First-class constraint

In Hamiltonian mechanics, let's say we have a symplectic manifold M with a smooth Hamiltonian over it. Let's also say we have a couple of constraints given as function equations f_i(x)=0 for n smooth functions {f_i}_i=1ⁿ such that everywhere on the constrained subspace, the n derivatives of the n functions are all linearly independent and also, the Poisson brackets {f_i,f_j} and {f_i,H} all vanish at the constrained subspace. Barring homotopical complications (or alternatively, looking at local charts), this means we can write $\{f_{i},f_{j}\}=\sum _{k}c_{ij}^{k}f_{k}$ for some smooth functions c_ij^k (there is a mathematical theorem on this) and $\{f_{i},H\}=\sum _{j}v_{i}^{j}f_{j}$ for some smooth functions v_i^j. Then, we say we have a first class constraint.

What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other ON the constrained subspace or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constaint flows all bring the point to another point on the constrained subspace. To borrow an idea from Lagrangian mechanics, the "Lagrange multipliers" associated with the constraints are all zero which means there is no modification of the dynamics associated with the first class constraints.

Examples

Gauge theories...