Local domain restriction for kick, drift, and leapfrog maps

Let $Q\subseteq {\mathbb{R}}^d$ be open, let $V\in C^1(Q,\mathbb{R})$, and let $M$ be a real symmetric positive-definite matrix. Use the force-kick $K_a$ and drift $D_b$ already defined in One-step leapfrog map as force and drift shears, with $U$ there replaced by $V$. The local kick is the restriction of $K_a$ to $Q\times {\mathbb{R}}^d$. The local drift is the restriction of $D_b$ to $\{(q,p):q\in Q,q+b{M}^{-1}p\in Q\}\mathrm{.}$

For a signed step $h$, the local leapfrog map is the restriction of $K_{h\mathrm{/}2}\circ D_h\circ K_{h\mathrm{/}2}$ to the states for which all three intermediate configurations lie in $Q$. Its output is defined only on that admitted set. This definition changes the configuration domain of the published map; it does not redefine the global leapfrog map or assert that the local map is complete.