All attractive central forces can produce circular orbits, which are naturally closed orbits. 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 are ignored here, since they do not produce circular orbits in general. The equation of motion for the radius r of a particle of mass mmoving in a central potentialV is given by motion equations where, and the angular momentumL = mr2ω is conserved. For illustration, the first term on the left is zero for circular orbits, and the applied inwards force equals the centripetal force requirementmrω2, as expected. The definition of angular momentum allows a change of independent variable from t to θ: giving the new equation of motion that is independent of time: This equation becomes quasilinear on making the change of variables and multiplying both sides by : 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 nor closed. Here we show that stable, exactly closed orbits can be produced only with an inverse-square force or radial harmonic oscillator potential. In the following sections, we show that those force laws do produce stable, exactly closed orbits. Define J as where f represents the radial force. The criterion for perfectly circular motion at a radius r0 is that the first term on the left be zero: where. The next step is to consider the equation for u under small perturbations from perfectly circular orbits. On the right, the J function can be expanded in a standard Taylor series: Substituting this expansion into the equation for u and subtracting the constant terms yields which can be written as where is a constant. β2 must be non-negative; otherwise, the radius of the orbit would vary exponentially away from its initial radius. If the right side may be neglected, the solutions are where the amplitude h1 is a constant of integration. For the orbits to be closed, β must be a rational number. What's more, it must be the same rational number for all radii, since β cannot change continuously; the rational numbers are totally disconnected from one another. Using the definition of J along with equation, where is evaluated at. Since this must hold for any value of u0, which implies that the force must follow a power law Hence, J must have the general form For more general deviations from circularity, η may be expanded in a Fourier series, e.g., We substitute this into equation and equate the coefficients belonging to the same frequency, keeping only the lowest-order terms. As we show below, h0 and h2 are smaller thanh1, being of order. h3, and all further coefficients, are at least of order. This makes sense, since must all vanish faster than h1 as a circular orbit is approached. From the cos term, we get where in the last step we substituted in the values of h0 and h2. Using equations and, we can calculate the second and third derivatives of J evaluated at u0: 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 and the radial harmonic-oscillator potential. The solution β = 0 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 can be derived from the general equation whose solution is the constant plus a simple sinusoid: where e, and θ0 are constants of integration. This is the general formula for a conic section that has one focus at the origin; e = 0 corresponds to a circle, e < 1 corresponds to an ellipse, e = 1 corresponds to a parabola, and e > 1 corresponds to a hyperbola. The eccentricity e is related to the total energyE : Comparing these formulae shows that E < 0 corresponds to an ellipse, E = 0 corresponds to a parabola, and E > 0 corresponds to a hyperbola. In particular, for perfectly circular orbits.
Harmonic oscillator
To solve for the orbit under a radial harmonic-oscillator potential, it's easier to work incomponentsr =. 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 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 = is closed because it repeats exactly after a period The system is also stable because small perturbations in the amplitudes and phases cause correspondingly small changes in the overall orbit.
