Arnold — Mathematical Methods of Classical Mechanics (Fast Track 1.11) — Audit + Gap Plan

Book: V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer Graduate Texts in Mathematics 60).

Russian original: Matematicheskie metody klassicheskoĭ mekhaniki, Nauka, Moscow, 1974.
1st English edition: Springer GTM 60, 1978 (translated by K. Vogtmann and A. Weinstein).
2nd English edition: Springer GTM 60, 1989 — the canonical reference, expanded with revised appendices and a substantially developed KAM appendix (Appendix 8). Chapter / appendix references in this plan are to the 2nd English edition.

Fast Track entry: 1.11 (the symplectic-geometry / classical-mechanics slot of the Dynamical-Systems-IV strand on the booklist; Arnold is paired there with Cannas da Silva Lectures on Symplectic Geometry and McDuff-Salamon Introduction to Symplectic Topology as the mechanics-first anchor of the strand).

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite). Output is a concrete punch-list of new units to write — and existing units to deepen — so that Mathematical Methods of Classical Mechanics is covered to the equivalence threshold (≥95% effective coverage; see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Not a full P1 audit (no line-number-level Problem inventory).

The audit surface is smaller than Hatcher but comparable to Cannas: Arnold §47 / §50 / §44 anchors already appear in the tier_anchors field of 05.05.03-generating-functions, 05.02.04-action-angle-coordinates, 05.02.05-cotangent-bundle, 05.02.03-integrable-system, and several more. Genuine gaps are concentrated in (1) Arnold-only material not in Cannas — KAM theory, Hamilton-Jacobi PDE, principle of least action, geodesic flow as Hamiltonian system, adiabatic invariants — and (2) the Lagrangian formalism / Euler-Lagrange equations, which Arnold develops as Part II but which Codex lacks as a dedicated unit.

Substantial overlap with the Cannas plan. The Hamiltonian core (Hamilton's equations, Poisson brackets, integrable systems, action-angle, moment map, symplectic reduction) is covered by both Arnold and Cannas; the Cannas plan (plans/fasttrack/cannas-da-silva-symplectic.md) already itemises priority-1 deepenings on these existing units. This Arnold plan does not double-count those deepenings. Where an item appears on both, this plan flags it explicitly and defers production to the Cannas batch. The Arnold-distinctive new-unit production is what this plan adds.

§1 What Arnold's book is for

Arnold is the canonical source for the mechanics-first, geometric presentation of classical mechanics. Where Goldstein / Landau-Lifshitz present mechanics in coordinates and reach symplectic geometry only as a late chapter (or never), Arnold inverts the hierarchy: the geometric objects (configuration manifold, tangent bundle, cotangent bundle, symplectic structure, Hamiltonian vector field) are introduced first and the coordinate calculations follow. The book has three Parts plus eight Appendices (Appendix 8 on KAM is itself ~80 pages, the size of a chapter).

Part I — Newtonian Mechanics (Ch. 1-2): Galilean relativity, Newton's equations, motion in 1D potential, central field, $N$ -body. Arnold treats this as setup, not load-bearing topic — dispatched in 60 pages so Lagrangian and Hamiltonian formulations can be developed in their proper geometric setting.
Part II — Lagrangian Mechanics (Ch. 3-5): Hamilton's principle of least action, Lagrange / Euler-Lagrange equations, motion on a Riemannian manifold (geodesic flow as Lagrangian system), Noether's theorem, oscillations, rigid body, Lagrange's top, configuration manifold and the tangent bundle as natural setting, fibre derivative.
Part III — Hamiltonian Mechanics (Ch. 6-10): symplectic manifold, Hamiltonian phase flow, Liouville's theorem (volume preservation), Poincaré recurrence, integral invariants of Poincaré-Cartan, Hamilton-Jacobi equation, generating functions, canonical transformations, action-angle, Liouville-Arnold integrability, perturbation theory, KAM, adiabatic invariants.
Appendices (1-8): Riemannian curvature; geodesics on Lie groups (Euler equations of an ideal fluid as geodesic on $Diff_{vol}$ !); symplectic structures on algebraic manifolds; contact structures; dynamical systems with symmetry; normal forms of quadratic Hamiltonians; normal forms of Hamiltonian systems near a fixed point; theory of perturbations / KAM theorem.

