the origin; e = 0 corresponds to a circle, 0 < 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 energy E (see Laplace–Runge–Lenz vector): 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 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. References Further reading Classical mechanics Orbits Physics theorems