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. (The solution β = 0 corresponds to a perfectly circular orbit.) If the right side may be neglected (i.e., for small perturbations), 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 (), 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 (i.e., when we cannot neglect the higher-order terms in the Taylor expansion of J ), 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 than h1, 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(βθ) 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 (β = 1) and the radial harmonic-oscillator potential (β = 2). The solution β = 0 corresponds to perfectly circular orbits, as noted above. Classical field potentials 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. This is the general formula for a conic section that has one focus at