Distinctive Arnold choices:

Geometric mechanics framing throughout. Configuration manifold is smooth from page one; Lagrangian is $L : T M \to R$ ; action is functional on paths in $M$ ; Euler-Lagrange equations are intrinsic. Not an upgrade from coordinate formulation — it is the formulation.
Exterior calculus before Hamiltonian formalism. §35 introduces differential forms, exterior derivative, Lie derivative, Cartan's magic formula — and only then defines the symplectic form on $T^{*} M$ .
Hamilton-Jacobi as generating function for the flow. Chapter 9 §47: $S (q, t)$ as action functional viewed as a function of the endpoint, satisfies $\partial_{t} S + H (q, \partial_{q} S, t) = 0$ , used to integrate Hamiltonian systems by separation of variables. The Hamilton-Jacobi equation is not a side topic — it is the engine for solving integrable systems and the bridge to geometric optics / WKB.
Liouville-Arnold integrable-systems theorem. Chapter 10 §49: if $H$ has $n$ Poisson-commuting integrals on a $2 n$ -dimensional symplectic manifold, with $d f_{i}$ independent on a compact connected level set $M_{c}$ , then $M_{c}$ is diffeomorphic to an $n$ -torus and on a tubular neighbourhood there exist action-angle coordinates $(I, θ)$ . Arnold's signature theorem in classical mechanics, jointly attributed in Cannas Lecture 24 as "Liouville-Arnold."
KAM theorem — apex appendix. Appendix 8 (~80 pages, 2nd ed.) presents Kolmogorov-Arnold-Moser perturbation theory: for a small Hamiltonian perturbation $H_{ϵ} = H_{0} + ϵ H_{1}$ of an integrable $H_{0}$ , most invariant tori of $H_{0}$ survive (positive measure, diophantine frequencies). Requires Newton iteration, small-divisor estimates. Gateway to Nekhoroshev / Aubry-Mather. Codex has zero KAM units — largest topic-level gap.
Foliations on cotangent bundles. Throughout Part III: $T^{*} M$ is foliated by Lagrangian submanifolds; level sets of Poisson-commuting integrals are Lagrangian; action-angle is the global statement; completely integrable systems are exactly those for which the foliation extends globally. Cannas covers the local Liouville theorem; Arnold's emphasis on the global foliation is Arnold-distinctive.
Variational principle of least action. Chapter 3 §13: a path is a motion iff it is a critical point of $S [q] = \int L (q, \overset{q}{˙}, t) d t$ . Codex's 03.04.08 variational-calculus covers Euler-Lagrange formalism in the differential-forms setting, but Hamilton's principle as the foundational principle of mechanics is not articulated as its own unit.
*Geodesic flow as Hamiltonian flow on $T^{*} M$ .* §17: geodesic equations on $(M, g)$ are Euler-Lagrange for $L = \frac{1}{2} ∣ v ∣^{2}$ , equivalently Hamiltonian flow for $H = \frac{1}{2} ∣ p ∣_{g^{- 1}}^{2}$ . The simplest non-trivial Hamiltonian system, running example throughout Part III.
Adiabatic invariants. §52: action variable $I = \oint p d q$ approximately conserved to all orders under slow parameter variation. Foundational for geometric / Berry phase, quasi-static thermodynamics.
Newton's equations dispatched as setup. Ch. 1-2 are 60 pages. Per Arnold's framing, Codex's lack of a Newtonian-mechanics unit is a deferrable gap — content is covered by the geometric units; only pedagogical setup is missing.
Examples-driven. Each chapter ends with worked computations: Kepler, harmonic oscillator, Lagrange's top, Euler equations on $S O (3)$ , three-body, ellipsoidal geodesics.
Originator-text status for some material. Arnold's Appendix 8 is the canonical synthesis of KAM (Kolmogorov 1954 / Arnold 1963 / Moser 1962 in Arnold's voice). The Liouville-Arnold upgrade is Arnold's. Originator-prose treatment of KAM and Liouville-Arnold must cite Arnold directly.

The book ends before symplectic topology in the modern sense (no Floer, no Gromov non-squeezing, no pseudoholomorphic curves). The Arnold conjecture is stated as a remark in §49, the bridge into the McDuff-Salamon programme.

§2 Coverage table (Codex vs Arnold)

Cross-referenced against the current 235-unit Codex corpus. ✓ = covered at Arnold-equivalent depth, △ = topic present but Codex unit shallower than Arnold's chapter (typically templated v0.5 Strand-B prose), ✗ = not covered.

Part I — Newtonian Mechanics (Ch. 1-2)

Arnold topic	Codex unit	Status	Note
Galilean group, Galilean relativity	—	✗	Deferred per Arnold's framing: setup, not load-bearing.
Newton's equation as 2nd-order ODE on $M$	—	✗	Setup. Could live in future `09.*` math-physics chapter.
Conservation of energy in 1D potential	—	✗	Setup.
Central-force motion, Kepler problem	—	✗	Worked example only — Arnold §8.
Motion in central field, virial theorem	—	✗	Worked example.
Conservation laws, $N$ -body	—	✗	Setup.

