The Bertrand theorem revisited

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The Bertrand theorem revisited
```The Bertrand theorem revisited
Yair Zarmi
Jacob Blaustein Institute for Desert Research, Sede-Boqer Campus, 84990,
Ben-Gurion University of the Negev, Israel and Physics Department, Beer-Sheva, 84105,
Ben-Gurion University of the Negev, Israel

The Bertrand theorem, which states that the only power-law central potentials for which the
bounded trajectories are closed are 1/r 2 and r 2 , is analyzed using the Poincaré–Lindstedt
perturbation method. This perturbation method does not generate secular terms and correctly
incorporates the effect of nonlinearities on the nature of periodic solutions. The requirement that the
orbits be closed implies that the theorem holds in each order of the expansion. © 2002 American
Association of Physics Teachers.

I. INTRODUCTION
The Bertrand theorem states that the only power-law central potentials for which the bounded trajectories are closed
are the 1/r 共gravitational and Coulomb兲 and r 2 共isotropic
three-dimensional harmonic oscillator兲.1 Proofs of this theorem can be found in many textbooks on classical mechanics

Planar motion is described by two degrees of freedom: r,
the distance of the orbiting particle from the origin, and ␸,
the angle between the position radius-vector and some initial
direction. Due to the conservation of angular momentum, the
two equations of motion can be reduced to a single one:
␮ dV
⬅J 共 u 兲
⫹u⫽⫺ 2
2
d␸
L du
d 2u

where V is the central potential. The term u on the left-hand
side of Eq. 共1兲 arises from the centrifugal potential. The constants on the right-hand side are ␮ , the reduced mass, and L,
the 共conserved兲 magnitude of the angular momentum.
Under appropriate conditions, Eq. 共1兲 describes oscillations of the radius, r, between two extreme values. The period of the oscillations is the angular sector, ⌬ ␸ , covered by
the orbiting point as the radius completes one full cycle between its smallest and largest values. For a bounded orbit to
be closed, ⌬ ␸ must be a rational fraction of 2 ␲ . With m and
n integers, we must have
n
⌬␸⫽ 2␲.
m

This condition ensures that the oscillatory motion of the
radius completes m full cycles when the angle ␸ completes n
revolutions in the plane.
Circular motion corresponds to a constant radius, r 0 , the
value of which is given by
u 0 ⫽J 共 u 0 兲 ,

If the orbit is not circular, then u⫽u 0 . The analysis is then
carried out in terms of x, the deviation of u from u 0 :
x⫽u⫺u 0 ,
␧⬅max共 兩 x 兩 兲 /u 0 Ⰶ1.

In nonlinear oscillatory systems, the period varies with the
amplitude of the oscillations. Because Eq. 共1兲 is a nonlinear
equation for u( ␸ ), one expects ⌬␸ to vary with the ampli446
Am. J. Phys. 70 共4兲, April 2002
http://ojps.aip.org/ajp/
tude of x. Hence, Eq. 共2兲 is bound to be violated, unless the
effect of the nonlinearity on ⌬␸ is eliminated. This idea is
the fundamental ingredient in all proofs of the theorem.
In one typical proof, ⌬␸ is computed in the extreme limits
of small and large values of x, which correspond to small and
large eccentricity, respectively. The requirement that ⌬␸ has
the same value in both limits and satisfies Eq. 共2兲 yields the
result of the theorem 共see, for example, Refs. 4 and 5兲.
Another proof employs a perturbation expansion in powers of x for small x 共see, for example, Ref. 3兲
J 共 u 兲 ⫽J 共 u 0 兲 ⫹

n⭓1
J n⫽
J nx n,
d nJ共 u 兲
du n

.
u⫽u 0
d 2x
d␸2
⫹ ␻ 20 x⫽

n⭓2
J n x n,
dJ 共 u 兲
␻ 20 ⫽1⫺J 1 ⫽1⫺
du

⫽1⫹
u⫽u 0
␮ d 2V
L 2 du 2

.
u⫽u 0
If we omit the nonlinear perturbation on the right-hand side
of Eq. 共6兲, the unperturbed motion is oscillatory, provided
that
␻ 20 ⬎0.

To ensure that the orbit of the actual planar motion is closed,
Eq. 共2兲 must be obeyed, leading to
␻ 20 ⫽ 共 m/n 兲 2 .

If we introduce the nonlinear perturbation, the angular velocity, ␻, deviates from its unperturbed value, ␻ 0 , and varies
continuously as the magnitude of x is increased. Hence, as
the amplitude is varied, ␻ goes through infinitely many irrational values. Therefore, for most values of the amplitude,
the orbit will not be closed. To ensure that the orbit is closed,
one must require that ␻ remains equal to ␻ 0 despite the
effect of the nonlinearity.
