7 A non-perturbative proof of Bertrand’s theorem
F C Santos ∗
V Soares†
A C Tort ‡
Instituto de F́ısica
Universidade Federal do Rio de Janeiro
Caixa Postal 68.528; CEP 21941-972 Rio de Janeiro, Brazil
September 17, 2018
Abstract
We discuss an alternative non-perturbative proof of Bertrand’s theorem that
leads in a concise way directly to the two allowed fields: the newtonian and the
isotropic harmonic oscillator central fields.
PACS: 45.50.Dd; 45.00.Pk
1 Introduction
In 1873, J. Bertrand[1] published a short but important paper in which he proved that
there are of only two central fields for which all orbits radially bounded are closed, namely:
The newtonian field and the isotropic harmonic oscillator field. Because of this additional
degenerescency it is no wonder that the properties of those two fields have been under
close scrutiny since Newton’s times. Newton addresses to the isotropic harmonic oscillator
in proposition X Book I, and to the inverse-square law in proposition XI [2]. Newton
shows that both fields give rise to an elliptical orbit with the difference that in the first
case the force is directed towards the geometrical centre of the ellipse and in the second
∗e-mail: filadelf@if.ufrj.br
†e-mail: vsoares@if.ufrj.br
‡e-mail: tort@if.ufrj.br.
http://arxiv.org/abs/0704.0575v1
case the force is directed to one of the foci. Bertrand’s result, also known as Bertrand’s
theorem, continues to fascinate old and new generations of physicists interested in classical
mechanics and unsurprisingly papers devoted to it continue to be produced and published.
Bertrand’s proof concise and elegant and contrary to what one may be led to think by a
number of perturbative demonstrations that can be found in modern literature, textbooks
and papers on the subject, it is fully non-perturbative. As examples of perturbative
demonstrations the reader can consult references [3, 4, 5]. We can also find in the literature
demonstrations that resemble the spirit of Bertrand’s original work as for example [6]. As
far as the present authors are aware of all those demonstrations have a restrictive feature,
i.e., they set a limit on the number of possibilities of the existence of central fields with
the property mentioned above to a finite number and finally show explicitly that among
the surviving possibilities only two, the newtonian and the isotropic harmonic oscillator,
are really possible.
In his paper, Bertrand proves initially by taking into consideration the equal radii limit
that a central force f(r) acting on a point-like body able of generating radially bounded
orbits must necessarily be of the form
f (r) = κ r(1/m
where r is the radial distance to center of force, κ is a constant and m a rational number.
Next, making use of this particular form of the law of force and considering also an
additional limiting condition, Bertrand finally shows that only for m = 1 and m = 1/2,
which correspond to Newton’s gravitational law of force
f (r) = −
and to the isotropic harmonic oscillator law of force
f (r) = −κ r,
respectively, we can have orbits with the properties stated in the theorem. However, we
can also prove that for these laws of force all bounded orbits are closed.
Here we offer an alternative non-perturbative proof of Bertrand’s theorem that leads
in a more concise way directly to the two allowed fields.
2 Bertrand’s theorem
In a central field one can introduce a potential function V (r), through the property
f = −∇V (r) . (1)
in such a way that the mechanical energy of a point-like body of mass µ
v2 + V (r) , (2)
is conserved. For radially bounded orbits there are two extreme radii rmax e rmin, the so
called apsidal points ra, that are determined by the condition ṙa = 0, and between which
the particle oscillates indefinitely. Moreover, the conservation of the angular momentum
of the particle under the action of a central field obliges the motion to take place on a
fixed plane and allows the introduction of the effective potential
U(r) = V (r) +
, (3)
with the help of which it is possible to reduce this problem to an equivalent unidimensional
one. This procedure can be found in several textbooks at the undergraduate and gradu-
ated level, see for example [7]. In terms of the effective potential orbits radially bounded
are characterised by apsidal distances rmax e rmin that satisfy the condition E = U (ra).
