We discuss physical experiments achievable via the monitoring of stellar dynamics near the massive black hole at the Galactic center with a diffraction-limited, next-generation, extremely large telescope (ELT). Given the likely observational capabilities of an ELT and what is currently known about the stellar environment at the Galactic center, we synthesize plausible samples of stellar orbits around the black hole. We use the Markov Chain Monte Carlo method to evaluate the constraints that the monitoring of these orbits will place on the matter content within the dynamical sphere of influence of the black hole. We express our results as functions of the number N of stars with detectable orbital motions and the astrometric precision δθ and spectroscopic precision δv at which the stellar proper motions and radial velocities are monitored. Our results are easily scaled to different telescope sizes and precisions. For N = 100, δθ = 0.5 mas, and δv = 10 km s^-1 (a conservative estimate of the capabilities of a 30 m telescope) we find that if the extended matter distribution enclosed by the orbits at 0.01 pc has a mass greater than ~10³ M, it will produce measurable deviations from Keplerian motion. Thus, if the concentration of dark matter at the Galactic center matches theoretical predictions, its influence on the orbits will be detectable. We also estimate the constraints that will be placed on the mass of the black hole and on the distance to the Galactic center and find that both will be measured to better than ~0.1%. We discuss the significance of knowing the distance to within a few parsecs and the importance of this parameter for understanding the structure of the Galaxy. We demonstrate that the lowest order relativistic effects, such as the prograde precession, will be detectable if δθ 0.5 mas. Barring the favorable discovery of a star on a highly compact, eccentric orbit, the higher order effects, including the frame dragging due to the spin of the black hole, will require δθ 0.05 mas. Finally, we calculate the rate at which monitored stars experience detectable nearby encounters with background stars. The encounters probe the mass function of stellar remnants that accumulate near the black hole. We find that ~30 such encounters will be detected over a 10 yr baseline for δθ = 0.5 mas.