We now expand x in a perturbative series
x共 ␸ 兲⫽

k⭓1
a k cos共 k ␻ 0 ␸ 兲 .
© 2002 American Association of Physics Teachers

446
The magnitude of a k is expected to decrease as k increases,
because they are generated in progressively higher orders of
the perturbation expansion. The requirement that ␻ 0 is a rational number yields the result of the theorem.
However, the expansion of Eq. 共9兲, if employed in a consistent manner in a perturbation analysis, yields unbounded

the perturbation expansion useless beyond short times of
O(1). Moreover, they generate an aperiodic approximation
to a physical system, whose motion is periodic. This wellknown problem and its, equally well known, resolution, will
be discussed in Sec. II.
II. POINCARÉ–LINDSTEDT EXPANSION
d 2x
d␶
2
⫹ ␻ 20 x⫽

n⭓2

J nx n.
Because Eq. 共10兲 describes the motion of an energy conserving system, the solution is expected to be periodic in ␶.
We now expand both the solution, x, and the angular velocity, ␻, in a power series in the small parameter ␧ 关see Eq.

x⫽␧ 共 x 0 ⫹␧x 1 ⫹␧ 2 x 2 ⫹␧ 3 x 3 ⫹••• 兲 ,

␻ ⫽ ␻ 0 ⫹␧ ␻ 1 ⫹␧ 2 ␻ 2 ⫹␧ 3 ␻ 3 ⫹•••.

In the naı̈ve method, all the ␻ n , n⭓1 are set to zero. Hence,
the angular velocity is not updated. With this choice, the
solution is expanded in terms of functions whose period coincides with the unperturbed one. The undesired consequences of this choice are discussed below.
If we substitute Eqs. 共11兲 and 共12兲 in Eq. 共10兲 and require
that the resulting equation holds in every order separately, we
obtain through third order in ␧,
d 2x 0
d␶2
⫹x 0 ⫽0,

⫺2
⫹
␻ 1 d 2x 0
1
⫹x
⫽⫺2
⫹
J x 2,
1
2 2 0
2
2
␻
0 d␶
d␶
2␻0
␻ 20
J 2x 0x 1⫹
1
6 ␻ 20
J 3 x 30 ,
Am. J. Phys., Vol. 70, No. 4, April 2002
1
␻ 20
J 2x 0x 2⫹
1
2 ␻ 20
J 3 x 20 x 1 ⫹
1
24␻ 20
J 4 x 40 .

As the starting point does not affect the analysis, ␶ 0 ⫽0 is
used in the following discussion. We substitute Eq. 共17兲 in
Eq. 共14兲 and obtain
d 2x 1
d␶2
⫹x 1 ⫽2
J 2 a 20
␻1
a 0 cos ␶ ⫹

␻0
4 ␻ 20

The homogeneous part of Eq. 共18兲 describes the motion of a
harmonic oscillator with angular velocity equal to unity. The
cos ␶ term on the right-hand side has the same angular velocity. Hence, it is a resonant term. If retained, it would
contribute to the solution of Eq. 共18兲 an aperiodic term of the
form
␻1
a ␶ sin ␶ .
␻0 0
This term is unbounded as a function of ␶, and is a secular
term. As the solution is expected to be periodic in ␶, such a
term would provide a poor approximation for the solution.
Hence, one must have
␻ 1 ⫽0.

Thus, in this order of the expansion, the naı̈ve choice is the
correct one.
Now that the resonant term has been removed, Eq. 共18兲
can be solved, yielding a periodic solution
x 1 ⫽a 11 cos ␶ ⫹a 12 sin ␶ ⫹
a 20 J 2
a 20 J 2
4␻0
12␻ 20
⫺
2
cos 2␶ .

If we use Eqs. 共17兲 and 共20兲 in Eq. 共15兲, the latter becomes

5J 22 a 30 J 3 a 30
␻2
J 2 a 0 a 11
⫹x 2 ⫽ 2 a 0 ⫹
⫹
cos ␶ ⫹
4
2
2
␻
0
d␶
24␻ 0
8␻0
2 ␻ 20
d 2x 2
⫹
␻ 2 d 2 x 0 ␻ 21 d 2 x 0
␻ 1 d 2x 1
⫹x
⫽⫺2
⫺
⫺2
2
␻ 0 d ␶ 2 ␻ 20 d ␶ 2
␻0 d␶2
d␶2
1
␻ 2 d 2x 1
␻ 1 d 2x 2
1
⫺2
⫹
J x2
␻0 d␶2
␻ 0 d ␶ 2 2 ␻ 20 2 1
The coefficients ␻ n will be chosen so as to guarantee that the
solution is bounded and periodic in ␶. Equation 共13兲 is solved
by

d 2x 2
⫹
␻ 3 d 2x 0
␻ 1 ␻ 2 d 2 x 0 ␻ 21 d 2 x 1
⫺2
⫺
␻0 d␶2
␻ 20 d ␶ 2 ␻ 20 d ␶ 2

⫹
d 2x 1
447
d␶2
⫹x 3 ⫽⫺2
x 0 ⫽a 0 cos共 ␶ ⫹ ␶ 0 兲 .
