Regular primary constraints and total generator

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and let $(M^{2n},\omega )$ be a smooth symplectic manifold. A finite family of smooth functions ${\chi }_1,\dots ,{\chi }_r$ defined on an open neighborhood $U\subseteq M$ is a regular primary constraint family on

$C:=\{z\in U:{\chi }_1(z)=\cdots ={\chi }_r(z)=0\}$

when $d{\chi }_1(z),\dots ,d{\chi }_r(z)$ are linearly independent at every $z\in C$. If $C$ is nonempty, the regular-level theorem A regular level set is an embedded submanifold gives $C$ its embedded-submanifold structure and $T_{z}C={\bigcap }_a\ker d{\chi }_a(z)$. If $C=\mathrm{\varnothing }$, use the unique empty smooth structure: the inclusion is an embedding and all pointwise tangent claims are vacuous; no dimension or codimension is inferred from the nonempty-level theorem. For functions $F,G$ on $U$, write $F\approx G$ when $F{\mathrm{\mid }}_C=G{\mathrm{\mid }}_C$; thus $F\approx 0$ means that $F$ vanishes on $C$, without requiring it to vanish off $C$.

Given an open interval $J\subseteq \mathbb{R}$, a smooth canonical Hamiltonian $H_c:J\times U\to \mathbb{R}$ and smooth multiplier functions $u^a:J\to \mathbb{R}$, the total generator is

$H_T(t,z):=H_c(t,z)+{\sum }_{a=1}^r{u}^a(t){\chi }_a(z)\mathrm{.}$

The functions $u^a$ are independent parameter functions; the definition does not assert that their equations determine them. If $H_c$ is an energy and ${\chi }_a$ has units $[{\chi }_a]$, then $u^a$ has units energy divided by $[{\chi }_a]$, so every term in $H_T$ has the same units. A later constraint, if generated by consistency, is not declared primary by this definition.