Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized momentum variable (such that $p=\dot{q}$, etc).

For instance, if degree is 1, then we can construct the curve as the phase space of a particle moving with constant velocity, and it is parameterized by $(x,0)$, so is given by the equation $y=0$, up to a choice of coordinates.

For $d=2$, a conic, we can use, for instance, a spring, whose position is $\cos(x)$, and so $(\cos x,\sin x)$ parameterizes a circle (ellipses are also fairly easy to see how to do this way, and parabolas are linearly accelerated particles, though I don't see hyperbolas immediately)

For $d=3$, it is well known that a simple pendulum has phase space an elliptic curve with roots the initial height, the length of the pendulum, and minus the length.

Are there other nice examples like this? Is there a natural physical system that realizes hyperbolas? Does every real elliptic curve arise in this way? How about quartic or higher curves? I assume there are physical systems that work, as every ODE is modeling SOME system, but are there well-known examples that arise naturally in physics?