The emergence of secular terms is avoided if instead of a
‘‘naı̈ve’’ expansion, one uses the method of Poincaré6 and
Lindstedt.7 共In Sec. 28 of Ref. 8, the method is used in the
analysis of anharmonic oscillations.兲 The method is described in detail in Refs. 9 and 10. Identical results are obtained in the methods of multiple-time-scales,11–13 normal
forms,14 –17 and averaging.18 The essence of the approach is
the fact that the nonlinear perturbation on the right-hand side
of Eq. 共6兲 changes the angular velocity, ␻ 0 , into an updated
angular velocity, ␻.
The natural variable for analyzing Eq. 共6兲 is ␶ ⫽ ␻␸ , rather
than ␻ 0 ␸ . In terms of ␶, Eq. 共6兲 becomes
␻2
d 2x 3
J 2 a 0 a 11

2 ␻ 20
J 3 a 30
24␻ 20
cos 2␶ ⫹
⫺
J 22 a 30
24␻ 40

J 2 a 0 a 12
2 ␻ 20
sin 2␶
cos 3␶ .

The ‘‘naı̈ve’’ choice of ␻ 2 ⫽0 leaves the resonant term 共proportional to cos ␶兲 intact, the result being that the solution of
Eq. 共21兲 for x 2 contains a term that is proportional to ␶. Such
an unbounded secular term spoils the periodic nature of the
approximate solution, and renders the perturbation expansion
Yair Zarmi
447
useless beyond short times of O(1). Thus, the cos ␶ term
a 20
␻ 2 ⫽⫺ 共 5J 22 ⫹3J 3 ␻ 20 兲
48␻ 30
⫺
a 0 a 12J 2
6 ␻ 20
sin 2␶ ⫹
a 0 a 11J 2
2 ␻ 20
⫺

192␻ 40
a 0 a 11J 2
6 ␻ 20
cos 3␶ .
J共 u 兲⫽
u 0⫽
␻ 3 ⫽⫺ 共 5J 22 ⫹3J 3 ␻ 20 兲
␻ 4 ⫽⫺
⫺
⫺

24␻ 0
2
2
2a 0 a 21⫹a 11
⫹a 12
⫹
48␻ 30
97a 40
13 824␻ 70
a 40 J 5
384␻ 0
19a 40 J 3
5760␻ 50

7a 40 J 23
480␻ 30
.
⫺
384␻ 30

The nonlinear term in Eq. 共10兲 generates corrections to the
frequency. These corrections depend on the amplitude in a
continuous manner. Consequently, even if ␻ 0 is rational, the
updated angular velocity, ␻, will be irrational in general. For
␻ to assume rational values for all amplitudes, it must remain
equal to ␻ 0 . Hence, all corrections that make ␻ different
from ␻ 0 must vanish,

In other words, one is seeking central potentials for which,
despite the nonlinearities that they generate, no resonant
terms appear in any order of the perturbative expansion. For
such potentials, the naı̈ve expansion does hold.
If we apply Eq. 共26兲 to Eq. 共22兲, we obtain

Equation 共27兲 ensures that ␻ 3 also vanishes, and simplifies
the requirement that ␻ 4 vanishes,
28J 23 ⫹35J 2 J 4 ⫹5J 5 ␻ 20 ⫽0.

To see what power-law central potentials are allowed by the
requirements of Eqs. 共8兲, 共27兲, and 共28兲, we write V(r) as
448
J 2⫽
Am. J. Phys., Vol. 70, No. 4, April 2002

1/共 s⫹2 兲

.
L2
J 4⫽

,
u0
J 3 ⫽⫺

u 20

u 30

,

,

u 40
.

If we substitute Eqs. 共31兲 and 共32兲 in Eq. 共8兲, we find
7a 40 J 2 J 4
III. BERTRAND THEOREM
␻ 2 ⫽ ␻ 3 ⫽ ␻ 4 ⫽•••⫽0.

␣s␮
J 5 ⫽⫺

u ⫺s⫺1 .
J 1 ⫽⫺ 共 s⫹1 兲 ,

,
3
L2
The derivatives J n are readily computed and given here
through n⫽5,
The procedure can be repeated in higher orders. Elimination
of resonant terms in each order determines the coefficients
␻ n in the expansion of ␻ 关Eq. 共12兲兴, thereby ensuring that the
approximate solution is bounded and periodic in ␶. The results for the updating of the angular velocity are given below
through O(␧ 4 ),
a 0 a 11
␣s␮
Equation 共4兲 then yields
cos 2␶

If we use the definition of J(u) 关see Eq. 共1兲兴, this form

.
