llcotp.com/textract/1-mechanics-48-separation-of-the-variables.txt

Separation of the variables
149
Thus the solution of the problem of the motion of a mechanical system by
the Hamilton-Jacobi method proceeds as follows. From the Hamiltonian,
we form the Hamilton-Jacobi equation, and find its complete integral (47.2).
Differentiating this with respect to the arbitrary constants a and equating
the derivatives to new constants B, we obtain S algebraic equations
asidar=Bt,
(47.4)
whose solution gives the co-ordinates q as functions of time and of the 2s
arbitrary constants. The momenta as functions of time may then be found
from the equations Pi = aslaqi.
If we have an incomplete integral of the Hamilton-Jacobi equation, depend-
ing on fewer than S arbitrary constants, it cannot give the general integral
of the equations of motion, but it can be used to simplify the finding of the
general integral. For example, if a function S involving one arbitrary con-
stant a is known, the relation asida = constant gives one equation between
q1, ..., qs and t.
The Hamilton-Jacobi equation takes a somewhat simpler form if the func-
tion H does not involve the time explicitly, i.e. if the system is conservative.
The time-dependence of the action is given by a term -Et:
S = So(g)-Et
(47.5)
(see 44), and substitution in (47.1) gives for the abbreviated action So(q)
the Hamilton-Jacobi equation in the form
(47.6)
$48. Separation of the variables
In a number of important cases, a complete integral of the Hamilton-
Jacobi equation can be found by "separating the variables", a name given to
the following method.
Let us assume that some co-ordinate, q1 say, and the corresponding
derivative asia appear in the Hamilton-Jacobi equation only in some
combination (q1, which does not involve the other co-ordinates, time,
or derivatives, i.e. the equation is of the form
(48.1)
where qi denotes all the co-ordinates except q1.
We seek a solution in the form of a sum:
(48.2)
150
The Canonical Equations
§48
substituting this in equation (48.1), we obtain
(48.3)
Let us suppose that the solution (48.2) has been found. Then, when it is
substituted in equation (48.3), the latter must become an identity, valid (in
particular) for any value of the co-ordinate q1. When q1 changes, only the
function is affected, and so, if equation (48.3) is an identity, must be a
constant. Thus equation (48.3) gives the two equations
(48.4)
= 0,
(48.5)
where a1 is an arbitrary constant. The first of these is an ordinary differential
equation, and the function S1(q1) is obtained from it by simple integration.
The remaining partial differential equation (48.5) involves fewer independent
variables.
If we can successively separate in this way all the S co-ordinates and the
time, the finding of a complete integral of the Hamilton-Jacobi equation is
reduced to quadratures. For a conservative system we have in practice to
separate only S variables (the co-ordinates) in equation (47.6), and when this
separation is complete the required integral is
(48.6)
where each of the functions Sk depends on only one co-ordinate; the energy
E, as a function of the arbitrary constants A1, As, is obtained by substituting
So = in equation (47.6).
A particular case is the separation of a cyclic variable. A cyclic co-ordinate
q1 does not appear explicitly in the Hamiltonian, nor therefore in the Hamilton-
Jacobi equation. The function (91, reduces to as/da simply, and
from equation (48.4) we have simply S1 = x1q1, so that
(48.7)
The constant a1 is just the constant value of the momentum P1 = asida
corresponding to the cyclic co-ordinate.
The appearance of the time in the term - Et for a conservative system
corresponds to the separation of the "cyclic variable" t.
Thus all the cases previously considered of the simplification of the integra-
tion of the equations of motion by the use of cyclic variables are embraced
by the method of separating the variables in the Hamilton-Jacobi equation.
To those cases are added others in which the variables can be separated even
though they are not cyclic. The Hamilton-Jacobi treatment is consequently
the most powerful method of finding the general integral of the equations of
motion.
§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