Bertrand's theorem
In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits.
[1][2] The first such potential is an inverse-square central force such as the gravitational or electrostatic potential: The second is the radial harmonic oscillator potential: The theorem is named after its discoverer, Joseph Bertrand.
The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius.
Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circular orbits in general.
For illustration, the first term on the left is zero for circular orbits, and the applied inwards force
equals the centripetal force requirement
The definition of angular momentum allows a change of independent variable from
: giving the new equation of motion that is independent of time: This equation becomes quasilinear on making the change of variables
(see also Binet equation): As noted above, all central forces can produce circular orbits given an appropriate initial velocity.
However, if some radial velocity is introduced, these orbits need not be stable (i.e., remain in orbit indefinitely) nor closed (repeatedly returning to exactly the same path).
Here we show that a necessary condition for stable, exactly closed non-circular orbits is an inverse-square force or radial harmonic oscillator potential.
In the following sections, we show that those two force laws produce stable, exactly closed orbits.
The criterion for perfectly circular motion at a radius
function can be expanded in a standard Taylor series: Substituting this expansion into the equation for
and subtracting the constant terms yields which can be written as where
If the right side may be neglected (i.e., for small perturbations), the solutions are where the amplitude
cannot change continuously; the rational numbers are totally disconnected from one another.
, which implies that the force must follow a power law Hence,
must have the general form For more general deviations from circularity (i.e., when we cannot neglect the higher-order terms in the Taylor expansion of
may be expanded in a Fourier series, e.g., We substitute this into equation (2) and equate the coefficients belonging to the same frequency, keeping only the lowest-order terms.
term, we get where in the last step we substituted in the values of
: Substituting these values into the last equation yields the main result of Bertrand's theorem: Hence, the only potentials that can produce stable closed non-circular orbits are the inverse-square force law (
corresponds to perfectly circular orbits, as noted above.
For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written The orbit u(θ) can be derived from the general equation whose solution is the constant
plus a simple sinusoid: where e (the eccentricity), and θ0 (the phase offset) are constants of integration.
To solve for the orbit under a radial harmonic-oscillator potential, it's easier to work in components r = (x, y, z).
The potential can be written as The equation of motion for a particle of mass m is given by three independent Euler equations: where the constant
must be positive (i.e., k > 0) to ensure bounded, closed orbits; otherwise, the particle will fly off to infinity.
The solutions of these simple harmonic oscillator equations are all similar: where the positive constants Ax, Ay and Az represent the amplitudes of the oscillations, and the angles φx, φy and φz represent their phases.
The resulting orbit r(t) = [x(t), y(y), z(t)] is closed because it repeats exactly after one period The system is also stable because small perturbations in the amplitudes and phases cause correspondingly small changes in the overall orbit.
