In graduate school, research areas such as General relativity, High-Energy, String Theory, etc exist. Classical Mechanics is a very important and big field in physics, but how come no one does research in classical mechanics? Is it already a conquered subject? I'm sure there is still research to be done on the optimal solving methods, N system pendulums, Hamiltonian dynamics, etc. Apologies if this sounds asinine.
You will generally find most "classical mechanics" (including solid/fluid mechanics/dynamics) is done in Applied Mathematics or engineering departments, rather than Physics departments.
Newtonian mechanics is an interesting field, but it is pretty well understood. Most of the research goes in to computational methods for designing new machines and structures.
There is in fact a ton of work in classical mechanics. A lot of falls under chaos theory, nonlinear systems, and complex systems, but if you go into Physica A or Physics Review E, you'll see a ton of papers in that area. It's the "google syndrome". You can find a lot of stuff on google, if you punch in the right search terms but you need someone to tell you the right search terms. You won't get that many useful hits with "classical mechanics" but punch in "non-linear systems" and a lot of useful stuff comes out.
It's not quite the same thing, but symplectic geometry grew out of considerations of Hamiltonian mechanics. So, you could say that some mathematicians study a very abstract form of classical mechanics.