Bertrande: Bertrand's Theorem
Mechanics, Chaos & Fluids arc, page 91 of 193. Back to The Pixelated Physics Universe
Who they are
Bertrande traces two orbits at once: a Kepler ellipse in one hand, an oscillator ellipse in the other. Every other force law frays into a rosette behind her.
Bertrande is a character of the Mechanics, Chaos & Fluids arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Joseph Bertrand proved it in the Comptes Rendus in 1873 (vol. 77, p. 849). For a power law f ∝ rⁿ the test gives β² = n + 3, so n = −2 or n = 1.
$$ \begin{gathered} F = -\frac{k}{r^{2}} \quad \text{or} \quad F = -k\,r \\ \beta^{2} = 3 + \frac{r\,f'(r)}{f(r)} \in \{1,\ 4\} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $F$ | radial force on the body (negative = attractive) | N |
| $r$ | distance from the centre of force | m |
| $k$ | force constant (Kepler: N·m²; oscillator: N/m) | N·m² or N/m |
| $f(r)$ | magnitude of the attractive central force | N |
| $f'(r)$ | derivative df/dr | N/m |
| $\beta$ | ratio of radial to orbital angular frequency for near-circular orbits | 1 |
Honest caveat: The theorem is about ALL bounded orbits. Other force laws do allow some closed orbits; only these two force every orbit to close.
Sources
- Santos, Soares & Tort (2007) An English translation of Bertrand (1873), C. R. Acad. Sci. 77, 849
- Goldstein, Poole & Safko (2002) Classical Mechanics, 3rd ed., Addison-Wesley, sec. 3.6 and appendix A
Crossovers
Bertrande meets Aurea (KAM Theorem). Bertrande says that only the inverse-square and linear forces close every bounded orbit. Aurea says that when such a system is slightly perturbed, most of its regular motion survives on invariant tori. Together they explain why the Solar System is so orderly.
Bertrande meets Lunelock (Tidal Locking). Bertrande's Kepler ellipses close because gravity goes as 1/r². Lunelock adds what pure point masses miss: tides raise bulges that slowly despin the Moon until it rotates once per orbit.
Wear the lore
Concept keywords: Bertrand's Theorem, Bertrande, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics