Orbit stability, closure, and Bertrand’s theorem

1. Bounded, closed, and stable describe different things

A planet-like trajectory can remain between two radii while its nearest point slowly turns around the center. It is bounded, but need not retrace a closed curve. A circular orbit is geometrically closed, but can be unstable. These words answer different questions.

Property	Physical question	Mathematical test in the present setting
Bounded	Does the trajectory remain in a finite spatial region?	Accessible radii have a finite upper bound
Unbounded	Can the radius grow without bound along the motion?	An accessible branch reaches infinity
Closed	Does the trajectory repeat its path and direction after a finite cycle?	Radial and angular cycles are commensurate
Open or non-closed	Does it fail to retrace after a finite cycle?	No common finite repeat; includes escape paths and non-closing rosettes
Stable circle	Do small radial perturbations remain small?	A local minimum of $V_{\rm eff}$ ; positive curvature gives linear stability
Unstable circle	Do some arbitrarily small radial perturbations grow?	Negative curvature gives a growing radial mode

Some texts reserve “open orbit” for unbounded trajectories. Here “non-closed” is used for bounded precessing trajectories to avoid that ambiguity. Neither energy sign nor a drawing of one revolution settles all these classifications.

2. Stability for a general attractive power law

At a circular radius, $\ell^2/\mu=r_c^3V'(r_c)$ . The effective-potential curvature is therefore

V''_{\rm eff}(r_c)=V''(r_c)+\frac{3\ell^2}{\mu r_c^4} =V''(r_c)+\frac{3V'(r_c)}{r_c}.

(19.1)

Write an attractive power-law force as $F(r)=-C r^s$ , $C>0$ , on a regular interval $r>0$ . Since $V'=Cr^s$ , $V''=Cs r^{s-1}$ . Hence

\boxed{V''_{\rm eff}(r_c)=C(s+3)r_c^{s-1}.}

(19.2)

The circular orbit is linearly stable for $s>-3$ and unstable for $s<-3$ . At $s=-3$ the linear test is marginal. Indeed $V=-C/(2r^2)$ and $V_{\rm eff}=[\ell^2/\mu-C]/(2r^2)$ . Only the tuned value $\ell^2=\mu C$ permits circles, and then $V_{\rm eff}$ is flat. A small nonzero radial velocity produces drift rather than restoring oscillation. It should not be described as ordinary stable confinement.

Figure 19.1 · Effective potentials with μ = ℓ = 1. A minimum restores small radial disturbances; a maximum amplifies them.

3. Apsides and near-circular frequencies

An apsis is an extreme radial point. For attraction, the nearest is periapsis and the farthest is apoapsis. Near a stable circle, let the mean angular frequency be $\Omega=\ell/(\mu r_c^2)$ . The circular condition implies

\Omega^2=\frac{V'(r_c)}{\mu r_c},\qquad \omega_r^2=\frac1\mu\left[V''(r_c)+\frac{3V'(r_c)}{r_c}\right].

(19.3)

Therefore

\boxed{\frac{\omega_r^2}{\Omega^2}=3+\frac{r_cV''(r_c)}{V'(r_c)}.}

(19.4)

For the power-law force this ratio is $s+3$ . During one small radial oscillation, the angle advances approximately $\Delta\theta=2\pi\Omega/\omega_r=2\pi/\sqrt{s+3}$ . This is the angle from one periapsis to the next. The periapsis-to-apoapsis angle is half as large; both are sometimes called an apsidal angle, so the convention must be stated.

A near-circular orbit closes if the frequency ratio is rational and the linear approximation remains adequate. For finite-amplitude motion, the exact angular advance is

\Delta\theta=2\int_{r_-}^{r_+}\frac{\ell\,dr}{r^2\sqrt{2\mu[E-V_{\rm eff}(r)]}}.

(19.5)

This follows from $d\theta/dr=\dot\theta/\dot r$ , integrating outward between turning radii and doubling. Closure requires $\Delta\theta/(2\pi)$ rational, so after some number of radial cycles the total angle is an integer multiple of $2\pi$ .

4. Bertrand's theorem

Statement. Among smooth attractive central potentials that admit stable circular orbits and their neighboring bounded motions on the radial domain under consideration, the only potentials for which every bounded non-collision orbit is closed are, up to additive constants,

\boxed{V(r)=-\frac{k}{r}\quad\text{or}\quad V(r)=\frac12\kappa r^2,\qquad k,\kappa>0.}

(19.6)

The force laws are inverse-square attraction and isotropic linear attraction. The qualification “every bounded orbit” is essential. Many other central potentials possess some closed trajectories, including circles. Bertrand's theorem identifies those with closure throughout the bounded family, not merely at one energy or angular momentum.

The theorem's complete necessity proof is beyond the intended BSc development. The near-circular frequency argument explains a necessary restriction, but is not a full proof: finite-amplitude apsidal angles must also be controlled. For example, other power laws can have a rational near-circular frequency ratio while their finite-amplitude orbits precess. The notes use the theorem as stated, give the local argument above, and verify closure for its two allowed cases.

For inverse-square attraction, the next chapter derives $r=p/(1+e\cos\theta)$ . Bounded non-collision motion has $0\leq e<1$ and repeats after $2\pi$ , establishing sufficiency for that potential.

For the isotropic oscillator, Cartesian equations are $\ddot x+\omega^2x=0$ , $\ddot y+\omega^2y=0$ , where $\omega^2=\kappa/\mu$ . Their general vector solution is

\mathbf r(t)=\mathbf a\cos\omega t+\mathbf b\sin\omega t.

(19.7)

This is the linear image of a unit circle in the two-dimensional plane spanned by constant vectors $\mathbf a,\mathbf b$ . It is an ellipse centered on the force center, or a degenerate line if those vectors are parallel. Every component repeats after $2\pi/\omega$ . Unlike the Kepler ellipse, its force center is the ellipse center rather than a focus.

5. Applications

Example 1 · Inverse-square and oscillator frequency ratios

For inverse-square force, $s=-2$ , so $\omega_r/\Omega=1$ and the periapsis-to-periapsis angle near a circle is $2\pi$ . For the oscillator, $s=1$ , so the ratio is $2$ and successive periapses are separated by $\pi$ . There are two closest approaches to the center during one oscillator ellipse. The exact Cartesian solution shows this is not merely a small-amplitude artifact.

Example 2 · A bounded non-closing near-circular orbit

For constant-magnitude attraction $F=-C$ , $s=0$ and $\omega_r/\Omega=\sqrt3$ . Its potential is $V=Cr$ , which grows at infinity, so finite-energy motion is bounded. Small radial oscillations advance through $2\pi/\sqrt3$ per cycle, an irrational fraction of a full revolution. The limiting near-circular pattern does not close and gives the local mechanism for a rosette. A particular finite-amplitude closure question must use the exact apsidal integral, not assert that the linearized ratio is exact at all amplitudes.

Example 3 · Positive energy with no escape

For an isotropic oscillator with $\mu=\kappa=1$ and $\ell=2$ , $V_{\rm eff}=r^2/2+2/r^2$ . Take $E=5$ . Setting $r^2=z$ gives $z/2+2/z=5$ , or $z^2-10z+4=0$ . Therefore $r_\pm=\sqrt{5\pm\sqrt{21}}$ . Both are finite although energy is positive. The reference and asymptotic form of the potential determine the meaning of energy sign.

Example 4 · A closed but unstable orbit

For $V=-a/r^3$ , the circle at $r_c=3\mu a/\ell^2$ is a perfectly closed geometric solution. However $V''_{\rm eff}(r_c)=-3a/r_c^5$ . The perturbation equation is $\ddot\eta=\lambda^2\eta$ , with $\lambda=\sqrt{3a/(\mu r_c^5)}$ , so $\eta=Ae^{\lambda t}+Be^{-\lambda t}$ . Any nonzero growing-mode coefficient eventually drives the radius away. Closure of the unperturbed circle does not establish stability.

6. What to retain about Bertrand's theorem

Retain the two potentials and the word “every.” Understand the competition between radial and angular periods, and be able to derive the local frequency ratio and show oscillator closure. Do not present the near-circular calculation as the missing global uniqueness proof. For the syllabus, the theorem's statement, physical meaning, allowed potentials, and this clearly delimited argument provide the appropriate scope.

Exercises

Hints and solutions are collected separately.

19.1 Give a physical distinction between bounded and closed.
19.2 Derive the power-law circular stability condition for $F=-Cr^s$ .
19.3 For $F=-C/r^{5/2}$ , determine local circular stability and the near-circular frequency ratio.
19.4 State Bertrand’s theorem with its crucial quantifier.
19.5 Why is a rational near-circular frequency ratio not a full proof of Bertrand’s theorem?
19.6 Prove closure for the isotropic harmonic potential in Cartesian coordinates.
19.7 Where is the force center in oscillator and Kepler ellipses?
19.8 What is the difference between periapsis-to-periapsis and periapsis-to-apoapsis angles?
19.9 Explain the marginal inverse-cube-force case.
19.10 Can a closed circular orbit be unstable?
19.11 For $F=-Cr$ , how many small radial cycles occur in one revolution?
19.12 Write the exact apsidal integral and its closure condition.

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