Evidently there is an intermediate point r0 where the effective potential has a minimum
that satisfies
U ′ (r0) = V
′ (r0)−
= 0. (4)
The angular displacement of the particle between two successive apsidal points, the apsidal
angle ∆θa, is determined by
∆θa =
∫ rmax
[E − U (r)]
. (5)
By considering the effective potential U as the independent variable and by making use
of the inverse function r (U), Tikochinsky[3] produced a very ingenious proof of Bertrand’s
theorem. The inversion of the equation (3), however, is not possible in all the domain
on which the radial coordinate r is defined because the function is not one-to-one in the
field of the real numbers. To circumvent this difficulty we define two one-to-one branches
of the function U (r), namely, one to the left and the other to the right of the point r0.
Then we introduce the inverse functions r1 = r1 (U) and r2 = r2 (U), defined to the left
and to the right of the point r0, respectively, see Figure 1 .
We express initially the angular displacement, equation when the particle moves from
the point of minimum radial distance rmin to the point r0 in terms of the variable U
∆θ1 =
[E − U ]
[E − U ]
. (6)
radial distance r
r0rmin
Figure 1: General form of the effective potential energy.
By the same token we will also have
∆θ2 =
[E − U ]
[E − U ]
, (7)
for the angular displacement from r0 to the point of maximum radial distance r2. Upon
adding up equations (6) and (7) we obtain the angular displacement between two succes-
sive apsidal points
∆θa =
F (U)
E − U
, (8)
where
F (U) =
. (9)
Equation (8) is Abel’s integral equation the solution of which can be found, for ex-
ample, in Landau’s well known book on classical mechanics [8]. A beautiful and straight-
forward solution of this equation is the one by Oldham and Spanier [9]. Abel’s solution
reads
∆θa (E)√
U − E
dE, (10)
where the explicit dependency of the apsidal angle on the energy was stressed.
If all bounded orbits are closed then the apsidal angle ∆θa (E), for these orbits, cannot
change when the energy changes in a continual manner otherwise the continual changes
would inevitably lead to open orbits. Taking this fact into account let us determine the
central potentials that produce the same apsidal angle for all radially bounded orbits.
After integrating equation (10) we obtain
2m∆θa
U − U0. (11)
Equation (11) was derived in Ref. [3] where a perturbative technique applied on a circular
orbit leads to Bertrand’s result. The functions r1 (U) and r2 (U) being the inverse function
of the function U (r) are not independent of each other, and combined as they are in
equation (11), do not allow an efficient manipulation and hide the unique inverse we are
looking for. At this point we perform an analytical continuation of the function U (r)
such that we can consider its inverse function r = r (U). Therefore we write
2µ∆θa
U − U0 + Φ(U, U0) , (12)
where Φ (U, U0) is an analytical function of the complex variable U in an open neighbor-
hood of U0 satisfying the condition Φ (U0, U0) = 1/r0, and whose analytical continuation
cannot have poles but can have other ramification points. Notice that it s not necessary
to make use of the symbol ± before the second term of equation 12) because the square
root has two branches. The positive sign corresponds to r < r0 and the negative one to
r > r0. Taking equation (3) into equation (12) we obtain
2m∆θa
r2V (r) +
− U0r2 + Φ(U, U0) . (13)
The left-hand side of the identity (13) represents a meromorphic function with a single
pole at r = 0 and the right hand side of this same identity contains several terms but
only one can spoil the analyticity of the complete function at some point not equal to
r = 0, namely the term that depends on the square root that generates a branch point at
r = r0. To avoid this it is mandatory to undo the branching effect inherent to the square
root. This is possible only if the radicand is the square of an analytical function with a
zero at r = r0. In this way we identify two possibilities for the potential V (r), to wit
V (r) = −
, newtonian potential, (14)
V (r) =
κ r2, isotropic harmonic oscillator potential; (15)
for which the apsidal angle is independent of the energy. We can calculate the corre-
sponding constant apsidal angles for those two potentials as follows.
