Regular minimum-energy paths on a declared collision-free tube

Let $d\ge 2$, let $C_{\mathrm{b}\mathrm{a}\mathrm{d}}\subseteq {\mathbb{R}}^d$ be the model's prescribed set of forbidden configurations, and let $U\subseteq {\mathbb{R}}^d\setminus C_{\mathrm{b}\mathrm{a}\mathrm{d}}$ be open. Call $U$ a declared collision-free tube when it is the supplied open domain on which the potential is defined; no physical interpretation of the choice of $U$ is implied. Let $V\in C^4(U,\mathbb{R})$, using the $C^k$ convention of $C^k$ maps and multi-index derivative notation in Euclidean space, and let $a,b\in U$. A regular path with fixed endpoints is $R\in C^1([0,1],U;{\mathbb{R}}^d)$ such that $R(0)=a$, $R(1)=b$, and $R^{\mathrm{\prime }}(s)\mathrm{\ne }0$ for every $s\in [0,1]$ (with one-sided derivatives at the endpoints). Its unit tangent and normal projector are

$t(s):=\frac{R^{\mathrm{\prime }}(s)}{\mathrm{\parallel }{R}^{\mathrm{\prime }}(s){\mathrm{\parallel }}_2},P_{t(s)}^{\perp }:=I-t(s)t(s{)}^T\mathrm{.}$

The path is a regular minimum-energy path (MEP) when

$P_{t(s)}^{\perp }\mathrm{\nabla }V(R(s))=0(0\le s\le 1),$

where $\mathrm{\nabla }V$ is the gradient in The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. Thus the gradient is tangent to the path; at points where it vanishes the equation still holds. The endpoints are fixed data and need not be local minima unless that is added as a hypothesis. This is a conditional mathematical definition and makes no claim about reaction rates, physical mechanisms, or uniqueness.