Linear evolution under a moving normal constraint

Statement

Let $n(t)\in\mathbb R^d\setminus\{0\}$ be $C^1$ , and let $A(t),g(t)$ be continuous. Write

P_n=I-\frac{n\otimes n}{|n|^2}.

The constrained equation $a'=Aa+g+\mu n$ , $n\cdot a=0$ , is equivalent, for initially transverse data, to

a'=P_n(Aa+g)-\frac{n(n'\cdot a)}{|n|^2}, \qquad \mu=-\frac{n'\cdot a+n\cdot(Aa+g)}{|n|^2}.

The projection alone is insufficient when the normal moves.

Verification

Differentiate $n\cdot a=0$ to obtain $n'\cdot a+n\cdot a'=0$ , then solve for $\mu$ . Conversely, the displayed evolution makes the derivative of $n\cdot a$ zero for every solution, so an initially zero constraint remains zero. The same formulas hold for complex $a,g$ by complex linear extension with the real normal $n$ .