of 7
arXiv:1008.1379v1 [physics.class-ph] 8 Aug 2010
Wheels within wheels: Hamiltonian dynamics as a hierarchy o
f action variables
Rory J. Perkins and Paul M. Bellan
Applied Physics, Caltech, Pasadena, CA 91125
(Date textdate)
In systems where one coordinate undergoes periodic oscilla
tion, the net displacement in any
other coordinate over a single period is shown to be given by d
ifferentiation of the action integral
associated with the oscillating coordinate. This result is
then used to demonstrate that the action
integral acts as a Hamiltonian for slow coordinates providi
ng time is scaled to the “tick-time”of the
oscillating coordinate. Numerous examples, including cha
rged particle drifts and relativistic motion,
are supplied to illustrate the varied application of these r
esults.
101
Hamiltonian dynamics is almost ubiquitous in physics and describes such
varied phenomena as celestial mechanics,
optics, fluid dynamics, quantum mechanics, and charged particle mo
tion in electromagnetic fields. Guiding center
theory, an approximation of Hamiltonian dynamics for charged part
icle motion in magnetic fields, describes the motion
of the particle’s cyclotron-orbit averaged position, or guiding cent
er [1]. The guiding center can be thought of as
a “quasi-particle” subject to new types of forces and manifesting
various drifts. We develop a general model, not
restricted to charged particle motion, of multi-dimensional system
s with a periodic variable and find drifts that cannot
be calculated using guiding center theory which becomes a limited exam
ple of the more general model. The model
shows that the action integral associated with the oscillatory coor
dinate acts as an effective Hamiltonian for the
remaining, slow coordinates providing time is measured in clock cycles o
f the oscillations. We note that Hamiltonian-
type aspects of action integrals have been previously discussed in s
pecific situations [2] but without developing a
general demonstration and relying on the detailed equations of mot
ion in their proofs. The model presented here
generates a hierarchy of “wheels-within-wheels” Hamiltonian syste
ms such that the action integral associated with
periodic motion at any level in the hierarchy acts as the Hamiltonian fo
r the next slower periodic motion.
Consider a 2D time-independent Hamiltonian system
H
(
ξ, P
ξ
, P
η
) with an ignorable coordinate
η
and where the
ξ
-motion is periodic, i.e.,
ξ
(
t
+ ∆
t
) =
ξ
(
t
), with no limit on the amplitude of
ξ. P
η
evolves trivially:
̇
P
η
= 0, but the
η
evolution is in general nontrivial. The period ∆
t
can be imagined as a clock tick over which
η
undergoes a net change
∆
η
. We claim that
∆
η
=
−
∂J/∂P
η
,
(1)
where
J
(
H, P
η
) =
∮
P
ξ
(
H, ξ, P
η
)
dξ
(2)
is the action integral [3] associated with
ξ.
Equation (1) means that if
J
(
H, P
η
) is known, the net change of
η
during
one period of
ξ
can be calculated without having to consider the potentially complicat
ed form of ̇
η.
To prove Eq. (1), first note that there is no contribution from diffe
rentiating the integral’s bounds, so
∂J
∂P
η
=
∮
∂P
ξ
∂P
η
dξ.
(3)
The differential of
H
is
dH
=
∂H
∂ξ
dξ
+
∂H
∂P
ξ
dP
ξ
+
∂H
∂P
η
dP
η
,
(4)
so
∂P
ξ
∂P
η
=
−
∂H/∂P
η
∂H/∂P
ξ
.
(5)
Using Eq. (5) and Hamilton’s equations in Eq. (3) gives
∂J
∂P
η
=
−
∮
∂H/∂P
η
∂H/∂P
ξ
dξ
=
−
∮
dη/dt
dξ/dt
dξ
=
−
∆
η.
(6)
If there are other ignorable coordinates in the system, then suita
bly adjusted versions of Eq. (6) apply to each of
them. Equation (1) generalizes the well-known theorem [3] that the
period of motion is given by a partial derivative
of
J
with respect to
H,
namely
∆
t
=
∂J/∂H.
