Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators

Nonlinear dynamical descent

Jeremy RodgersSection 9 of 14Original PDF ↗Publication DOI ↗

Nonlinear dynamical descent

The criterion $PLQ=0$ has an exact nonlinear counterpart, obtained by replacing the linear quotient map by a surjective submersion and the linear generator by a smooth vector field.

Assumption 8.1.

$S$ and $T$ are smooth manifolds, $p:S\to T$ is a smooth surjective submersion (so $Dp_{s}:T_{s}S\to T_{p(s)}T$ is surjective for every $s\in S$ , and $p$ is onto), and $X$ is a smooth vector field on $S$ .

The hypothesis that $p$ be a surjective submersion is not cosmetic. It guarantees (i) that $T$ is a genuine smooth manifold and $p^{-1}(t)$ an embedded submanifold for each $t$ , and (ii) the existence of smooth local sections, which is what makes the smoothness argument in Theorem 8.2 work. We make no claim about arbitrary set-theoretic quotients: a quotient of a manifold by an arbitrary equivalence relation need not be a smooth manifold at all, and nothing below should be read as asserting otherwise.

Theorem 8.2 (Nonlinear projectability criterion).

Under Assumption 8.1, there exists a smooth vector field $Y$ on $T$ with $Dp\circ X=Y\circ p , \qquad\text{i.e.}\qquad Dp_{s}X_{s}=Y_{p(s)}\ \ \text{for every }s\in S,\tag{8.1}$ if and only if $Dp$ applied to $X$ is constant along every fiber: $p(s)=p(s')\ \Longrightarrow\ Dp_{s}X_{s}=Dp_{s'}X_{s'} \qquad\text{in }T_{p(s)}T .\tag{8.2}$ When it exists, $Y$ is unique.

Proof. Necessity. Suppose $Y$ exists and $p(s)=p(s')=:t$ . Then by (8.1), $Dp_{s}X_{s}=Y_{t}=Dp_{s'}X_{s'}$ , and both sides lie in the same tangent space $T_{t}T$ , so (8.2) holds.

Sufficiency. Assume (8.2). For $t\in T$ , use surjectivity of $p$ to pick $s\in p^{-1}(t)$ and define $Y_{t}:=Dp_{s}X_{s}\ \in T_{t}T .\tag{8.3}$ By (8.2) this is independent of the chosen $s\in p^{-1}(t)$ , so $Y:T\to TT$ is a well-defined section of $TT$ : $Y_{t}\in T_{t}T$ by construction. Equation (8.1) holds by definition.

Smoothness. Fix $t_{0}\in T$ and $s_{0}\in p^{-1}(t_{0})$ . Because $p$ is a submersion, the local normal form for submersions [8, Thm. 4.12] provides an open neighbourhood $W\ni t_{0}$ in $T$ and a smooth local section $\sigma:W\to S$ with $p\circ\sigma=\mathrm{id}_{W}$ and $\sigma(t_{0})=s_{0}$ . On $W$ we then have, from (8.3) applied with $s=\sigma(t)\in p^{-1}(t)$ , $Y|_{W}=Dp\circ X\circ\sigma ,\tag{8.4}$ which is a composition of smooth maps ( $\sigma$ smooth, $X$ smooth as a section of $TS$ , $Dp:TS\to TT$ smooth as the tangent map of a smooth map), hence smooth on $W$ . Different choices of section give the same $Y$ on the overlap by fiber constancy (8.2), so these local formulas patch to a globally well-defined smooth field. Since $t_{0}$ was arbitrary, $Y$ is smooth on $T$ .

Uniqueness. If $Y$ and $Y'$ both satisfy (8.1), then for any $t\in T$ pick $s\in p^{-1}(t)$ (possible by surjectivity); then $Y_{t}=Dp_{s}X_{s}=Y'_{t}$ . ◻

Remark 38 (The nonlinear analogue of $PLQ\neq0$ ).

Failure of (8.2) says: there are two source states $s,s'$ with the same readout $p(s)=p(s')$ whose projected velocities $Dp_{s}X_{s}\neq Dp_{s'}X_{s'}$ differ. This is verbatim the phenomenon of Theorem 6.4, where $p$ is the linear map $P$ , the fibers are the affine subspaces $x_{0}+\mathcal{H}_{Q}$ , and the projected velocity at $x_{0}+z_{0}$ is $PL(x_{0}+z_{0})=PLP\,x_{0}+PLQ\,z_{0}$ , which is fiber-constant precisely when $PLQ=0$ . Thus the linear closure criterion is the linearization, and $\href{#eq:fiberconst}{(8.2)}$ its exact generalization: the flow descends to the observable manifold exactly when equal present readouts force equal projected velocities. Nothing in Theorem 8.2 produces a memory kernel; in the nonlinear setting the exact reduction identity of Theorem 6.2 has no elementary counterpart, and the theorem should be read as supplying only the sharp criterion for descent, not a reduction formula.