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§48
Separation of the variables
151
To make the variables separable in the Hamilton-Jacobi equation the
co-ordinates must be appropriately chosen. We shall consider some examples
of separating the variables in different co-ordinates, which may be of
physical interest in connection with problems of the motion of a particle in
various external fields.
(1) Spherical co-ordinates. In these co-ordinates (r, 0, ), the Hamiltonian is
and the variables can be separated if
U
=
where a(r), b(a), c(b) are arbitrary functions. The last term in this expression
for U is unlikely to be of physical interest, and we shall therefore take
U = a(r)+b(8)/r2.
(48.8)
In this case the Hamilton-Jacobi equation for the function So is
1
Since the co-ordinate is cyclic, we seek a solution in the form So
Pot+S1(T)+S2(9), obtaining for the functions S1(r) andS 2(0) the equations
(day)
=
E.
Integration gives finally
S = -
(48.9)
The arbitrary constants in (48.9) are Pp, B and E; on differentiating with
respect to these and equating the results to other constants, we have the
general solution of the equations of motion.
(2) Parabolic co-ordinates. The passage from cylindrical co-ordinates
(here denoted by p, o, 2) to parabolic co-ordinates E, N, o is effected by the
formulae
1(-n),pv(En).
(48.10)
The co-ordinates & and n take values from 0 to 00; the surfaces of constant
$ and n are easily seen to be two families of paraboloids of revolution, with
152
The Canonical Equations
§48
the z-axis as the axis of symmetry. The equations (48.10) can also be written,
in terms of
r = =
(48.11)
(i.e. the radius in spherical co-ordinates), as
$ = r++,
= r Z.
(48.12)
Let us now derive the Lagrangian of a particle in the co-ordinates $, n, o.
Differentiating the expressions (48.10) with respect to time and substituting
in the Lagrangian in cylindrical co-ordinates
L =
we obtain
L
=
(48.13)
The = and
the Hamiltonian is
(48.14)
The physically interesting cases of separable variables in these co-ordinates
correspond to a potential energy of the form
(48.15)
The equation for So is
2
=
E.
The cyclic co-ordinate can be separated as a term PoO. Multiplying the equa-
tion by m(s+n) and rearranging, we then have
Putting So = P&O + S2(n), we obtain the two equations
-B,
§48
Separation of the variables
153
integration of which gives finally
S
=
dn.
(48.16)
Here the arbitrary constants are Ps, B and E.
(3) Elliptic co-ordinates. These are E, n, o, defined by
(48.17)
The constant o is a parameter of the transformation. The co-ordinate $ takes
values from 1 to 80, and n from - 1 to + 1. The definitions which are geo-
metrically clearest+ are obtained in terms of the distances r1 and r2 to points
A1 and A2 on the z-axis for which 2 = to: r1 = V[(2-0)2+p2],
r2 = Substitution of (48.17) gives
= o(s-n), r2 = o(+n),
(48.18)
& = (r2+r1)/2o, n = (r2-r1)/2o. =
Transforming the Lagrangian from cylindrical to elliptic co-ordinates, we
find
L
=
(48.19)
The Hamiltonian is therefore
H
=
(48.20)
The physically interesting cases of separable variables correspond to a
potential energy
(48.21)
where a() and b(n) are arbitrary functions. The result of separating the
variables in the Hamilton-Jacobi equation is
S
=
1-n2
t The surfaces of constant $ are the ellipsoids = 1, of which A1 and
A2 are the foci; the surfaces of constant n are the hyperboloids 22/02/2-22/02(1-n2 = 1,
also with foci A1 and A2.
154
The Canonical Equations
§49
PROBLEMS
PROBLEM 1. Find a complete integral of the Hamilton-Jacobi equation for motion of a
particle in a field U = a/r-Fz (a combination of a uniform field and a Coulomb field).
SOLUTION. The field is of the type (48.15), with a(f)=a1F,b(n)a+Fn2 Formula
(48.16) gives
S
=
with arbitrary constants Po, E,B. The constant B has in this case the significance that the one-
valued function of the co-ordinates and momenta of the particle
B
is conserved. The expression in the brackets is an integral of the motion for a pure Coulomb
field (see $15).
PROBLEM 2. The same as Problem 1, but for a field U = ai/r +az/r2 (the Coulomb field
of two fixed points at a distance 2a apart).
