Bertrand's theorem states

Among central force potentials with bound orbits, there are only two types of central force potentials with the property that all bound orbits are also closed orbits, the inverse-square force potential and the harmonic oscillator potential.

Especially the notion of "closed orbits" reminds me of Lyapunov stability, a prominent concept of Chaos theory. Is there a connection between Bertrands theorem and Chaos theory? Can Bertrands theorem be derived using methods from Chaos theory?

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    $\begingroup$ Bertrand's theorem is concerned with the two-body system, which is not chaotic. Am I missing something? I mean, the demonstrations of Bertrand's theorem heavily rely on the problem being two-body. $\endgroup$ – user154997 Jun 3 '17 at 22:14
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    $\begingroup$ @LucJ.Bourhis Bertrand's theorem is basically proving that the two body system is not chaotic (for inverse square and harmonic oscillator potential), because all bound orbits are closed and so the motion is periodic and not chaotic. But to prove non-chaotic behavior of a system one usually would use methods of Chaos theory like Lyapunov stability. But Bertrands theorem was found before the "invention" of chaos theory. So I'm asking if Bertrands theorem could also be proofed using modern methods from Chaos theory. $\endgroup$ – asmaier Jun 5 '17 at 13:19

The methods which one uses to understand or even construct Bertrand's theorem do indeed belong to the wide class of methods which are used also in chaos theory, but the connection is rather loose. I will describe how you can understand even this loose connection.

When we have a dynamical system in classical mechanics, we need to analyze whether it is integrable or not. Integrable means that along the trajectories of the system, we have enough integrals of motion such as angular momentum or energy so that we can express all the future positions and velocities in terms of the initial data in terms of quadratures (possibly implicit formulas featuring definite integrals).

Consider the example of a particle moving in 1 dimension in a time-independent potential $V(x)$. For this particle, we know that it conserves energy along its motion $$E = \frac{1}{2}m \left( \frac{dx}{dt} \right)^2 + V(x)$$ We can thus express its velocity in terms of the initial energy and position $$\frac{d x}{ dt} = \sqrt{2 (E - V)/m}$$ As long as $dx/dt \neq 0$ you can formally rewrite this system as $$\frac{d x}{\sqrt{2 (E - V)/m}} = dt$$ And after integration you end up with $$\int_{x_0}^{x} \frac{d x}{\sqrt{2 (E - V)/m}} = t - t_0$$ this is what would be called a solution in terms of quadratures.

For a higher-dimensional classical-mechanical (Hamiltonian) systems, we have theorems such as Liouville-Arnold's theorem which tells us that if a suitable set of integrals of motion exists, then there exists a solutions in terms of quadratures. More specifically, for a system of $N$ degrees of freedom, one needs a set of $N$ integrals of motion which are independent and in involution via the Poisson bracket.

For bound motion, this ends up in the motion breaking up into $N$ independent periodic motions with $N$ generally independent "fundamental" frequencies. Consider a particle moving in a plane and a central potential:

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The motion breaks up into 1) a periodic oscillation in the distance from the centre (the periastron and apoastron), and 2) the periodic rotation around the center. If these two frequencies do not match, we get a quasi-periodic motion, a trajectory which is not periodic in the sense of reaching the same point and velocity at a finite time but which still breaks up into a regular product of periodic motions.

However, if there is not a sufficient amount of integrals of motion, the motion gets chaotic. It is notoriously difficult to analytically prove that there is not a sufficient number of integrals of motion for the given set of trajectories in a dynamical system and that these trajectories will thus necessarily get chaotic. There are some methods such as the Melnikov integral (sometimes called Poincaré-Melnikov-Arnold integral) which can show that small layers of chaos in the phase space exist. (Poincaré used something of this sort to prove chaoticity of the three body problem.) But actual investigations in chaos are mostly built around numerical or semi-numerical approaches.

But now I am getting to Bertrand's theorem; you can actually sort of prove Bertrand's theorem by thinking about integrability.

The reason is that when you have a system which is "too integrable", the number of fundamental frequencies reduces. For instance, the motion of a particle in three dimensions will generically have three fundamental frequencies, but the motion in a spherically symmetric and stationary potential will have just two. This is due to the fact that stationarity will give you an energy integral of motion (due to Noether's theorem), and three angular momenta arise due to three rotation symmetries (think Euler angles, not polar coordinates $\vartheta, \varphi$). Not all the angular momenta are in involution but it does not stop them from acting as a constraint which restricts the freedom in which the fully integrable trajectory can wobble. Specifically, the conservation of the whole angular momentum vector restricts the motion to a plane as shown above. So, in spherical systems you end up with an "overly integrable" system which has effectively two degrees of freedom and only two fundamental frequencies.

Now comes the motion in the $1/r, r^2$ potentials. These have special additional symmetries which make the posses additional integrals of motion.