For the newtonian potential the effective potential, equation (3), is given by
U = −
. (16)
Solving equation (16) with respect to 1/r we obtain
. (17)
Making use of equation (4) with the effective potential given by equation (16) we find
r0 = ℓ
2/(µκ) and the corresponding minimum energy U0 = −µκ2/(2ℓ2). Now we can
recast equation (17) into the form
U − U0. (18)
Comparing equation (12) with equation (18) we can finally determine the apsidal angle
for the newtonian potential which reads
∆θa = π. (19)
The procedure employed with the newtonian potential can be also applied with a little
bit more of effort to the case of the isotropic harmonic oscillator. The effective potential
is now given by
kr2 +
. (20)
This equation is a quartic equation in 1/r, biquadratic more precisely, and its solution is
given by
− 1. (21)
Factoring out the right hand side of the equation (21) we have
U − U0 +
U + U0, (22)
where now we have made use of the relations r2
µκ and U0 = ℓ
κ/µ. Comparing
equations (12) and (22) we obtain
2µ∆θa
, ∴ ∆θa =
. (23)
We can see that both potentials for which the apsidal angle is constant the orbits are
closed. For the newtonian case the radius oscillates only once in a complete cycle and for
the oscillator case the radius oscillates twice.
3 Final Remarks
In this brief paper we derived Bertrand’s theorem in a non-perturbative way. We have
shown that simple analytical function techniques applied to the problem of finding the only
central fields that allow an entire class of bounded, closed orbits with a minimum number
of restrictions leads in a concise, straightforward way directly to the two allowed fields.
We believe that the derivation discussed here is a valid alternative to a non-perturbative
proof of Bertrand’s theorem and can be presented at the undergraduate and graduate
level or assigned as a problem for classroom discussion.
References
[1] Bertrand J 1873 C.R. Acad. Sci. Paris 77 849
[2] Newton I 1687 Philosophiae Naturalis Principia Mathematica (London: Royal Soci-
ety). English translation by A Motte revised by F Cajori 1962 (University of Cali-
fornia Press, Berkeley CA)
[3] Tikochinsky Y 1988 Am. J. Phys. 56 1073
[4] Brown L S 1978 Am. J. Phys. 46 930
[5] Zarmi Y 2002 Am. J. Phys. 70 446
[6] Arnol’d V I 1976 Les Méthodes Mathématiques de la Mécanique Classique (Mir:
Moscou)
[7] Goldstein H, Poole C and Safko J 2002 Classical Mechanics 3rd edn (Reading:
Addison-Wesley)
[8] Landau L and Lifchitz E 1969 Mècanique 3e èdition revue (Mir: Moscou)
[9] Oldham K B and Spanier J 1974 The Fractional Calculus (London: Academic Press)
	Introduction
	Bertrand's theorem
	Final Remarks

  We discuss an alternative non-perturbative proof of Bertrand's theorem that
leads in a concise way directly to the two allowed fields: the newtonian and
the isotropic harmonic oscillator central fields.

Introduction
In 1873, J. Bertrand[1] published a short but important paper in which he proved that
there are of only two central fields for which all orbits radially bounded are closed, namely:
The newtonian field and the isotropic harmonic oscillator field. Because of this additional
degenerescency it is no wonder that the properties of those two fields have been under
close scrutiny since Newton’s times. Newton addresses to the isotropic harmonic oscillator
in proposition X Book I, and to the inverse-square law in proposition XI [2]. Newton
shows that both fields give rise to an elliptical orbit with the difference that in the first
case the force is directed towards the geometrical centre of the ellipse and in the second
∗e-mail: filadelf@if.ufrj.br
†e-mail: vsoares@if.ufrj.br
‡e-mail: tort@if.ufrj.br.
http://arxiv.org/abs/0704.0575v1
case the force is directed to one of the foci. Bertrand’s result, also known as Bertrand’s
theorem, continues to fascinate old and new generations of physicists interested in classical
mechanics and unsurprisingly papers devoted to it continue to be produced and published.
Bertrand’s proof concise and elegant and contrary to what one may be led to think by a
number of perturbative demonstrations that can be found in modern literature, textbooks
and papers on the subject, it is fully non-perturbative. As examples of perturbative
demonstrations the reader can consult references [3, 4, 5]. We can also find in the literature
demonstrations that resemble the spirit of Bertrand’s original work as for example [6]. As
far as the present authors are aware of all those demonstrations have a restrictive feature,
i.e., they set a limit on the number of possibilities of the existence of central fields with
the property mentioned above to a finite number and finally show explicitly that among
the surviving possibilities only two, the newtonian and the isotropic harmonic oscillator,
are really possible.