SOLUTION. This field is of the type (48.21), with a($) = (a1+az) /o, = (a1-az)n/o.
From formula (48.22) we find
S
=
The constant B here expresses the conservation of the quantity
B = cos 01+ cos 02),
where M is the total angular momentum of the particle, and 01 and O2 are the angles shown in
Fig. 55.
12
r
The
20
a
FIG. 55
$49. Adiabatic invariants
Let us consider a mechanical system executing a finite motion in one dimen-
sion and characterised by some parameter A which specifies the properties of
the system or of the external field in which it is placed, and let us suppose that
1 varies slowly (adiabatically) with time as the result of some external action;
by a "slow" variation we mean one in which A varies only slightly during the
period T of the motion:
di/dt < A.
(49.1)
§49
Adiabatic invariants
155
Such a system is not closed, and its energy E is not conserved. However, since
A varies only slowly, the rate of change E of the energy is proportional to the
rate of change 1 of the parameter. This means that the energy of the system
behaves as some function of A when the latter varies. In other words, there
is some combination of E and A which remains constant during the motion.
This quantity is called an adiabatic invariant.
Let H(p, q; A) be the Hamiltonian of the system, which depends on the
parameter A. According to formula (40.5), the total time derivative of the
energy of the system is dE/dt = OH/dt = (aH/dx)(d)/dt). In averaging this
equation over the period of the motion, we need not average the second
factor, since A (and therefore i) varies only slowly: dE/dt = (d)/dt)
and in the averaged function 01/01 we can regard only P and q, and not A, as
variable. That is, the averaging is taken over the motion which would occur
if A remained constant.
The averaging may be explicitly written
dE dt
According to Hamilton's equation q = OHOP, or dt = dq - (CH/OP). The
integration with respect to time can therefore be replaced by one with respect
to the co-ordinate, with the period T written as
here the $ sign denotes an integration over the complete range of variation
("there and back") of the co-ordinate during the period. Thus
dq/(HHap)
(49.2)
dt $ dq/(HHdp)
As has already been mentioned, the integrations in this formula must be
taken over the path for a given constant value of A. Along such a path the
Hamiltonian has a constant value E, and the momentum is a definite function
of the variable co-ordinate q and of the two independent constant parameters
E and A. Putting therefore P = p(q; E, 1) and differentiating with respect
to A the equation H(p, q; X) )=E, we have = 0, or
OH/OP ax ap
t If the motion of the system is a rotation, and the co-ordinate q is an angle of rotation ,
the integration with respect to must be taken over a "complete rotation", i.e. from 0 to 2nr.
156
The Canonical Equations
§49
Substituting this in the numerator of (49.2) and writing the integrand in the
denominator as ap/dE, we obtain
dt
(49.3)
dq
or
dt
Finally, this may be written as
dI/dt 0,
(49.4)
where
(49.5)
the integral being taken over the path for given E and A. This shows that, in
the approximation here considered, I remains constant when the parameter A
varies, i.e. I is an adiabatic invariant.
The quantity I is a function of the energy of the system (and of the para-
meter A). The partial derivative with respect to energy is given by 2m DI/DE
= $ (ap/dE) dq (i.e. the integral in the denominator in (49.3)) and is, apart from
a factor 2n, the period of the motion:
(49.6)
The integral (49.5) has a geometrical significance in terms of the phase
path of the system. In the case considered (one degree of freedom), the phase
space reduces to a two-dimensional space (i.e. a plane) with co-ordinates
P, q, and the phase path of a system executing a periodic motion is a closed
curve in the plane. The integral (49.5) taken round this curve is the area
enclosed. It can evidently be written equally well as the line integral
I = - $ q dp/2m and as the area integral I = II dp dq/2m.
As an example, let us determine the adiabatic invariant for a one-dimen-
sional oscillator. The Hamiltonian is H = where w is the
frequency of the oscillator. The equation of the phase path is given by the
law of conservation of energy H(p, q) = E. The path is an ellipse with semi-
axes (2mE) and V(2E/mw2), and its area, divided by 2nr, is
I=E/w.
(49.7)
t It can be shown that, if the function X(t) has no singularities, the difference of I from a
constant value is exponentially small.
§49
Adiabatic invariants
157
The adiabatic invariance of I signifies that, when the parameters of the
oscillator vary slowly, the energy is proportional to the frequency.