Equation 共21兲 may now be solved for x 2 , yielding a solution
that is periodic in ␶,
x 2 ⫽a 21 cos ␶ ⫹a 22 sin ␶ ⫹
V 共 r 兲 ⫽ ␣ r s ⫽ ␣ u ⫺s .
␻ 20 ⫽s⫹2⫽

m
n
2

.
We require that ␻ 2 and ␻ 4 vanish and find that the only
solutions common to both are

s⫽⫺2,⫺1,⫹2.

are not solutions of the requirement that ␻ 2 ⫽0.兲 The same
conclusion is reached in higher orders. The case s⫽⫺2 corresponds to ␻ 0 ⫽0, namely, no oscillatory behavior, and
hence is excluded. As a result, the only power-law central
potentials that admit closed orbits are the 1/r and the r 2
potentials.
From the viewpoint of perturbation theory, the Bertrand
theorem has the following significance. In general, the nonlinearities generated by a central potential in Eq. 共10兲 lead to
the appearance of resonant terms in the dynamical equations
in all, or most, orders of the expansion. To avoid the emergence of unbounded and aperiodic secular terms in the perturbative expansion, one must account for the update in the
angular velocity, ␻. The only power-law central potentials
for which no resonant terms emerge in the perturbative expansion are the two unique potentials singled out by the
theorem. Because these potentials do not generate resonant
terms, there is also no updating of the angular velocity, and
the naı̈ve expansion yields the correct results.
1
J. Bertrand, ‘‘Mécanique analytique,’’ C. R. Acad. Sci. 77, 849– 853

2
E. J. Whittaker, A Treatise on the Analytical Dynamics of Particles and
Rigid Bodies 共Cambridge University Press, Cambridge, MA, 1961兲, pp.
80–93.
Yair Zarmi
448
H. Goldstein, Classical Mechanics 共Addison Wesley, New York, 1981兲,
App. A, pp. 601– 605.
4
J. L. McCauley, Classical Mechanics 共Cambridge University Press, Cambridge, MA, 1997兲, pp. 134 –138.
5
J. V. José and E. J. Saletan, Classical Dynamics 共Cambridge University
Press, Cambridge, MA, 1998兲, pp. 88 –92.
6
H. Poincaré, New Methods of Celestial Mechanics, originally published as
Les Méthodes Nouvelles de la Méchanique Célèste 共Gauthier-Villars,
Paris, 1892兲 共American Institute of Physics, New York, 1993兲.
7
A. Lindstedt, ‘‘Über die integration einer für die störungstheorie wichtigen
differentialgleichung,’’ Astron. Nachr. 103, 211–220 共1882兲.
8
L. D. Landau and E. M. Lifshitz, Mechanics 共Pergamon, Oxford, 1960兲.
9
A. H. Nayfeh, Introduction to Perturbation Techniques 共Wiley, New York,
1981兲.
10
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for
Scientists and Engineers 共McGraw-Hill, New York, 1978兲.
3
11
J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics 共Springer-Verlag, New York, 1985兲.
12
J. A. Murdock, Perturbations Theory and Methods 共Wiley, New York,
1991兲.
13
J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation
Methods 共Springer-Verlag, New York, 1996兲.
14
G. I. Hori, ‘‘Theory of general perturbations with unspecified canonical
variables,’’ Publ. Astron. Soc. Jpn. 18, 287–296 共1966兲.
15
G. I. Hori, ‘‘Theory of general perturbations for non-canonical systems,’’
Publ. Astron. Soc. Jpn. 23, 567–587 共1971兲.
16
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential
Equations 共Springer-Verlag, New York, 1988兲.
17
P. B. Kahn and Y. Zarmi, Nonlinear Dynamics: Exploration Through Normal Forms 共Wiley, New York, 1998兲.
18
J. A. Sanders and F. Verhulst, Averaging Method in Nonlinear Dynamical
Systems 共Springer-Verlag, New York, 1985兲.
Celestial Globe. This Celestial Globe at Oberlin College is by Gilman Joslin 共1804-ca.1886兲. Joslin worked in many fields in addition to making globes: he
was one of the first Americans to make a daguerreotype and was engaged in shipbuilding. The celestial globe showed the stars forming the constellations and
also the mythical figures associated with them. 共D. J. Warner, ‘‘Geography of Heaven and Earth—III,’’ Rittenhouse, 2, 共1988兲, pp. 88 – 89兲 共Photograph and
notes by Thomas B. Greenslade, Jr., Kenyon College兲
449
Am. J. Phys., Vol. 70, No. 4, April 2002
Yair Zarmi
449
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