(7)
Equation (7) resembles Eq. (1) because (
t,
−
H
) form a pair of canonical coordinates in extended phase space, so
Eq. (7) is a special case of the theorem presented here. The drift
, or net time evolution, of
η
is clearly
∆
η
∆
t
=
−
∂J/∂P
η
∂J/∂H
,
(8)
which generalizes the particle drifts associated with guiding center t
heory.
102
We now relax the requirement that
η
is ignorable and allow the oscillations to evolve adiabatically. We do so by
coupling the original Hamiltonian, now denoted as
H
loc
, to an external system
H
ext
that is otherwise isolated. This
gives a total Hamiltonian
H
(
ξ, P
ξ
, η, P
η
) =
H
loc
(
ξ, P
ξ
, η, P
η
) +
H
ext
(
η, P
η
)
.
(9)
We presume the system behaves as follows. First, the
ξ
-oscillation is described by the
local
Hamiltonian
H
loc
in
which
η
and
P
η
play the role of slowly varying parameters:
dξ/dt
=
∂H
loc
/∂P
ξ
and
dP
ξ
/dt
=
−
∂H
loc
/∂ξ.
Second,
the “parametric” coordinates
η
and
P
η
are described by the
total
Hamiltonian
H
, so
dη/dt
=
∂H/∂P
η
and
dP
η
/dt
=
−
∂H/∂η
. The local and external systems exchange energy, but the tota
l energy
E
=
E
loc
(
t
) +
E
ext
(
t
) is conserved
since the entire system is isolated.
J
is defined as in Eq. (2) but is now also a function of
η
. As in Ref. [4], we assume
it is a good approximation to hold the parametric coordinates
η
and
P
η
fixed while evaluating the
ξ
action integral.
J
is an adibatic invariant and is thus conserved. Furthermore,
J
depends only on
H
loc
,
i.e.
J
=
J
(
H
loc
, η, P
η
) =
J
(
H
−
H
ext
(
η, P
η
)
, η, P
η
)
,
because
H
loc
is sufficient to prescribe the
ξ
dynamics. A proof analogous to that of Eq. (1)
shows
∂J
∂η
= ∆
P
η
,
∂J
∂P
η
=
−
∆
η.
(10)
Note that
J
=
J
(
H
−
H
ext
(
η, P
η
)
, η, P
η
) depends on
η
and
P
η
both implicitly through
H
ext
and also explicitly.
Accordingly, ∆
η
and ∆
P
η
have two terms: one term comes from the explicit dependence and is
the drift of the
system; the second term comes from the implicit dependence and is t
he slow change of
H
ext
.
Equations (10) have the makings of a Hamiltonian system with
−
J
serving as the Hamiltonian. They are
precisely
Hamiltonian as follows. We define discretized derivatives
dη/dt
= ∆
η/
∆
t
and
dP
η
/dt
= ∆
P
η
/
∆
t
that capture the
net rates of change of
η
and
P
η
.
Upon invocation of a rescaled time
τ
normalized by the
ξ
-period:
dτ
=
dt/
∆
t,
(11)
Eqs. (10) become
dη
dτ
=
∂
∂P
η
(
−
J
)
,
dP
η
dτ
=
−
∂
∂η
(
−
J
)
.
(12)
Thus,
−
J
is the Hamiltonian for the averaged system provided time is measured
in units of ∆
t.
It should be noted
that
τ
is the angle variable conjugate to
J
.
Alternatively, we can obtain a Hamiltonian for the
ξ
-averaged system by solving
J
=
J
(
H
loc
, η, P
η
) for
H
loc
=
H
loc
(
J, η, P
η
) which upon inserting in Eq. (9) gives
H
=
H
loc
(
J, η, P
η
) +
H
ext
(
η, P
η
)
.
(13)
Solution of Eq. (13) for
J
gives
J
=
J
(
H, η, P
η
)
.
The differential of
J
using this latter form is
dJ
=
∂J
∂H
dH
+
∂J
∂η
dη
+
∂J
∂P
η
dP
η
.
