First-class constraint (${\phi }_i\approx 0$)

A first-class constraint has Poisson brackets with all constraints that vanish on the constraint surface. For a regular constrained Hamiltonian system, first-class constraints generate gauge transformations; each independent constraint and its gauge direction remove one canonical pair.

The Poisson brackets of first-class constraints close on the constraints. When the constraints are independent and the coefficients are constant, the Jacobi identity for the Poisson bracket makes the coefficients structure constants of a Lie algebra.