Extension criterion for first class functions

Let $T^*Q$ be the phase space, let $\mathscr{C}$ be the constraint set, and let $\mathfrak{P}$ be the collection of integral curves in $\mathscr{C}$ of Hamiltonian vector fields of constraint functions. A function on $\mathscr{C}$ is constant along an integral curve when its restriction to that curve is constant.

Extension criterion for first class functions. A function $f$ on the constraint set $\mathscr{C}$ extends to a first class function if and only if it is constant along all the integral curves of $\mathfrak{P}$.

This gives a characterization of the restrictions to the constraint set of first class functions in terms of the curves generated by constraint Hamiltonians. The source presents the assertion as a theorem following from the closedness of the constraint set, although the parser supplied no independent resolution status.