(14)
Equation (14) determines
∂H/∂P
η
=
−
(
∂J/∂P
η
)
/
(
∂J/∂H
) etc., so using Eqs. (10)
∂H
∂P
η
=
−
∂J/∂P
η
∂J/∂H
=
−
−
∆
η
∆
t
=
dη
dt
(15)
∂H
∂η
=
−
∂J/∂η
∂J/∂H
=
−
∆
P
η
∆
t
=
−
dP
η
dt
.
(16)
Thus,
H
, written as Eq. (13), generates the discretized derivatives. The
term
H
loc
(
J, η, P
η
) is an adiabatic potential [5],
the residue of averaging the periodic
ξ
-motion. For systems approximating a harmonic oscillator,
J
= 2
πH
loc
/ω
(
η, P
η
),
so the adiabatic potential is
H
loc
=
Jω/
2
π
, showing that
J
acts like an electrostatic charge and
ω
(
η, P
η
) acts like
an electrostatic potential. The magnitude of this “
J
-charge” depends on the amplitude of the
ξ
-oscillation. The use
of
−
J
(
H, η, P
η
) as a Hamiltonian with normalized time
τ
and the use of
H
(
J, η, P
η
) with regular time are entirely
equivalent. Practically, though, there are techniques to evaluate
J
directly [6], so using
−
J
as the Hamiltonian spares
one from inverting
J
for
H
, which might not be analytically feasible.
We now provide examples illustrating applications. Fig. 1 shows an elect
ron moving in the
rz
-plane and subjected
to the magnetic field of a current-carrying wire aligned along the
z
-axis. The
z
coordinate corresponds to
η
and is
103
FIG. 1. For planar electron motion outside a current-carryi
ng wire, the axial displacement ∆
z
can be derived from the radial
action variable.
ignorable; the radial motion is periodic and not ignorable because of t
he magnetic field gradient. The electron displaces
itself an axial distance ∆
z
with every gyration as shown in Fig. 1. Using the characteristic veloc
ity
β
=
μ
◦
Ie/
2
πm
[7], the Hamiltonian is
H
=
1
2
mv
2
=
P
2
r
2
m
+
(
P
z
−
mβ
ln (
r/R
))
2
2
m
,
(17)
where
R
is an arbitrary constant of integration.
J
can be evaluated exactly [8] using the substitution cos
θ
=
(
P
z
−
mβ
ln (
r/R
))
/mv
and the integral representation of the modified Bessel function
I
n
(
x
) =
π
−
1
∫
π
0
e
x
cos
θ
cos(
nθ
)
dθ
[9]
so that
J
=
∮
P
r
dr
= 2
πmvr
gc
I
1
(
v
β
)
,
(18)
where
r
gc
=
R
exp(
P
z
/mβ
).
r
gc
, plotted as a dashed line in Fig. 1, is the radial position at which the
z
-velocity
vanishes, as can be checked from
P
z
=
mv
z
+
mβ
ln
r/R. J
generalizes the first adiabatic invariant
μ
=
mv
2
⊥
/
2
B
[1]
and reduces to 2
π
(
m/e
)
μ
when
v
≪
β
, which for this system is the condition for the guiding center approx
imation to
hold [7]. ∆
z
and ∆
t
can be computed using Eqs. (1) and (7) and noting that
v
=
√
2
H/m.
The exact drift velocity,
computed without appealing to the guiding center approximation, is
v
d
=
∆
z
∆
t
ˆ
z
=
−
v
I
1
(
v/β
)
I
0
(
v/β
)
ˆ
z.
(19)
Equation (19) holds for orbits of all energies even when the guiding c
enter approximation fails. The
v
≪
β
limit of
Eq. (19) reduces to the grad B drift [10] of the guiding center appr
oximation.
Next we show how
J
can be used as a Hamiltonian to give the magnetic mirror force [10]. For
a magnetic field
mainly in the
z
direction, the cyclotron motion is essentially harmonic oscillation at th
e gyrofrequency
ω
=
qB/m
in
the perpendicular direction, so we identify
η
with
z
and
H
ext
with
P
2
z
/
2
m.
Then
J
= 2
π
H
loc
ω
= 2
π
H
−
P
2
z
/
2
m
qB/m
= 2
π
mv
2
⊥
/
2
qB/m
,
(20)
which equals
μ
except for the factor 2
πm/q.
