PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Time-dependent evolution satisfies the two-time cocycle law

Statement

Let $Ψ_{t, s}$ be a local evolution operator for a smooth time-dependent vector field. Whenever both sides are defined,

$Ψ_{r, t} \circ Ψ_{t, s} = Ψ_{r, s} .$

Facts & Assumptions

Given: A local evolution operator $Ψ_{t, s}$ for a smooth time-dependent vector field $X_{t}$ .

[L1]

Local evolution operators give unique solutions of the time-dependent initial-value problem (Time-dependent vector fields have local smooth evolution operators).

Proof

technique · direct

1.1L1given

Fix admissible times $r, t, s$ and a point $p$ for which all maps are defined. The curve $α (u) : = Ψ_{u, t} (Ψ_{t, s} (p))$ solves $\dot{α} (u) = X_{u} (α (u))$ and satisfies $α (t) = Ψ_{t, s} (p)$ .

2.1L1step 1.1

The curve $β (u) : = Ψ_{u, s} (p)$ solves the same differential equation and has the same value at $u = t$ . By uniqueness in [L1], $α (u) = β (u)$ wherever both are defined.

3.1step 2.1∎

Evaluating step 2.1 at $u = r$ gives $Ψ_{r, t} (Ψ_{t, s} (p)) = Ψ_{r, s} (p)$ . Therefore the two-time cocycle law holds.

Depends on

Time-dependent vector fields have local smooth evolution operators

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marius Crainic, Rui Loja Fernandes, and Ioan Marcut, Lectures on Poisson Geometry (standard reference, not scraped)