Part II — Lagrangian Mechanics (Ch. 3-5)

Arnold topic	Codex unit	Status	Note
Variational principle (Hamilton's least action)	`03.04.08` variational-calculus	△	Euler-Lagrange machinery is in `03.04.08`; least action as foundational principle of mechanics is not articulated as own unit.
Euler-Lagrange equations, derivation	`03.04.08` (mention)	△	Differential-forms / calculus-of-variations setting. Mechanics-first framing — Lagrangian on $T M$ as equations of motion — not covered.
Lagrangian on tangent bundle $L : T M \to R$	—	✗	Gap. Arnold §16. The intrinsic Lagrangian-on- $T M$ setup.
Legendre transform, fibre derivative $T M \to T^{*} M$	—	✗	Gap. Arnold §15. The bridge from Lagrangian to Hamiltonian formalism.
Geodesic flow as Lagrangian system on Riemannian manifold	—	✗	Gap. Arnold §17.
Noether's theorem (continuous symmetry → conservation law)	—	✗	Gap. Arnold §20. Foundational across math-physics.
Oscillations, normal modes, Lagrange's small-oscillation theorem	—	✗	Gap. Arnold Ch. 5 — worked example.
Rigid body, Euler's theorem, Lagrange's top	—	✗	Gap. Arnold §27-§30 + Appendix 2 (Euler equations as geodesic on $S O (3)$ ).

Part III — Hamiltonian Mechanics (Ch. 6-10)

Bulk of Codex's existing symplectic chapter lives here.

Arnold topic	Codex unit	Status	Note
Symplectic vector space, symplectic form	`05.01.01` symplectic-vector-space	△	Templated; Cannas-plan depth gap.
Symplectic manifold, Darboux's theorem	`05.01.02`, `05.01.04` darboux-theorem	△	Same Darboux depth gap as Cannas plan.
Cotangent bundle, Liouville form $θ = p d q$	`05.02.05` cotangent-bundle	△	`tier_anchors` cite Arnold §37-§44. Naturality of $θ$ and geodesic-flow worked example absent.
Hamiltonian vector field, $ι_{X_{H}} ω = d H$	`05.02.01` hamiltonian-vector-field	△	Templated.
Liouville's theorem (phase volume preservation)	—	✗	Gap. Arnold §16 / §38. $L_{X_{H}} ω^{n} = 0$ .
Poincaré recurrence theorem	—	✗	Gap. Arnold §16. Volume preservation ⇒ recurrence.
Poisson bracket, Jacobi identity	`05.02.02` poisson-bracket	△	Templated.
Integral invariants of Poincaré-Cartan	—	✗	Gap. Arnold §44. The 1-form $p d q - H d t$ on extended phase space. Arnold-distinctive, Cannas-absent.
Hamilton-Jacobi equation $\partial_{t} S + H (q, \partial_{q} S, t) = 0$	`05.05.03` generating-functions	△	Generating-function-as-symplectomorphism in `05.05.03`; Hamilton-Jacobi PDE for the flow (action as function of endpoint, separation of variables, Jacobi's complete-integrals theorem) absent. Arnold-distinctive depth gap.
Canonical transformations, generating functions	`05.05.03` generating-functions	△	Recently shipped; covers symplectomorphism-from-function. The four classical types $S_{1}, S_{2}, S_{3}, S_{4}$ not all worked.
Integrable system (Liouville)	`05.02.03` integrable-system	△	Templated; depth gap.
Action-angle variables (local statement)	`05.02.04` action-angle-coordinates	△	Templated. On Cannas priority-1 deepening list.
Liouville-Arnold theorem (compact level set ⇒ torus, global action-angle)	`05.02.04` (claimed)	△	Title invokes "action-angle"; theorem not actually proved. On Cannas priority-1 deepening list.
KAM theorem (Kolmogorov-Arnold-Moser, persistence of diophantine tori)	—	✗	Gap (priority-1, apex unit). Not in Codex anywhere. Largest topic-level gap.
Adiabatic invariants	—	✗	Gap (priority-2). Arnold §52.
Perturbation theory (Lindstedt series, Birkhoff normal form)	—	✗	Gap. Arnold §51 + Appendix 7. Setup for KAM.

