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Lecture 14: Appendix A The general central force problem
We can apply the ideas of Lecture 14 to the general central force problem and (as
promised at the beginning of the course) ask which central potential can yield
periodic (clock-like) behavior. We consider a very general form for the potential
0
ln ,
j
j
j
j
j j
k
U k k
(14.A.1)
with 0
j
and 2
j
, 0 , for bounded motion. The question is, which terms
in this expression can yield periodic trajectories independently of the specific initial
conditions and the specific force constants
j
k , i.e ., we are interested in when stable
periodic behavior arises. This question can be stated in terms of the integral
max
min
2
z
L d
W
E V
(14.A.2)
where we ask under what conditions does this integral yield a rational result
independent of the specific values of , ,
z j
E L k. We can study this problem by
focusing first on circular orbits where
0
E V ,
0
V 0
,
0
V 0
and
p 0
. This is a single frequency problem with
2
z 0
L
. Next we consider
small perturbations around this solution and determine when these perturbed orbits
are independent of the initial conditions and force constants and then when they yield
rational values in Eq. (14.A.2). As usual with such linearized perturbation problems
we will start with a harmonic oscillator problem. We can simplify the expressions by
choosing a rescaled variable
z
x L , setting 1 ( i.e ., rescaling all energies by the
reduces mass) and defining a new potential
2
z
x L
w x U
x
(14.A.3)
Now we are interested in the quantity
max
min
x
x
dx
E w x
(14.A.4)
which is the half period of a particle described by the 1-D (effective) Hamiltonian
2
x
h w x E
(14.A.5)
By construction, we are near a circular orbit at
0 z 0
x L where
2
0
0 0
x x
w x w x w x
(14.A.6)
Thus the frequency of the perturbed radial motion is
0
0
w x
(14.A.7)
and we want to know when this quantity is independent of initial conditions and force
constants. Using the expression for the potential in Eq. (14.A.3) we have
expression is independent of the force constant and its sign, we need only consider
U
( 2 for bounded motion) and find
1 2
U U
U
U U
W
(14.A.10)
So a potential with a single power can produce a winding number that is independent
of the initial conditions and the force constant. However, the result is periodic only
if W above is rational. Thus the allowed powers ( 2 ) are -1, 2, 7, 14,
(
2
n 2 , with n an integer).
Next we ask what happens for larger perturbations around the circular orbit. Since
the result in Eq. (14.A.10) is required to be stable with respect to changes in the
energy E , we might as well consider large changes such that the resulting expressions
are relatively simple. For the case U k
, k , 0 we consider the limit E .
In this limit we expect that the
2
x 2 term in
w x will dominate at the maximum
turning point
max
x x ,
2
max max
E w x x 2. We can scale out this behavior
with the change of variables
max
y x x (
z z max
L x L yx ),
2
2
max
z
w x L
y k
E y x
(14.A.11)
where we expect, in the E limit, that
min min max
y x x 0. With these
definitions of the variables for the E limit and assuming that the
2
y terms
dominates ( y fixed,
max
x ), we have from Eq. (14.A.4) that
max
min
1
1
1
2 0
0
sin
x
E
x
E
dx
W
E w x
dy
y
y
(14.A.12)
From Eq. (14.A.10) we see that this corresponds to 2 and means that this is the
only positive exponent that gives the same winding number both at the energy of the
circular orbit and at arbitrarily large energy. Thus we conclude that the only potential
with a positive exponent that yields periodic motion independent of the initial
conditions, i.e ., the value of E , is the isotropic harmonic oscillator.
To study the negative exponent case, U k
, 0 2 we consider the opposite
limit E 0. We have
2
max min
min
1 2
max
z
z
x x
w x k
L
w x w x E
x
k
x
L
Note that here
min
x 0 implies that where we can consider that the orbits
close. Again we use
max
y x x and noting that
2
max max
2
z
x kx L
, we have
2 2
2 max
z
k x x
w x x y y
L
(14.A.13)
, or better it’s inverse , must be an infinitely valued function. For periodic
behavior we have W m n and the orbit function is an m -valued function of if n is
even and a 2 m -valued function of if n is odd. [You can confirm this result by
considering the various orbits illustrated in the Appendix to this lecture. The “multi-
valuedness” of
is given by how many times you intersect the orbit in moving
from
min
to
max
at a fixed value of .] For the Kepler (and harmonic oscillator)
problem m = 1 with n even and the orbit function is single-valued, i.e ., analytic. This
analyticity arises from the existence of another conserved quantity, which allows the
problem to be solved algebraically (and analytically) rather than via an integral. Here
we discuss the Kepler problem. A similar strategy works for the oscillator problem.
Quite generally for the central force problem we have
2
2
p r p U r r
p L U r r L U r r r r
U r rr r r rr
U r rr rr
d r
U r r
dt r
d
U r r r
dt
(14.A.15)
Since the total angular momentum is conserved for a central force problem, we can
write
d p L dt p L
. For the Kepler potential we have
2
U k / r
so that we
can rewrite Eq. (14.A.15) as
d d
p L k r
dt dt
d
p L kr
dt
(14.A.16)
Thus we have a new conserved vector quantity,
A p L kr ,
(14.A.17)
called the Laplace-Runge-Lenz, or just Runge-Lenz, vector. Note that, since the
angular momentum is orthogonal to the radius vector, we have
A L 0.
(14.A.18)
The Runge-Lenz vector lies in the plane orthogonal to the angular momentum. It has
a magnitude defined by (using helpful vector identities)
2 2 2
2
2 2 2 2
2
2 2 2 2
2
2 2 2
2 2 2
ˆ 2
ˆ 2
2
2
2
A p L p L kr p L k
p L p L kL r p k
L
p L k k
r
p k
L k
r
EL k
(14.A.19)
It is now easy to find the orbit function in terms of this conserved quantity and the
angular momentum (using the form of the scalar triple product – symmetric under
cyclic permutations),
2
r A r p L kr L r p kr
L kr
(14.A.20)
As before we choose to define cylindrical coordinates with the z-axis aligned with L
and the x-axis along the projection of A
into the
x y , plane. Then, for motion in
that plane as required by angular momentum conservation, Eq. (14.A.20) reads
