Scaled classical RATTLE step map

Use the data of Regular holonomic phase space and constrained dynamics. For a consistent input $(q,p)\in \mathcal{M}$, put $v=M^{-1}p$ and ${\lambda }_{\ast }={\mu }_{\ast }=\frac{1}{2}A(q{)}^{-1}(D^{2}c(q)[v,v]-J(q)M^{-1}\mathrm{\nabla }V(q))\mathrm{.}$ For nonzero signed $h$, form the sequential scaled residual equations $F_q(\lambda )=h^{-2}c\text{\hspace{0.17em}⁣}(q+h{M}^{-1}(p-\frac{h}{2}\mathrm{\nabla }V(q)-hJ(q{)}^T\lambda ))=0,$ $F_p(\mu ;\lambda )=h^{-1}J(q_1)M^{-1}{p}_1=0,$ where $q_1,p_{1\mathrm{/}2},p_1$ are given by the updates below. The selected multiplier pair, when it exists uniquely, is the local root branch whose limits as $h\to 0$ are ${\lambda }_{\ast }$ and ${\mu }_{\ast }$. This defines a partial exact step map; no root-existence assertion is part of this definition.

For such a selected pair $\lambda ,\mu \in {\mathbb{R}}^m$, define $p_{1\mathrm{/}2}=p-\frac{h}{2}\mathrm{\nabla }V(q)-hJ(q{)}^T\lambda ,q_1=q+h{M}^{-1}{p}_{1\mathrm{/}2},$ $p_1=p_{1\mathrm{/}2}-\frac{h}{2}\mathrm{\nabla }V(q_1)-hJ(q_1{)}^T\mu \mathrm{.}$ The position and momentum equations are respectively $c(q_1)=0$ and $J(q_1)M^{-1}{p}_1=0$. At $h=0$ define the step to be the identity.

Here $\lambda ,\mu$ are force-level multipliers: the corresponding constraint impulses are $h{J}^T\lambda$ and $h{J}^T\mu$. This scaling keeps the limiting multiplier equations nonsingular as $h\to 0$. The position equation is SHAKE; the final momentum equation is the RATTLE tangency correction.