential energy tends toward infinity except along the
thalweg. As ρ→∞, the potential energy tends toward zero.
For large enough values of ρ, one can define an asymptotic region where particles
are free from the influence of the dipole potential. A particle that is scattered by the
dipole potential originates in the asymptotic region. It enters the reaction region (the
111
H =
1
2m
p −
e
c
A(r)
2
,
(4.11)
where r and p are the position and momentum, respectively, of the particle, A(r) =
A 0
−
y
r 3 ˆ
x +
x
r 3 ˆ
y
is the magnetic vector potential, and c is the speed of light (this
system was first studied extensively in Stomer 1907 and later in Contopoulis and
Vlahos 1975 and Dragt and Finn 1976). We can write the Hamiltonian in terms of
cylindrical coordinates, (ρ, φ, z), where x = ρcos(φ) and y = ρsin(φ). Then, the
Hamiltonian takes the form
H =
1
2m
p
2
ρ + p
2
z +
p φ
ρ
−
αρ
r 3
2
,
(4.12)
where α = eA 0 /c and r =
ρ 2 + z 2 . It is easy to see that the angular momentum
p φ is a constant of the motion, reflecting the fact that the system has azimuthal
symmetry about the z-axis. All the interesting motion occurs in the (ρ, z) plane.
We can write the Hamiltonian in terms of dimensionless momenta and coordinates (Dragt and Finn 1976). We shall assume that p φ > 0. Let ρ = ρ o α/p φ ,
z = z o α/p φ , p ρ = p o
ρ p 2
φ /α, p z = p o
z p 2
φ /α, and H = H o p 4
φ /mα 2 , where
(p o
ρ , p o
z , ρ o , z o , H o ) are dimensionless. Then let p o
ρ →p ρ , etc. The Hamiltonian, in
terms of dimensionless coordinates and momenta, takes the form
H =
1
2
(p
2
ρ + p
2
z ) + V (ρ, z) = E,
(4.13)
where E is the total energy, V (ρ, z) is the effective potential,
V (ρ, z) =
1
2
1
ρ
−
ρ
r 3
2
,
(4.14)
and all quantities are now dimensionless. The only controllable parameter is the
total energy E.
It is useful to look more carefully at the potential energy V (ρ, z). A contour plot
showing lines of constant V (ρ, z) is given in Fig. 4.9. There is an oddly shaped
potential energy bowl that lies to the left of the point (ρ = 2, z = 0). The lowest
value of the potential energy, V (ρ, z) = 0, is in the bowl (and at ρ = 0) and occurs
along a continuous line, ρ 2 − (ρ 2 + z 2 ) 3/2 = 0, called the thalweg, which intersects
the z-axis at ρ = 0 and ρ = 1. A saddle point occurs at point P S = (ρ = 2, z = 0).
The value of the potential energy at the saddle point is V (ρ, z) =
1
32 = 0.03125.
The saddle point provides a passageway from the outer region, ρ > 2, to the interior
of the bowl. As ρ→0, the potential energy tends toward infinity except along the
thalweg. As ρ→∞, the potential energy tends toward zero.
For large enough values of ρ, one can define an asymptotic region where particles
are free from the influence of the dipole potential. A particle that is scattered by the
dipole potential originates in the asymptotic region. It enters the reaction region (the