The Dirac analysis of constrained Hamiltonian mechanics is one of the conventional precursors to the quantization of classical systems. In this paper the analysis is reformulated in the language of exterior differential systems, starting from the Lagrangian, moving through the generation of primary and secondary constraints and leading to the construction of symmetry generators for gauge symmetries. This reformulation extends the procedure to non-coordinate systems. A computer algebra implementation of the procedure in REDUCE is also described.