Regular holonomic phase space and constrained dynamics

Let $U\subseteq {\mathbb{R}}^d$ be open, let $c:U\to {\mathbb{R}}^m$ be smooth (that is, $C^{\mathrm{\infty }}$) with $1\le m<d$, let $M$ be a constant symmetric positive-definite $d\times d$ matrix, and let $V:U\to \mathbb{R}$ be smooth. A holonomic constraint is the position condition $c(q)=0$. Write $J(q)=Dc(q)$, let $Q=\{q\in U:c(q)=0\}$, and define its regular part by $Q_{\mathrm{r}\mathrm{e}\mathrm{g}}=\{q\in Q:\mathrm{rank}J(q)=m\}$. For $q\in Q_{\mathrm{r}\mathrm{e}\mathrm{g}}$, $A(q)=J(q)M^{-1}J(q{)}^T$ is positive definite. On a compact regular chart $K\subset Q_{\mathrm{r}\mathrm{e}\mathrm{g}}$, $A^{-1}$ is uniformly bounded: ${\sup }_{q\in K}\mathrm{\parallel }A(q{)}^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}<\mathrm{\infty }$.

A position-momentum pair $(q,p)$ is consistent when $q\in Q_{\mathrm{r}\mathrm{e}\mathrm{g}}$ and $J(q)M^{-1}p=0$. The regular constrained phase manifold is $\mathcal{M}=\{(q,p)\in U\times {\mathbb{R}}^d:c(q)=0,J(q)M^{-1}p=0,q\in Q_{\mathrm{r}\mathrm{e}\mathrm{g}}\}\mathrm{.}$ The ambient canonical two-form is ${\omega }_0={\sum }_{i=1}^{d}d{q}_i\wedge d{p}_i$; its restriction to $\mathcal{M}$ is denoted ${\omega }_{\mathcal{M}}$.

For $v=M^{-1}p$, the constrained Hamilton equations are $\dot{q}=v,\dot{p}=-\mathrm{\nabla }V(q)-J(q{)}^T\lambda ,$ with multiplier $\lambda =A(q{)}^{-1}(D^{2}c(q)[v,v]-J(q)M^{-1}\mathrm{\nabla }V(q))\mathrm{.}$ Here $D^{2}c(q)[v,v]\in {\mathbb{R}}^m$ has components $v^T{D}^2{c}_j(q)v$. This multiplier is the one that keeps both $c(q)=0$ and $J(q)M^{-1}p=0$ satisfied.