In his paper, Bertrand proves initially by taking into consideration the equal radii limit
that a central force f(r) acting on a point-like body able of generating radially bounded
orbits must necessarily be of the form
f (r) = κ r(1/m
where r is the radial distance to center of force, κ is a constant and m a rational number.
Next, making use of this particular form of the law of force and considering also an
additional limiting condition, Bertrand finally shows that only for m = 1 and m = 1/2,
which correspond to Newton’s gravitational law of force
f (r) = −
and to the isotropic harmonic oscillator law of force
f (r) = −κ r,
respectively, we can have orbits with the properties stated in the theorem. However, we
can also prove that for these laws of force all bounded orbits are closed.
Here we offer an alternative non-perturbative proof of Bertrand’s theorem that leads
in a more concise way directly to the two allowed fields.
2 Bertrand’s theorem
In a central field one can introduce a potential function V (r), through the property
f = −∇V (r) . (1)
in such a way that the mechanical energy of a point-like body of mass µ
v2 + V (r) , (2)
is conserved. For radially bounded orbits there are two extreme radii rmax e rmin, the so
called apsidal points ra, that are determined by the condition ṙa = 0, and between which
the particle oscillates indefinitely. Moreover, the conservation of the angular momentum
of the particle under the action of a central field obliges the motion to take place on a
fixed plane and allows the introduction of the effective potential
U(r) = V (r) +
, (3)
with the help of which it is possible to reduce this problem to an equivalent unidimensional
one. This procedure can be found in several textbooks at the undergraduate and gradu-
ated level, see for example [7]. In terms of the effective potential orbits radially bounded
are characterised by apsidal distances rmax e rmin that satisfy the condition E = U (ra).
Evidently there is an intermediate point r0 where the effective potential has a minimum
that satisfies
U ′ (r0) = V
′ (r0)−
= 0. (4)
The angular displacement of the particle between two successive apsidal points, the apsidal
angle ∆θa, is determined by
∆θa =
∫ rmax
[E − U (r)]
. (5)
By considering the effective potential U as the independent variable and by making use
of the inverse function r (U), Tikochinsky[3] produced a very ingenious proof of Bertrand’s
theorem. The inversion of the equation (3), however, is not possible in all the domain
on which the radial coordinate r is defined because the function is not one-to-one in the
field of the real numbers. To circumvent this difficulty we define two one-to-one branches
of the function U (r), namely, one to the left and the other to the right of the point r0.
Then we introduce the inverse functions r1 = r1 (U) and r2 = r2 (U), defined to the left
and to the right of the point r0, respectively, see Figure 1 .
We express initially the angular displacement, equation when the particle moves from
the point of minimum radial distance rmin to the point r0 in terms of the variable U
∆θ1 =
[E − U ]
[E − U ]
. (6)
radial distance r
r0rmin
Figure 1: General form of the effective potential energy.
By the same token we will also have
∆θ2 =
[E − U ]
[E − U ]
, (7)
for the angular displacement from r0 to the point of maximum radial distance r2. Upon
adding up equations (6) and (7) we obtain the angular displacement between two succes-
sive apsidal points
∆θa =
F (U)
E − U
, (8)
where
F (U) =
. (9)
Equation (8) is Abel’s integral equation the solution of which can be found, for ex-
ample, in Landau’s well known book on classical mechanics [8]. A beautiful and straight-
forward solution of this equation is the one by Oldham and Spanier [9]. Abel’s solution
reads
∆θa (E)√
U − E
dE, (10)
where the explicit dependency of the apsidal angle on the energy was stressed.
If all bounded orbits are closed then the apsidal angle ∆θa (E), for these orbits, cannot
change when the energy changes in a continual manner otherwise the continual changes
would inevitably lead to open orbits. Taking this fact into account let us determine the
central potentials that produce the same apsidal angle for all radially bounded orbits.