If
B
depends on
z
then Eqs. (15) and (16) become
dz
dt
=
−
∂J/∂P
z
∂J/∂H
=
P
z
/m
qB/m
qB
m
=
P
z
m
(21)
dP
z
dt
=
∂J/∂z
∂J/∂H
=
−
qJ
2
πm
∂B
∂z
=
−
μ
∂B
∂z
,
(22)
establishing the magnetic mirror force without considering the micro
scopic motion.
A slightly different approach retrieves the grad B drift. Suppose
B
=
B
z
(
x
)ˆ
z,
so
A
=
A
y
(
x
)ˆ
y
with
B
z
=
∂A
y
/∂x
.
We define the
x
component of the guiding center as the position
x
gc
where
v
y
vanishes:
P
y
=
qA
y
(
x
gc
). Setting
104
η
=
y
, Eq. (15) applied to Eq. (20) gives
dy
dt
=
−
∂J/∂P
y
∂J/∂H
=
μ
(
∂B
z
∂x
)
x
gc
∂x
gc
∂P
y
.
(23)
We then use
∂x
gc
/∂P
y
= (
qB
(
x
gc
))
−
1
, obtained by differentiating
P
y
=
qA
y
(
x
gc
) with respect to
P
y
. Equation (23)
thus becomes
dy
dt
=
μ
qB
z
(
∂B
z
∂x
)
x
gc
(24)
which is the grad
B
drift.
A surprising application arises in relativistic mechanics, where it is foun
d that in crossed electric and magnetic fields
E
=
E
ˆ
x
and
B
=
B
ˆ
z
with
E < Bc
a particle’s
z
-velocity is modulated by the cyclotron motion, in contrast to the no
n-
relativistic situation where
v
z
is constant and independent of the cyclotron motion. The modulatio
n arises from the
periodic addition and subtraction of the
E
×
B
drift to the cyclotron velocity, which modulates
γ
=
(
1
−
v
2
/c
2
)
−
1
/
2
and hence the particle’s effective mass;
v
z
then varies because
v
z
=
P
z
/γm
and
P
z
is invariant as
z
is ignorable.
Using the relativistic canonical momenta
P
=
mγ
v
+
q
A
with
A
=
Bx
ˆ
y
, the electrostatic potential
φ
=
−
Ex,
and
the relativistic Hamiltonian
H
=
c
√
(
P
−
q
A
)
2
+
m
2
c
2
+
qφ,
it is found that the relativistic
x
-action
J
=
∮
P
x
dx
evaluates to
J
2
π
=
(
BP
y
+
HE/c
2
)
2
2
q
(
B
2
−
E
2
/c
2
)
3
/
2
+
(
H
2
/c
2
−
P
2
z
−
P
2
y
−
m
2
c
2
)
2
q
(
B
2
−
E
2
/c
2
)
1
/
2
.
(25)
Calculating ∆
z
and ∆
t
using Eqs. (1) and (7) gives the
E
-dependent
z
-drift velocity
v
d
=
∆
z
∆
t
=
B
2
−
E
2
/c
2
BEP
y
/c
2
+
B
2
H/c
2
P
z
.
(26)
As shown in Fig. (2), this
v
z
drift has been verified by direct numerical integration of the relativ
istic equation of
motion
d
(
γm
v
)
/dt
=
q
(
E
+
v
×
B
) which shows that the modulation of
v
z
is typically spiky as
γ
≈
1 for a short
interval during each cyclotron period and then
γ
≫
1 for the remaining fraction of the cyclotron period. Clearly, this
analysis generalizes to force-drifts by replacing
E
with
F
/q
.
FIG. 2. A particle undergoing relativistic
E
×
B
motion in the
xy
-plane with
E
= 0
.
95
Bc
and initial momentum
P
x
= 0,
P
y
= 0
.
7
mc
, and
P
z
= 0
.
3
mc
. (a) The
z
-velocity (from numerical integration of the relativistic
equation of motion) is non-
constant, spiking when the particle’s
xy
-motion slows down so that
γ
≈
1. (b) Solid line is the numerically integrated
z
-position;
dashed line, calculated using Eq. (26), captures the
z
-drift motion.