Appendices — Arnold-distinctive material

Appendix	Codex unit	Status	Note
App. 1: Riemannian curvature	`03.05.09` curvature; cross-strand `03.*`	△	Topic present; Arnold's mechanics-flavoured exposition not separately reproduced.
App. 2: Geodesics on Lie groups; Euler-Arnold equations on $Diff_{vol}$	—	✗	Gap. Arnold's celebrated ideal-fluid-as-geodesic observation. Arnold-distinctive.
App. 3: Symplectic structures on algebraic manifolds	partial in `04.*`, `05.01.02`	△	Standard examples ( $C P^{n}$ as Kähler) present.
App. 4: Contact structures	`05.10.01` contact-manifold	✓	Topic shipped.
App. 5: Dynamical systems with symmetry	`05.04.01` moment-map; `05.04.02` reduction	△	Cannas-anchored; on Cannas priority-1 list.
App. 6: Williamson's normal form for quadratic Hamiltonians	—	✗	Gap.
App. 7: Normal forms near fixed point / closed trajectory; Birkhoff	—	✗	Gap. Setup for KAM.
App. 8: KAM theorem	—	✗	Gap (priority-1, apex unit). ~80 pages of Arnold.

Topics Arnold covers as remarks / pointers (no equivalence-coverage required)

Topic	Codex unit	Note
Arnold conjecture (fixed-point lower bound)	`05.08.01` arnold-conjecture	✓ Codex has the unit; Floer downstream in `05.08.02-04`.
Symplectic topology (Gromov non-squeezing)	`05.07.01-02`	✓ Cannas / McDuff-Salamon territory.

Aggregate coverage estimate

Theorem layer: ~55% topic-level, ~30% Arnold-equivalent proof-depth. After priority-1: ~65%; priority-1+2: ~85%; + priority-3 + Cannas-deferred: ~93%.
Exercise layer: Arnold's ~250 Problems vs. Codex's templated 7-question block. Defer to dedicated exercise-pack pass.
Worked-example layer: ~20% covered. Kepler, oscillator, Lagrange top, ellipsoidal geodesics absent.
Notation layer: ~85% aligned. $ω = d p \land d q = - d θ$ , $θ = p d q$ , $L$ Lagrangian, $H$ Hamiltonian, $X_{H}$ , ${,}$ , $(I, θ)$ — all match. No notation/arnold.md needed.
Sequencing layer: ~70%. Hamiltonian flow OK; Lagrangian-to- Hamiltonian bridge (Legendre / fibre derivative) missing.
Intuition layer: ~40%. Mechanics-first intuition partially captured but not at Arnold's depth.
Application layer: ~50%. Kepler, rigid body, geodesic flow, three-body partially in Cannas-anchored Connections sections.

§3 Gap punch-list (priority-ordered)

The Codex symplectic chapter is mature on the Hamiltonian side; Arnold exposes gaps on (a) the Lagrangian side (entire 05.00-* chapter to write), (b) the perturbation / KAM block (entire 05.09-* chapter to write, with KAM as apex), and (c) a handful of Hamilton-Jacobi / geodesic-flow / Liouville-recurrence / Poincaré-Cartan / adiabatic- invariant items absent from Cannas.

Recommended slot ranges:

New 05.00-lagrangian-mechanics/ chapter (Lagrangian foundation).
New 05.09-perturbation/ chapter (KAM, normal forms, adiabatic).
Extensions to existing 05.02-hamiltonian/ and 05.05-lagrangian/.

