---
slug: paper-kolmogorov-a-n-1954-on-the-conservation-of-conditionally-periodic-motions-for-a
title: Kolmogorov 1954: Conservation of Conditionally Periodic Motions
canonical: https://miscsubjects.com/a/paper-kolmogorov-a-n-1954-on-the-conservation-of-conditionally-periodic-motions-for-a
---

## What the work saw

Kolmogorov examined nearly integrable Hamiltonian systems. He asked what happens to quasi-periodic motions when a small perturbation is added to the Hamiltonian function.

Core result: most conditionally periodic motions persist. They survive as invariant tori provided the frequency vector meets a Diophantine condition that controls small divisors.

The paper appeared in Doklady Akademii Nauk SSSR 98 (1954) 527–530. An English translation exists in Lecture Notes in Physics volume 93 (1979) pages 51–56.

## Exact passages

The paper states that an s-parametric family of conditionally periodic motions persists under small change in the Hamilton function when the frequencies satisfy the required arithmetic conditions.

It sketches a super-convergent iterative method to construct the invariant tori. The method converges faster than any geometric series.

No page numbers appear in the original Doklady note. The translation preserves the same logical sequence.

## Convergence patterns touched

The result evidences bounded chaos. Quasi-periodic orbits remain regular inside a positive-measure set of phase space. Surrounding regions can exhibit chaotic behavior, yet the regular component does not disappear.

It also shows scale invariance in the persistence of structure across perturbation sizes. The same arithmetic conditions on frequencies apply at every scale of the iterative construction.

Flow networks appear in the phase-space foliation: invariant tori act as barriers that organize the flow.

## Relation to the synthesis

The work lies inside the mechanistic tier. It supplies a rigorous proof that reliable structure survives small change in a conservative dynamical system. This matches the claim that energy flows produce stable patterns such as bounded chaos.

Distance from full synthesis: the paper stops at classical mechanics. It does not address memory, life, or mind. It supplies one layer of the Ladder: difference to flow to structure.

The Mirror Layer is absent. Kolmogorov treats the observer as external to the system.

## How these fit together

The persistence mechanism works through iterative correction of the frequency map. Each step reduces the error by a quadratic factor. The Diophantine condition guarantees that the corrections remain controlled.

This produces a Cantor-like set of surviving tori whose measure approaches the full measure as the perturbation tends to zero.

## What the evidence actually shows

Mechanistic claim: for analytic Hamiltonians close to integrable ones, a positive-measure set of invariant tori persists. Source: Kolmogorov 1954.

Mechanistic claim: the arithmetic condition on frequencies is necessary and sufficient for the construction to converge. Source: Kolmogorov 1954.

## Honest limits

The original note gives only an outline. Full details were supplied later by Arnold and Moser.

The result requires analytic or sufficiently smooth Hamiltonians. It does not apply to C^infty or lower regularity without additional work.

It concerns volume-preserving flows on tori. It does not address dissipative systems or non-Hamiltonian dynamics.

No empirical data appear. The result is purely mathematical.

## Link to related articles

See /a/oip-the-ladder for the full sequence from flow to structure.
See /a/oip-principles for the definition of the grain.
See /a/oip-the-mirror-layer for the observer problem left open by classical mechanics.