Kepler motion provides a non-relativistic and non-electromagnetic e
xample. The radial action is [3]
J
r
=
−
2
π
|
P
φ
|
+ 2
π
√
mmMG/
√
2
|
H
|
,
(27)
105
where
P
φ
is the conserved angular momentum. Equation (1) gives ∆
φ
=
±
2
π
depending on the sign of
P
φ
, immediately
proving that bounded Kepler orbits are always closed.
We now present a purely mechanical system which exhibits the equiva
lent of “magnetic”mirroring. Consider a
non-relativistic particle in a long groove where the width of the groov
e varies with position. The Hamiltonian is
H
=
P
2
x
2
m
+
P
2
y
2
m
+
1
2
κx
2
(
1 +
αy
2
)
+
λ
2
y
2
;
(28)
where
y
is the distance along the groove and
x
is the distance across the groove. Presuming that the
y
-position
changes slowly relative to the oscillations across the groove (i.e.,
|
α
|
and
|
λ
|
are small compared to
κ
) the
y
-dependent
frequency of
x
-oscillation is
ω
(
y
) =
√
κ
m
√
1 +
αy
2
.
(29)
We identify
H
loc
=
P
2
x
/
2
m
+
mω
(
y
)
2
x
2
/
2
,
so the
x
-action is
J
= 2
πH
loc
/ω
(
y
)
,
and Eq. (13) becomes
H
=
P
2
y
2
m
+
λ
2
y
2
+
ω
(
y
)
2
π
J.
(30)
Equation (16) gives an average force
−
(
J/
2
π
)
∂ω/∂y
=
−
yJκα/
2
πmω
in the
y
-direction due to the increase in
x
-oscillation energy where the groove narrows. This is a restoring fo
rce and, if sufficiently strong, can overwhelm
the contribution from
λ.
A negative
λ
corresponds to a potential hill, and if
J
= 0 the particle falls down the hill.
However, if
J
is sufficiently large and
α
is positive, the particle does not fall down but instead oscillates abou
t the top
of the hill! This mechanical analog of a magnetic mirror has been verifie
d by direct numerical integration as shown
in Fig. 3.
FIG. 3. A particle in a thin saddle-like groove can undergo os
cillatory motion due to narrowing of the groove.
x
is the direction
across the groove,
y
along the groove, and
z
the vertical.
H
is given by Eq. (28) with
m
=
κ
=
α
= 1 and
λ
=
−
.
01, and the
particle starts at
x
=
y
= 0 with
v
x
=
.
25 and
v
y
= 1.
For oscillatory
y
-motion, Eq. (30) admits an action integral in the
y
-direction, which we denote by
K
, that acts as
a Hamiltonian for the
x
-averaged system. This is a two-tier heirachy of action variables, o
r a wheel within a wheel.
For the reduced system,
J
is a conserved quantity, so we develop an analog of Eq. (1):
∂K
∂J
=
∮
∂P
y
(
H, J, y
)
∂J
dy
=
∮
1
∂J/∂P
y
dy
(31)
=
∮
−
1
dy/dτ
dy
=
−
∆
τ,
(32)
where we use Eq. (14) to evaluate
∂P
y
/∂J
and Eq. (12) to evaluate
∂J/∂P
y
. Since
τ
counts
x
-cycles,
−
∂K/∂J
gives
the number of
x
-cycles per
y
-cycle. If this quantity is a rational number, the trajectory is clos
ed. This is of interest
106
when quantizing the system, as there is sometimes a one-to-one co
rrespondence between periodic classical trajectories
and quantum energy levels [11].
Acknowledgments:
Supported by USDOE and NSF.
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Cosmical Electrodynamics
(Oxford, 1963).
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Plasma Physics and the Problem of Controlled Thermonuclear
Reactions
(Pergamon Press, 1959); T. G.
Northrop and E. Teller, Physical Review,
117
, 215 (1960); R. B. White and M. S. Chance, Physics of Fluids,
27
, 2455
(1984).
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