One-step leapfrog map as force and drift shears

Let $d\ge 1$, let $U:{\mathbb{R}}^d\to \mathbb{R}$ be continuously differentiable, and let $M$ be a real symmetric positive-definite $d\times d$ matrix. Write $\mathrm{\nabla }U$ for its Euclidean gradient, and let $ϵ>0$.

For $a,b\in \mathbb{R}$, define the force kick and drift shear $K_a(q,p)=(q,p-a\mathrm{\nabla }U(q)),D_b(q,p)=(q+b{M}^{-1}p,p)\mathrm{.}$ The one-step leapfrog map is ${\mathrm{\Phi }}_{ϵ}=K_{ϵ\mathrm{/}2}\circ D_{ϵ}\circ K_{ϵ\mathrm{/}2},$ where composition is read right to left. For a finite integer $L\ge 1$, its $L$-step map is ${\mathrm{\Phi }}_{ϵ}^L$. Define the momentum flip $R(q,p)=(q,-p)$ and the flipped proposal map $T_{ϵ,L}=R\circ {\mathrm{\Phi }}_{ϵ}^L$.

Each kick and drift is a continuous bijection with inverse $K_{-a}$ or $D_{-b}$, respectively. These finite maps use only $\mathrm{\nabla }U$ and do not require a solution of Hamilton's ODE. The classical derivative of a force kick is not part of this definition: for merely $U\in C^1$, $\mathrm{\nabla }U$ need not be differentiable.