  A standard tenet of canonical quantum gravity is that evolution generated by
a Hamiltonian constraint is just a gauge transformation on the phase space and
therefore does not change the physical state. The basis for this belief is a
theorem of Dirac that identifies primary first-class constraints as generators
of physically irrelevant motions. We point out that certain assumptions on
which Dirac based his argument do not hold for reparametrization invariant
systems, and show that the primary Hamiltonian constraint of these systems does
generate physical motion. We show explicitly how the argument fails for systems
described by Jacobi's principle, which has a structure closely resembling that
of general relativity. We defer discussion of general relativity and the
implications for quantum gravity to a later paper.
