= Solution
Put $\delta=\phi_2-\phi_1$. If both coordinate <amplitudes> are nonzero, eliminate the common <phase> to obtain
$$
\boxed{\left(\frac{x}{x_0}\right)^2+\left(\frac{y}{y_0}\right)^2-2\frac{xy}{x_0y_0}\cos\delta=\sin^2\delta.}
$$
For $\sin\delta\ne0$, this is an <ellipse> centred at the origin. The following special cases exhaust the possible nontrivial shapes:
* $\delta=0$ or $\pi$ gives straight-line <simple harmonic motion>, including an axis-aligned line if one coordinate <amplitude> vanishes.
* $\delta=\pm\pi/2$ gives an axis-aligned <ellipse>; if additionally $x_0=y_0$, it is a <circle> traversed in one of the two orientations.
* Other <phase differences> give tilted <ellipses>.
The degenerate zero-<amplitude> solution is the stationary origin. There are no general multi-loop patterns here because the two <angular frequencies> are equal. Another useful description is $\mathbf r=\mathbf a\cos\omega t+\mathbf b\sin\omega t$, the image of a <circle> under a <linear map>.
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021-q8-orbits-energy.png]
{title=Undamped planar <harmonic oscillator> trajectories and <potential energy> for the two specified initial conditions}
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