Priority 1 — apex gap (KAM) and load-bearing Lagrangian infrastructure

05.09.01 KAM theorem (Kolmogorov-Arnold-Moser). Apex unit. Arnold Appendix 8 anchor; Kolmogorov 1954 (Dokl. Akad. Nauk SSSR 98), Arnold 1963 (Russ. Math. Surveys 18), Moser 1962 (Nachr. Akad. Wiss. Göttingen) as joint originator-citations. Three-tier; ~2500-3000 words at master tier. Master section: diophantine condition $∣ ⟨ k, ω ⟩ ∣ \geq γ ∣ k ∣^{- τ}$ ; Newton-iteration scheme; small-divisor bounds; statement that for sufficiently small $ϵ$ the surviving tori have positive Lebesgue measure tending to full as $ϵ \to 0$ . Highest priority — largest single topic-level gap in the symplectic chapter.
05.00.01 Lagrangian on the tangent bundle. Arnold §16 anchor; Lagrange 1788 Mécanique analytique (originator of analytical form), Arnold 1974 (originator of geometric tangent-bundle form). Three-tier; ~1800 words. Master section: $L : T M \to R$ ; action $S [γ] = \int L (γ, \overset{γ}{˙}) d t$ ; Euler-Lagrange $\frac{d}{d t} \partial_{v} L - \partial_{q} L = 0$ . Worked example: $L = \frac{1}{2} m ∣ \overset{q}{˙} ∣^{2} - V (q)$ recovers Newton.
05.00.02 Hamilton's principle of least action. Arnold §13 anchor; Maupertuis 1744 / Lagrange 1788 / Hamilton 1834 (joint originator-citation). Three-tier; ~1500 words. Statement: $q (t)$ is a motion iff critical point of $S [q]$ . Master section: derivation of Euler-Lagrange from $δ S = 0$ ; least action as foundational rather than derived.
**05.00.03 Legendre transform / fibre derivative $T M \to T^{*} M$ .** Arnold §15 anchor; Legendre 1787 originator-citation. Three-tier; ~1600 words. Master section: $\mathbb{F}L : (q, v) \mapsto (q, \partial_v L) $i s a d i f f eo m or p hi s m f or h y p er - r e g u l a r$ L$; $H (q, p) = p \cdot v - L$ . Worked example: $L = \frac{1}{2} ∣ v ∣^{2} - V$ ⇒ $H = \frac{1}{2} ∣ p ∣^{2} + V$ . Bridge from 05.00-* Lagrangian block to existing 05.02-* Hamiltonian block.
05.00.04 Noether's theorem. Arnold §20 anchor; Noether 1918 Invariante Variationsprobleme (Nachr. Königl. Ges. Wiss. Göttingen) — originator, mandatory originator-prose treatment. Three-tier; ~2000 words. Statement: one-parameter group of symmetries of a Lagrangian system gives a conserved quantity; in the Hamiltonian setting, this becomes the moment map. Worked examples: translation ⇒ momentum, rotation ⇒ angular momentum, time-translation ⇒ energy. Foundational across math-physics; used downstream by gauge theory, QFT.
**05.02.06 Geodesic flow as Hamiltonian flow on $T^{*} M$ .** Arnold §17 + App. 1 anchor; Jacobi 1837 originator-citation. Three-tier; ~1700 words. Master section: geodesic equations as Euler-Lagrange for $L = \frac{1}{2} ∣ v ∣_{g}^{2}$ , equivalently Hamiltonian flow on $T^{*} M$ for $H = \frac{1}{2} ∣ p ∣_{g^{- 1}}^{2}$ . Worked examples: round $S^{n}$ (integrable); flat torus (integrable); ellipsoid (Jacobi's integrable case — anchors Liouville-Arnold).
Deepen 05.02.04 action-angle-coordinates (Liouville-Arnold). Already on Cannas priority-1 list (Cannas item 7). Replace templated proof with actual Liouville-Arnold statement and proof. No double-count — deferred to Cannas batch; flagged here for honest overlap accounting.

Priority 2 — Arnold-distinctive Hamiltonian-mechanics depth (absent from Cannas)

05.05.04 Hamilton-Jacobi equation. Arnold §47 anchor; Hamilton 1834 / Jacobi 1837 Vorlesungen über Dynamik joint originator-citation. Three-tier; ~2000 words. Statement: $S (q, t)$ solves $\partial_{t} S + H (q, \partial_{q} S, t) = 0$ , generates the Hamiltonian flow. Jacobi's theorem on complete integrals: complete integral $S (q, t; α)$ depending on $n$ parameters $α_{i}$ gives flow via $\partial_{α} S = β$ , $p = \partial_{q} S$ . Master section: WKB / geometric optics interpretation; separation of variables (Kepler, central force); bridge to 05.05.03.
05.02.07 Liouville's theorem (phase volume preservation). Arnold §16 / §38 anchor; Liouville 1838 originator. Three-tier; ~1300 words. Statement: Hamiltonian flow preserves $ω^{n} / n!$ . Master section: one-line Cartan-formula proof; ergodic-theory consequences. Used silently throughout existing units; promotion closes a foundational gap.
05.02.08 Poincaré recurrence theorem. Arnold §16 anchor; Poincaré 1890 Sur le problème des trois corps originator, mandatory originator-prose. Three-tier; ~1300 words. Statement: measure-preserving transformation on finite-measure space sends a.e. point arbitrarily close to itself infinitely often; specialised to Hamiltonian flow on compact energy surface. Master section: proof from Liouville; Boltzmann reply to Loschmidt.
05.09.02 Adiabatic invariants. Arnold §52 anchor; Einstein 1911 / Ehrenfest 1916 originators. Three-tier; ~1800 words. Statement: for $H (q, p; λ)$ with slowly varying $λ (ϵ t)$ , $I = \frac{1}{2 π} \oint p d q$ is conserved to all orders in $ϵ$ on time scales $1/ ϵ$ . Master section: averaging proof; Arnold-Henrard generalisation. Bridge to geometric / Berry phase.
05.09.03 Birkhoff normal form / perturbation theory. Arnold §51 + App. 7 anchor; Birkhoff 1927 Dynamical Systems originator. Three-tier; ~1800 words. Statement: $H$ near non-resonant elliptic fixed point can be transformed to $H = H_{0} (I) + O (I^{N})$ for any $N$ . Lindstedt series here. Setup for KAM (item 1).
05.02.09 Integral invariants of Poincaré-Cartan. Arnold §44 anchor; Cartan 1922 Leçons sur les invariants intégraux originator. Three-tier; ~1400 words. Statement: 1-form $θ - H d t$ on $T^{*} M \times R$ has integral over closed contour invariant under Hamiltonian flow. Cannas-absent, Arnold-distinctive.

