Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?
From the right perspective, maybe.
(image from Wikipedia)
I'm not exactly sure how to frame this as a trajectory problem, but certainly there is stuff moving and a catenary is traced!
We have a square moving horizontally at a constant speed, and rotating at "the right" constant angular velocity (I'm not certain the angular velocity is fixed, but I suspect it is). Throughout a given quarter rotation starting with a vertex of the square at the bottom, the point directly below the radius will trace out an inverted catenary.
As I've shown in a previous answer, the focus of a parabola rolling on a straight line traces a catenary. Similarly, the directrix of the same rolling parabola will envelope another catenary, a reflection of the one being traced by the focus.
Here is a modern (as in done with the current version of Mathematica) version of the cartoon I did for that previous answer:
A freely suspended chain or string forms a catenary.
Neglecting air resistance, and assuming constant gravity, the trajectory of anything will be a parabola. If there is air resistance, trajectories of roughly spherical objects become a lot more complicated, and are not described easily using nice geometric terms. However, the trajectories do involve the hyperbolic cosine function, which traces out the catenary curve. See this page for details. In spherical classical gravity, trajectories are conic sections, especially hyperbolas and ellipses.
It may be possible to get a true catenary if you had an object with strange aerodynamic properties, or with a very precise arrangement of objects forming a gravitational field. But in either case, the catenary trajectory would be entirely contrived.
The relativistic trajectory of an object under the influence of a constant force field perpendicular to its initial direction of motion is a catenary. The trajectory reduces to a parabola in the non-relativistic limit.
(Eg : motion of electron under constant electric field)