After integrating equation (10) we obtain
2m∆θa
U − U0. (11)
Equation (11) was derived in Ref. [3] where a perturbative technique applied on a circular
orbit leads to Bertrand’s result. The functions r1 (U) and r2 (U) being the inverse function
of the function U (r) are not independent of each other, and combined as they are in
equation (11), do not allow an efficient manipulation and hide the unique inverse we are
looking for. At this point we perform an analytical continuation of the function U (r)
such that we can consider its inverse function r = r (U). Therefore we write
2µ∆θa
U − U0 + Φ(U, U0) , (12)
where Φ (U, U0) is an analytical function of the complex variable U in an open neighbor-
hood of U0 satisfying the condition Φ (U0, U0) = 1/r0, and whose analytical continuation
cannot have poles but can have other ramification points. Notice that it s not necessary
to make use of the symbol ± before the second term of equation 12) because the square
root has two branches. The positive sign corresponds to r < r0 and the negative one to
r > r0. Taking equation (3) into equation (12) we obtain
2m∆θa
r2V (r) +
− U0r2 + Φ(U, U0) . (13)
The left-hand side of the identity (13) represents a meromorphic function with a single
pole at r = 0 and the right hand side of this same identity contains several terms but
only one can spoil the analyticity of the complete function at some point not equal to
r = 0, namely the term that depends on the square root that generates a branch point at
r = r0. To avoid this it is mandatory to undo the branching effect inherent to the square
root. This is possible only if the radicand is the square of an analytical function with a
zero at r = r0. In this way we identify two possibilities for the potential V (r), to wit
V (r) = −
, newtonian potential, (14)
V (r) =
κ r2, isotropic harmonic oscillator potential; (15)
for which the apsidal angle is independent of the energy. We can calculate the corre-
sponding constant apsidal angles for those two potentials as follows.
For the newtonian potential the effective potential, equation (3), is given by
U = −
. (16)
Solving equation (16) with respect to 1/r we obtain
. (17)
Making use of equation (4) with the effective potential given by equation (16) we find
r0 = ℓ
2/(µκ) and the corresponding minimum energy U0 = −µκ2/(2ℓ2). Now we can
recast equation (17) into the form
U − U0. (18)
Comparing equation (12) with equation (18) we can finally determine the apsidal angle
for the newtonian potential which reads
∆θa = π. (19)
The procedure employed with the newtonian potential can be also applied with a little
bit more of effort to the case of the isotropic harmonic oscillator. The effective potential
is now given by
kr2 +
. (20)
This equation is a quartic equation in 1/r, biquadratic more precisely, and its solution is
given by
− 1. (21)
Factoring out the right hand side of the equation (21) we have
U − U0 +
U + U0, (22)
where now we have made use of the relations r2
µκ and U0 = ℓ
κ/µ. Comparing
equations (12) and (22) we obtain
2µ∆θa
, ∴ ∆θa =
. (23)
We can see that both potentials for which the apsidal angle is constant the orbits are
closed. For the newtonian case the radius oscillates only once in a complete cycle and for
the oscillator case the radius oscillates twice.
3 Final Remarks
In this brief paper we derived Bertrand’s theorem in a non-perturbative way. We have
shown that simple analytical function techniques applied to the problem of finding the only
central fields that allow an entire class of bounded, closed orbits with a minimum number
of restrictions leads in a concise, straightforward way directly to the two allowed fields.
We believe that the derivation discussed here is a valid alternative to a non-perturbative
proof of Bertrand’s theorem and can be presented at the undergraduate and graduate
level or assigned as a problem for classroom discussion.
References
[1] Bertrand J 1873 C.R. Acad. Sci. Paris 77 849
[2] Newton I 1687 Philosophiae Naturalis Principia Mathematica (London: Royal Soci-
ety). English translation by A Motte revised by F Cajori 1962 (University of Cali-
fornia Press, Berkeley CA)
[3] Tikochinsky Y 1988 Am. J. Phys. 56 1073
[4] Brown L S 1978 Am. J. Phys. 46 930
[5] Zarmi Y 2002 Am. J. Phys. 70 446
[6] Arnol’d V I 1976 Les Méthodes Mathématiques de la Mécanique Classique (Mir:
Moscou)
[7] Goldstein H, Poole C and Safko J 2002 Classical Mechanics 3rd edn (Reading:
Addison-Wesley)
[8] Landau L and Lifchitz E 1969 Mècanique 3e èdition revue (Mir: Moscou)
[9] Oldham K B and Spanier J 1974 The Fractional Calculus (London: Academic Press)
	Introduction
	Bertrand's theorem
	Final Remarks