Priority 3 — depth gaps on existing units, examples, applications

Deepen 05.02.05 cotangent-bundle. Already on Cannas priority-3 list (item 17). Compute $ω = - d θ$ ; naturality of $θ$ ; geodesic-flow worked example. Deferred to Cannas batch.
Deepen 05.05.03 generating-functions. Add Master-tier table of $S_{1}, S_{2}, S_{3}, S_{4}$ ; Jacobi identity for canonical transformations; bridge to Hamilton-Jacobi (item 8). No new ID.
05.09.04 Williamson's normal form for quadratic Hamiltonians. Arnold App. 6 anchor; Williamson 1936 (Amer. J. Math. 58) originator. Three-tier; ~1400 words. Quadratic Hamiltonian on $R^{2 n}$ is symplectically equivalent to a sum of oscillators with eigenfrequencies from spectrum. Optional; closes classification gap.
05.00.05 Worked Lagrangian examples (oscillator, central force, rigid body). Arnold Ch. 5 anchor. Three-tier; ~1800 words. Beginner: harmonic oscillator. Intermediate: central potential, Kepler's laws. Master: Euler equations of rigid body as geodesic flow on $S O (3)$ left-invariant metric; Lagrange's top. Provides worked-example layer Codex currently lacks.
05.00.06 Galilean group and Newtonian mechanics setup. Arnold Ch. 1 anchor. Three-tier; ~1500 words. Optional / low priority — Arnold himself dispatches as setup. Defer unless a flow needs the Newtonian-to-Lagrangian bridge as a step.

Priority 4 — Appendix material, optional / advanced

05.09.05 Geodesics on Lie groups / Euler-Arnold equations. Arnold App. 2 anchor; Arnold 1966 (Ann. Inst. Fourier 16) originator. Master-only; ~1500 words. Euler equations of ideal fluid as geodesic flow on $Diff_{vol} (M)$ . Originator-prose; bridges to Arnold-Khesin (FT 1.12).
05.09.06 Nekhoroshev estimates / exponential stability. Nekhoroshev 1977 originator. Master-tier; ~1500 words. Strengthening of KAM to deterministic exponential bound on action drift. Optional sequel to item 1.
05.00.07 Survey: classical-mechanics examples (three-body, Lagrange points, Hill's lunar theory). Master-only; ~1000 words. Defer unless curriculum expands into celestial mechanics.

§4 Implementation sketch

Minimum Arnold-equivalence batch = priority 1 only (items 1-7): 6 new units + 1 deferred deepening. Estimates:

~3 hours per typical new unit; ~5 hours for KAM (item 1) — large and requiring originator research.
Priority 1: 5 typical × 3 h + 1 KAM × 5 h = ~20 hours.
Priority 1+2: ~20 + 6 new × 3 h = ~38 hours.
Priority 1+2+3 (excl. deferred items 7, 14): ~38 + 3 new × 3 h + 1 deepening × 1.5 h = ~49 hours.

At 4-6 production agents in parallel, priority-1+2 fits in 3-4 days with one integration agent. KAM is the bottleneck; warrants its own dedicated agent invocation with extended token budget.

Batch structure.

Batch A (Lagrangian foundation, items 2-5, ~12 h): opens new 05.00-lagrangian-mechanics/ chapter. Internal references; produce together. Must precede item 6.
Batch B (Geodesic / Hamilton-Jacobi / Liouville / recurrence, items 6, 8, 9, 10, ~13 h): extends 05.02-hamiltonian/ and 05.05-lagrangian/. After or in parallel with Batch A.
Batch C (Perturbation block, items 1, 11, 12, 13, ~14 h): opens new 05.09-perturbation/ chapter (item 13 in 05.02-*). After Batch B (KAM cites Birkhoff cites Poincaré recurrence).
Optional Batch D (priority-3+4, items 15-21, ~15 h): after Cannas priority-1+2 batch lands.