The equations of motion of a closed system with constant parameters
may be reformulated in terms of I. Let us effect a canonical transformation
of the variables P and q, taking I as the new "momentum". The generating
function is the abbreviated action So, expressed as a function of q and I. For
So is defined for a given energy of the system; in a closed system, I is a func-
tion of the energy alone, and so So can equally well be written as a function
So(q, I). The partial derivative (So/dq)E is the same as the derivative
( for constant I. Hence
(49.8)
corresponding to the first of the formulae (45.8) for a canonical trans-
formation. The second of these formulae gives the new "co-ordinate",
which we denote by W:
W = aso(q,I)/aI.
(49.9)
The variables I and W are called canonical variables; I is called the action
variable and W the angle variable.
Since the generating function So(q, I) does not depend explicitly on time,
the new Hamiltonian H' is just H expressed in terms of the new variables.
In other words, H' is the energy E(I), expressed as a function of the action
variable. Accordingly, Hamilton's equations in canonical variables are
i = 0,
w = dE(I)/dI.
(49.10)
The first of these shows that I is constant, as it should be; the energy is
constant, and I is so too. From the second equation we see that the angle
variable is a linear function of time:
W = (dE/dI)t + constant.
(49.11)
The action So(q, I) is a many-valued function of the co-ordinate. During
each period this function increases by
(49.12)
as is evident from the formula So = Spdq and the definition (49.5). During
the same time the angle variable therefore increases by
Aw = (S/I) =
(49.13)
t The exactness with which the adiabatic invariant (49.7) is conserved can be determined by
establishing the relation between the coefficients C in the asymptotic (t + 00) expressions
q = re[c exp(iw+t)] for the solution of the oscillator equation of motion q + w2(t) q = 0.
Here the frequency w is a slowly varying function of time, tending to constant limits w as
t
+ 00. The limiting values of I are given in terms of these coefficients by I = tw+/c+l2.
The solution is known from quantum mechanics, on account of the formal resemblance
between the above equation of motion and SCHRODINGER'S equation 4" + k2(x) 4 = 0 for
one-dimensional motion of a particle above a slowly varying (quasi-classical) "potential
barrier". The problem of finding the relation between the asymptotic (x + 00)
expressions
for & is equivalent to that of finding the "reflection coefficient" of the potential barrier; see
Quantum Mechanics, $52, Pergamon Press, Oxford 1965.
This method of determining the exactness of conservation of the adiabatic invariant for an
oscillator is due to L. P. PITAEVSKII. The relevant calculations are given by A. M. DYKHNE,
Soviet Physics JETP 11, 411, 1960. The analysis for the general case of an arbitrary finite
motion in one dimension is given by A.A. SLUTSKIN, Soviet Physics JETP 18, 676, 1964.
158
The Canonical Equations
§50
as may also be seen directly from formula (49.11) and the expression (49.6)
for the period.
Conversely, if we express q and P, or any one-valued function F(p, q) of
them, in terms of the canonical variables, then they remain unchanged when
W increases by 2nd (with I constant). That is, any one-valued function F(p, q),
when expressed in terms of the canonical variables, is a periodic function of W
with period 2.
$50. General properties of motion in S dimensions
Let us consider a system with any number of degrees of freedom, executing
a motion finite in all the co-ordinates, and assume that the variables can be
completely separated in the Hamilton-Jacobi treatment. This means that,
when the co-ordinates are appropriately chosen, the abbreviated action
can be written in the form
(50.1)
as a sum of functions each depending on only one co-ordinate.
Since the generalised momenta are Pi = aso/dqi = dSi/dqi, each function
Si can be written
(50.2)
These are many-valued functions. Since the motion is finite, each co-ordinate
can take values only in a finite range. When qi varies "there and back" in this
range, the action increases by
(50.3)
where
(50.4)
the integral being taken over the variation of qi just mentioned.
Let us now effect a canonical transformation similar to that used in 49,
for the case of a single degree of freedom. The new variables are "action vari-
ables" Ii and "angle variables"
w(a(q
(50.5)
+ It should be emphasised, however, that this refers to the formal variation of the co-
ordinate qi over the whole possible range of values, not to its variation during the period of
the actual motion as in the case of motion in one dimension. An actual finite motion of a
system with several degrees of freedom not only is not in general periodic as a whole, but
does not even involve a periodic time variation of each co-ordinate separately (see below).
