  (abridged) This thesis presents theoretical and statistical techniques
broadly related to systems of dynamically-interacting particles composed of
several different types of populations. They are applied to observations of
dense star clusters (SCs) in order to study gravitational interactions between
stars. We present a new analytic method of quantifying the frequency of
encounters involving single, binary and triple stars. With this technique, we
have shown that dynamical encounters involving triple stars occur commonly in
at least some SCs, and that they are likely to be an important dynamical
channel for stellar mergers to occur. We have also used our techniques to
analyze observational data for a large sample of SCs taken from the ACS Survey
for Globular Clusters. The results of this analysis are as follows: (1) We have
compiled a homogeneous catalogue of stellar populations for every cluster in
our sample, including main-sequence (MS), red giant branch, horizontal branch
and blue straggler (BS) stars. (2) With this catalogue, we have quantified the
effects of the cluster dynamics in determining the relative sizes and spatial
distributions of these stellar populations. (3) These results are particularly
interesting for BSs since they provide compelling evidence that they are
descended from binary stars. (4) Our analysis of the MS populations is
consistent with a remarkably universal initial stellar mass function in old
massive SCs in the Milky Way. This is a new result with important implications
for our understanding of star formation in the early Universe and, more
generally, the history of our Galaxy. Finally, we describe how our techniques
are ideally suited for application to a number of other outstanding puzzles of
modern astrophysics, including chemical reactions in the interstellar medium
and mergers between galaxies in galaxy clusters.
