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The classical isotropic harmonic oscillator has the same angular frequency $\Omega >0$ in every Cartesian direction. Its equation of motion is $\ddot{\mathbf{r}}=-{\Omega }^2\mathbf{r}$, with solution $\mathbf{r}=\mathbf{A}\cos \Omega t+\mathbf{B}\sin \Omega t$. The trajectory lies in the span of $\mathbf{A},\mathbf{B}$ and is an ellipse centred on the force centre, degenerating to a line for dependent vectors. For nonzero angular momentum, $r^2$ oscillates at $2\Omega$ unless the trajectory is exactly circular. This distinguishes radial and Cartesian frequencies.

Assume positive, pairwise distinct masses and generic nonzero $\omega ,\lambda$. Let $p_{x_i}=m_i{\dot{x}}_i$, and similarly for $y,z$. The three uncoupled $x$ oscillators each conserveUniform translation of all $y_i$ conserves $P_y={\sum }_i{p}_{y_i}$. The $z_1,z_2$ interaction permits their common translation, while $z_3$ is entirely free, giving two independent momenta $P_{z,12}=p_{z_1}+p_{z_2}$ and $p_{z_3}$. The interacting $z_1,z_2$ subsystem is autonomous and has its own conserved energy. A useful basic collection is thereforeThe total energy is generally not conserved. With $\kappa =\lambda (m_1-m_2)$, its derivative is $\dot{H}=-\kappa \sin (\kappa t)U_{12}(y_1-y_2)$, which is generically nonzero.

Unequal masses do not remove every rotation charge in the oscillator sector. Put $X_i=\sqrt{m_i}{x}_i$, $P_i=p_{x_i}/\sqrt{m_i}$. This is a canonical transformation and makes that sector an isotropic harmonic oscillator. Its rotations conserveThese mass-weighted oscillator symmetries exist already at distinct masses. The energies and three rotation charges are not all functionally independent; the three-oscillator sector has at most five independent time-independent integrals on regular trajectories.

There are further simple explicitly time-dependent constants, such as $x_i\cos \omega t-({\dot{x}}_i/\omega )\sin \omega t$ and $x_i\sin \omega t+({\dot{x}}_i/\omega )\cos \omega t$, and center-of-mass boost constants. For example $MY-t{P}_y$ is constant for $M={\sum }_i{m}_i$ and $Y={\sum }_i{m}_i{y}_i/M$. These illustrate why an unrestricted raw number of conserved functions is not meaningful.

The two positive powers below require $\beta >0$; the negative power requires $\beta <0$, so that the force is attractive. To leading order, $r\simeq r_0[1+ϵ\cos (\sqrt{k+2}\varphi +\delta )]$. The azimuthal advance from one radial maximum to the next isFor $k=2$, ${\omega }_r/\Omega =2$: there are two radial oscillations per revolution. The isotropic harmonic oscillator has orbits that are exact ellipses centred on the force centre, and the perturbed circle is such an ellipse.

For $k=7$, ${\omega }_r/\Omega =3$: the leading perturbation has three radial lobes per revolution, with consecutive radial maxima separated by $2\pi /3$. Its small-amplitude sketch is a rounded three-lobed orbit.

For $k=-7/4$, ${\omega }_r/\Omega =1/2$: the radius completes one oscillation while the body makes two revolutions. The leading curve closes after $\Delta \varphi =4\pi$, with a double-wound, generally self-crossing shape; the nearest and furthest points occur in the same spatial direction one revolution apart.

The rational ratios for $k=7$ and $k=-7/4$ describe closure of the linearized near-circular curves. They do not imply that every finite-amplitude orbit closes exactly: higher-order changes in the radial-to-orbital frequency ratio generally produce apsidal precession. Bertrand's theorem identifies the harmonic potential $k=2$ and the Kepler potential $k=-1$ as the exceptional attractive central potentials with all nearby bounded orbits closed.

Choose the orientation $J>0$. To leading order in the radial disturbance, the azimuthal angular frequency is ${\omega }_{\theta }=J/(m{r}_0^2)$, soFor either orientation the positive frequency ratio is ${\omega }_r/|{\omega }_{\theta }|$. At $k=-1$, the ratio is $1$: there is one radial oscillation per revolution. The bound Kepler orbit is an ellipse with the force centre at a focus, with one closest and one furthest approach per revolution. Its exact polar coordinates equation isAt $k=2$, the ratio is $2$: there are two radial oscillations per revolution. The isotropic harmonic oscillator trajectory is an ellipse centred on the force centre. An exact parametrization isHere $r^2=a^2{\cos }^2\Omega t+b^2{\sin }^2\Omega t$ repeats twice during one full circuit. The two different placements of the force centre distinguish these ellipses. These are the two all-bound-orbit cases of Bertrand's theorem.