Mathematica is a great high-level functional language with powerful symbolic and numeric programming capabilities. I use it for analytical work and visualization. Below are some neat samples of what one is able to do...
The Mathematica animation at right shows the solution to a 2D wave equation with diffusion:
where u is a function of time and spatial Cartesian coordinates x and y, o is the oscillation constant, w is the diffusion constant, and v is the propogation speed of the wave. The solution is a nice first-order approximation to an expanding water surface wave. The notebook can be found here.
The image at left shows equal density contours in the (R,z) plane for a Satoh density distribtuion (Binney and Tremaine, pg. 46). One can take advantage of the Vector Analysis package in Mathematica to easily solve and map equations in various curvilinear coordinate systems.
George Collins wrote a nice Celestial Mechanics book, available on the web.