Originator-prose targets. Arnold is partly originator (the Liouville-Arnold action-angle upgrade is his; the geodesic-on- Diff_vol is his; KAM synthesis is his voice). Joint originator- citations:

KAM (item 1): Kolmogorov 1954, Arnold 1963, Moser 1962. Cite all three; Appendix 8 is canonical synthesis.
Lagrangian-on- $T M$ (2): Lagrange 1788 + Arnold 1974.
Hamilton's principle (3): Maupertuis 1744 / Lagrange 1788 / Hamilton 1834.
Legendre (4): Legendre 1787; Hamilton 1834 / Arnold 1974 for geometric form.
Noether (5): Noether 1918. Originator-prose mandatory — paraphrase her one-page proof.
Geodesic flow (6): Jacobi 1837; Arnold §17 / App. 1.
Hamilton-Jacobi (8): Hamilton 1834 / Jacobi 1837.
Liouville's theorem (9): Liouville 1838.
Poincaré recurrence (10): Poincaré 1890. Originator-prose mandatory.
Adiabatic invariants (11): Einstein 1911 / Ehrenfest 1916 / Arnold §52.
Birkhoff (12): Birkhoff 1927.
Poincaré-Cartan (13): Poincaré 1899 / Cartan 1922.

Each priority-1 unit's Master section cites originator + Arnold.

Notation crosswalk. Aligned per §2. No notation/arnold.md needed. Pin sign convention $ω = d p \land d q = - d θ$ , $θ = p d q$ in Master sections of items 4 and 8.

DAG edges. New prerequisites for priority-1+2:

05.00.01 ← {03.02.01 smooth-manifold, 03.04.08 variational- calculus}
05.00.02 ← 05.00.01
05.00.03 ← {05.00.01, 05.02.05 cotangent-bundle}
05.00.04 ← {05.00.02, 03.03.02 group-action}; 05.00.04 → 05.04.01 moment-map (Noether → moment-map: the Hamiltonian moment map is the Noether-conserved quantity).
05.02.06 ← {05.00.03, 05.02.05, 05.02.01}
05.02.07 ← {05.01.02, 05.02.01, 03.04.04}
05.02.08 ← 05.02.07
05.05.04 ← {05.05.03, 05.02.04}
05.09.01 ← {05.09.03, 05.02.04, 05.02.08}
05.09.02 ← 05.02.04
05.09.03 ← {05.02.04, 05.05.04}
05.02.09 ← {05.02.05, 05.05.03}

Chapter structure. Two new chapters + extensions to two existing:

New 05.00-lagrangian-mechanics/: items 2-5 (and 17-18, 21).
New 05.09-perturbation/: items 1, 11, 12 (and 16, 19, 20).
Extend 05.02-hamiltonian/: items 6, 9, 10, 13.
Extend 05.05-lagrangian/: item 8.

Composite Cannas + Arnold batch. Production order:

Cannas priority-1 (Moser, Weinstein, Darboux + action-angle deepening — closes technique gap).
Arnold Batch A (Lagrangian foundation).
Arnold Batch B (Geodesic-HJ-Liouville-recurrence).
Arnold Batch C (KAM block — depends on Cannas action-angle deepening from step 1).
Cannas priority-2 + Arnold priority-3+4 interleave as background.

Cross-references. Arnold KAM (item 1) ↔ Liouville-Arnold deepening (item 7, in Cannas batch): KAM is perturbation of Liouville-Arnold foliated picture. Arnold Noether (item 5) → moment-map (05.04.01): moment map is Noether-conserved quantity for group action.

§5 What this plan does NOT cover

Line-number Problem inventory across three Parts and eight Appendices. Defer unless punch-list expands.
Arnold's ~250 Problems vs. Codex's templated 7-problem block. Dedicated Arnold-exercise-pack family (05.00.E1 Lagrangian, 05.02.E1 Hamiltonian, 05.09.E1 perturbation) is P3-priority follow-up.
Cannas-overlap items: deepening of 05.02.04 action-angle (Cannas priority-1 item 7) and 05.02.05 cotangent-bundle (Cannas priority-3 item 17) not duplicated here. Referenced in §3 (items 7, 14) but production owned by Cannas plan.
McDuff-Salamon territory: Floer-theoretic Arnold conjecture (05.08.02-04), Gromov non-squeezing, $J$ -holomorphic curves. Arnold mentions only as remarks.
Heavy classical-mechanics computation: full Laplace-Runge-Lenz, full three-body, Hill's lunar theory, Lagrange points L1-L5. Goldstein / Landau-Lifshitz territory; Arnold dispatches as worked examples. Item 21 is pointer-stub only.
Topological methods in hydrodynamics (FT 1.12). Item 19 is stub; full Arnold-Khesin coverage is its own per-book plan.
Statistical mechanics / ergodic theory beyond Poincaré recurrence. Lives in Codex 08-stat-mech/.
Geometric quantisation (Kostant-Souriau). Arnold mentions in passing (App. 3); deferred to a future plan.
notation/arnold.md standalone file. Crosswalk decisions in §4 are sufficient.
Worked-example densification across the existing 21 symplectic units (curriculum-wide v0.5-Strand-B pattern). Items 6 and 17 partially fill the gap.

