First class constraint

Definition

Let $V \subseteq \R^{2n}$ be open, with coordinates $(q, p)$ and the Poisson bracket $\{f, g\} = \sum_{i=1}^n \big(\frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}\big)$. Let $\phi_1, \ldots, \phi_m \colon V \to \R$ be smooth constraint functions, and let $\Sigma = \set{x \in V : \phi_1(x) = \cdots = \phi_m(x) = 0}$ be the constraint surface. Assume that the gradients $\nabla \phi_1, \ldots, \nabla \phi_m$ are linearly independent at each point of $\Sigma$.

1. A smooth function $f$ on $V$ is first class when $\{f, \phi_a\} = 0$ on $\Sigma$ for every $a = 1, \ldots, m$.
2. A constraint $\phi_a$ is first class when it is a first class function, and second class otherwise.