§50
General properties of motion in S dimensions
159
where the generating function is again the action expressed as a function of
the co-ordinates and the Ii. The equations of motion in these variables are
Ii = 0, w = de(I)/I, which give
I=constant,
(50.6)
+ constant.
(50.7)
We also find, analogously to (49.13), that a variation "there and back" of
the co-ordinate qi corresponds to a change of 2n in Wi:
Awi==2m
(50.8)
In other words, the quantities Wi(q, I) are many-valued functions of the co-
ordinates: when the latter vary and return to their original values, the Wi
may vary by any integral multiple of 2. This property may also be formulated
as a property of the function Wi(P, q), expressed in terms of the co-ordinates
and momenta, in the phase space of the system. Since the Ii, expressed in
terms of P and q, are one-valued functions, substitution of Ii(p, q) in wi(q, I)
gives a function wilp, q) which may vary by any integral multiple of 2n
(including zero) on passing round any closed path in phase space.
Hence it follows that any one-valued function F(P, q) of the state of the
system, if expressed in terms of the canonical variables, is a periodic function
of the angle variables, and its period in each variable is 2nr. It can be expanded
as a multiple Fourier series:
(50.9)
ls==
where l1, l2, ls are integers. Substituting the angle variables as functions
of time, we find that the time dependence of F is given by a sum of the form
(50.10)
lg==
Each term in this sum is a periodic function of time, with frequency
(50.11)
Since these frequencies are not in general commensurable, the sum itself is
not a periodic function, nor, in particular, are the co-ordinates q and
momenta P of the system.
Thus the motion of the system is in general not strictly periodic either as a
whole or in any co-ordinate. This means that, having passed through a given
state, the system does not return to that state in a finite time. We can say,
t Rotational co-ordinates (see the first footnote to 49) are not in one-to-one relation
with the state of the system, since the position of the latter is the same for all values of
differing by an integral multiple of 2nr. If the co-ordinates q include such angles, therefore,
these can appear in the function F(P, q) only in such expressions as cos and sin , which
are in one-to-one relation with the state of the system.
160
The Canonical Equations
§50
however, that in the course of a sufficient time the system passes arbitrarily
close to the given state. For this reason such a motion is said to be conditionally
periodic.
In certain particular cases, two or more of the fundamental frequencies
Wi = DE/DI are commensurable for arbitrary values of the Ii. This is called
degeneracy, and if all S frequencies are commensurable, the motion of the
system is said to be completely degenerate. In the latter case the motion is
evidently periodic, and the path of every particle is closed.
The existence of degeneracy leads, first of all, to a reduction in the number
of independent quantities Ii on which the energy of the system depends.
If two frequencies W1 and W2 are such that
(50.12)
where N1 and N2 are integers, then it follows that I1 and I2 appear in the energy
only as the sum n2I1+n1I2.
A very important property of degenerate motion is the increase in the
number of one-valued integrals of the motion over their number for a general
non-degenerate system with the same number of degrees of freedom. In the
latter case, of the 2s-1 integrals of the motion, only s functions of the state
of the system are one-valued; these may be, for example, the S quantities I
The remaining S - 1 integrals may be written as differences
(50.13)
The constancy of these quantities follows immediately from formula (50.7),
but they are not one-valued functions of the state of the system, because the
angle variables are not one-valued.
When there is degeneracy, the situation is different. For example, the rela-
tion (50.12) shows that, although the integral
WIN1-W2N2
(50.14)
is not one-valued, it is so except for the addition of an arbitrary integral
multiple of 2nr. Hence we need only take a trigonometrical function of this
quantity to obtain a further one-valued integral of the motion.
An example of degeneracy is motion in a field U = -a/r (see Problem).
There is consequently a further one-valued integral of the motion (15.17)
peculiar to this field, besides the two (since the motion is two-dimensional)
ordinary one-valued integrals, the angular momentum M and the energy E,
which hold for motion in any central field.
It may also be noted that the existence of further one-valued integrals
leads in turn to another property of degenerate motions: they allow a complete
separation of the variables for several (and not only one+) choices of the co-
t We ignore such trivial changes in the co-ordinates as q1' = q1'(q1), q2' = 92'(92).