§6 Acceptance criteria for FT equivalence (Arnold)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, the book is at equivalence-coverage when:

≥95% of Arnold's named theorems map to Codex units at Arnold-equivalent proof depth (currently ~30%; after priority-1 ~65%; after priority-1+2 ~85%; after priority-3 + Cannas-deferred ~93%; after priority-4 ~96%). KAM (item 1) is the single largest contributor.
≥80% of Arnold's Problems have a Codex equivalent (currently ~5%; closing requires the Arnold-exercise-pack pass per §5).
≥90% of Arnold's worked examples reproduced (currently ~20%; priority-1+2 batch + item 17 brings to ~65%; remainder needs optional worked-example densification pass).
Notation alignment recorded inline in rewritten Master sections; no separate notation/arnold.md.
For every chapter dependency in Arnold (Part II → Part III via Legendre; §47 → §49 via Hamilton-Jacobi as integrability tool; §50 → App. 8 via action-angle as unperturbed system for KAM), there is a corresponding prerequisites arrow chain in Codex's DAG. The Lagrangian → Legendre → Hamiltonian chain in particular must be unbroken after the priority-1 batch.
Pass-W weaving connects the new units (05.00.01-04, 05.02.06-09, 05.05.04, 05.09.01-03) to the existing 05.01-symplectic-linear/, 05.02-hamiltonian/, 05.04-moment-reduction/, 05.05-lagrangian/, and the prerequisite 03.04-differential-forms/ chapters via lateral connections.

The 6 priority-1 new units (items 1-6) + the deferred Cannas-batch deepening (item 7) close the load-bearing-technique gap and ship the KAM apex unit. The 6 priority-2 items (items 8-13) close Arnold-distinctive Hamiltonian depth absent from Cannas. The 8 priority-3+4 items (items 14-21) are depth-completion, examples, and survey pointers; they bring proof-depth from ~85% to ~96% but are not strictly required for sign-off.

Honest scope. Mid-sized equivalence gap: symplectic chapter is mature on the Hamiltonian side (Arnold cited as anchor on most existing units) but has a structural gap on the Lagrangian side (entire 05.00-* chapter to write) and a single largest gap in the perturbation / KAM block (entire 05.09-* chapter to write, KAM as apex). Priority-1 batch is 6 new units + 1 deferred deepening — larger than Cannas priority-1 (3-4 new) but smaller than Hatcher priority-1+2 (10 units).

Apex unit designation. Item 1 (05.09.01 KAM) is designated an apex unit — at the boundary of the curriculum's depth, requiring originator-research synthesis (three joint originators across two languages and three decades), substantial master-tier exposition (~3000 words), bridging to modern dynamical-systems literature. Apex units are produced with extended agent budgets and stricter quality-sampling thresholds (per docs/specs/ORCHESTRATION_PROTOCOL.md §8). The Arnold KAM unit joins a small list across the curriculum (e.g., 03.09.10 Atiyah-Singer in the Lawson-Michelsohn batch).

Composite Cannas + Arnold batch (restated). Cannas Lectures 22-24 and Arnold Part III overlap on Liouville-Arnold and action-angle. Producing Cannas priority-1 and Arnold priority-1 in coordinated succession yields a ~10-unit mechanics-equivalence block closing both books' core gaps together. Shared 05.02.04 deepening owned by Cannas batch; Arnold-distinctive new units (KAM, Lagrangian foundation, Hamilton-Jacobi PDE, geodesic flow, Noether, Liouville volume, Poincaré recurrence) owned by this Arnold batch. After both land, the symplectic-mechanics block is at ≥85% Arnold-equivalent and ≥85% Cannas-equivalent simultaneously.

Arnold — *Mathematical Methods of Classical Mechanics* (Fast Track 1.11) — Audit + Gap Plan