For the $r^2$ it is easy, because the dynamics of the particle in this field corresponds to a Cartesian product of motions of three harmonic oscillators which gives you three integrals of motion corresponding to three energies of the independent oscillators. You know that the fundamental frequencies are the frequencies of the oscillators and thus there will be only one fundamental frequency if you set the stiffness the same in every direction. However, a different point of view on this property is to see that when the stiffness of the oscillators is the same in every direction, the system gets spherically symmetric, we get the additional angular-momentum integrals of motion, and this reduces the number of fundamental frequencies.

For the $1/r$ potential this story is a little bit more difficult since the additional symmetry mixes space and time transformations. The simple consequence is the conservation of the so-called Lenz-Runge vector which once again leads to hyperintegrability and the reduction of the number of fundamental frequencies. But by its very definition, if there is only one fundamental frequency, the motion must be strictly periodic and has to close after one loop.

Nevertheless, this does not prove that the $r^2,1/r$ potentials are the only ones with this property. What Joseph Bertrand actually did is that he has shown that if the potential is any other than $1/r, r^2$ potential, there will always be some trajectories which do not close.

Every spherically symmetric analytical potential which has at least some bound motion will also have circular orbits (which are closed). Joseph Bertrand studied orbits which are very close to circular orbits and showed that already these slightly perturbed orbits will not close unless these potentials are $r^2, 1/r$. The wikipedia page on this topic is well written.

Perturbing orbits and seeing what is their behavior is actually one of the foundational methods of chaos theory and is even the basis of the definition of a Lyapunov exponent, the prime measure of chaos. However, it is easy to see that in Bertrand's theorem these methods are used in a completely different context and to a completely different end.

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  • $\begingroup$ Do I understand correctly that in a central potential/spherical symmetric system/2d system there cannot be chaotic motion, just quasiperiodic motion because angular momentum in such systems is conserved? And on top of this Bertrand proved that if in addition the potential is of type $1/r$ or $r^2$ then the motion is always periodic? $\endgroup$ – asmaier Jun 16 '17 at 21:22
  • $\begingroup$ Yes, this is exactly the point I am trying to make. $\endgroup$ – Void Jun 16 '17 at 22:05
  • $\begingroup$ So the motion of a planar double pendulum is not chaotic, but quasi-periodic? $\endgroup$ – asmaier Jun 16 '17 at 23:37
  • $\begingroup$ No, sorry, this is a misunderstanding, spherically symmetric systems will always be integrable, and 2D systems will be integrable if they have an additional integral of motion. The point of spherically symmetric (stationary) potentials is that the conservation of the whole angular momentum vector restricts the motion to the plane. But a plane is 2D and thus the number of frequencies reduces to 2. $\endgroup$ – Void Jun 16 '17 at 23:51
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    $\begingroup$ The Poincaré-Bendixson theorem talks about a general dynamical system which does not need to have the structure of a classical-mechanical system. The dimension of a general dynamical system can be defined as the number of variables you have to give to specify its state uniquely. In classical-mechanical systems with motion in $D$ coordinates the full state is additionally characterized by $D$ velocities and its dynamical-system (phase-space) dimension is thus $2D$. The double pendulum has $D=2$ and thus phase-space dimension $4$ which already allows for chaos. $\endgroup$ – Void Jun 17 '17 at 9:16

Demonstrating Bertrand's theorem has attracted lots of interest, and there is quite a bunch of proofs using various methods. My library of papers feature one article [1] which would qualify I think, but it is in French. Here is the published abstract in English

"When a point mass undergoes a central, attractive, gradient force, there exists a one parameter family of circular periodic orbits. Bertrand's theorem asserts that if all the orbits close to these circular orbits are periodic then the potential is Newtonian (i.e. proportional to $1/r$, where $r$ is the distance to the fixed centre of attraction) or elastic (i.e. proportional to $r^2$) (J. Bertrand. Comptes Rendus 77 (1873), 849–853). Following an idea of Michael Herman, we compute the first two Birkhoff invariants of this system along the circular trajectories for a generic potential; then we show how to derive Bertrand's theorem."

and here is a translation of mine of the phrase introducing the developments in the paper:

"In this demonstration, among the non-newtonians and non-harmonic potentials, what prevents the property of having only periodic orbits, comes either for the $1/r^2$ potential from the existence of a strict Lyapunov function, or, for a generic potential, from the existence of motions with two incommensurate frequencies (one precession frequency and one revolution frequency)"

The phrasing "strict Lyapunov function" is a literal translation which I hope makes sense in English: my knowledge of this field is flimsy!

[1] Jacques Féjoz and Laurent Kaczmarek, Sur le théorème de Bertrand (d'après Michael Herman), Ergodic Theory and Dynamical Systems 24 (2004), 1583-1589 https://doi.org/10.1017/S0143385704000434

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