Souriau — *Structure of Dynamical Systems: A Symplectic View of Physics* (Fast Track 2.11) — Audit + Gap Plan

Book: Jean-Marie Souriau, Structure of Dynamical Systems: A Symplectic View of Physics, Progress in Mathematics 149 (Birkhäuser, Boston 1997, xxxiv + 406 pp.). English translation by C.-H. Cushman-de Vries; original French Structure des systèmes dynamiques, Dunod, Paris 1970; reissued by Éditions Jacques Gabay 2008.

Fast Track entry: 2.11 on the booklist (docs/catalogs/FASTTRACK_BOOKLIST.md line 71, listed as "BUY"; statistical-physics block alongside Landau-Lifshitz 2.10 and Baxter 2.12). The book is also called out in the FASTTRACK_COVERAGE_ROADMAP.md §2 endgame batch and in the FASTTRACK_EQUIVALENCE_PLAN.md stat-physics list.

PDF status — REDUCED PASS. No local PDF in reference/fasttrack-texts/, reference/book-collection/free-downloads/, or ~/Downloads. Birkhäuser / Springer page returned 404 from WebFetch; the title is BUY-only and not freely posted. This plan is produced from canonical background knowledge of Souriau's book (the standard reference treatment in Marsden-Ratiu, Guillemin-Sternberg, Abraham-Marsden, Woodhouse — all of which cite Souriau extensively chapter-by-chapter) plus the existing Codex symplectic-chapter inventory. A full P1 line-number inventory is deferred until a copy is acquired.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of units to write and existing units to deepen so that Structure of Dynamical Systems is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

The audit surface is smaller than Cannas or Hatcher in raw new-unit count but conceptually distinctive: Souriau's book is the originator-text for the modern moment-map / coadjoint-orbit / geometric- quantisation / symplectic-statistical-mechanics package. Most of its modern content has already been shipped via the Cannas-anchored 05-symplectic/ chapter (21 units across 7 sub-chapters), the Marsden-Weinstein-anchored reduction audit (Cycle 4-7), and adjacent KAM / integrable / contact units. What is missing is Souriau's originator-distinctive content: (i) the Souriau cocycle and non-equivariant moment maps, (ii) the symplectic / Lie-group formulation of Gibbs statistical mechanics, (iii) the original classification of elementary particles by coadjoint orbits of Galilei and Poincaré, and (iv) the Bose-Fermi alternative read off from the topology of the spin coadjoint orbit. These four blocks are the Souriau-specific deepenings and small new-unit additions this plan tracks.

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§1 What Souriau's book is for

Souriau's Structure of Dynamical Systems (SDS hereafter) is the originator-text for the modern symplectic-geometric formulation of physics. Where Arnold Mathematical Methods of Classical Mechanics (FT 1.0a) gives the mechanics-first, KAM-friendly symplectic textbook, and Cannas da Silva (FT 1.11) gives the modern lecture-note synthesis, Souriau gives the philosophical-foundational treatment in which the phase space of every physical system — classical particle, classical spin, classical or quantum elementary particle, statistical-mechanics state — is a symplectic (or pre-symplectic) manifold with a Hamiltonian action of a Lie group, and physical observables are recovered as the moment map.

The historical importance is hard to overstate. Souriau introduced the moment map (he called it the "moment géométrique") independently of Kostant and earlier than the modern Marsden-Weinstein 1974 formulation; he introduced the Souriau cocycle that measures the failure of equivariance of a Hamiltonian action and showed that it classifies the central extensions of the Lie algebra needed for a faithful representation on phase space; he classified the elementary classical systems of the Galilei and Poincaré groups as coadjoint orbits, with mass and spin emerging as Casimir invariants of the orbit, before the quantum-mechanical Wigner / Bargmann classification was rephrased symplectically; he gave the symplectic-statistical-mechanics framework in which a Gibbs state on a symplectic manifold with a Hamiltonian $G$-action is determined by a generalised temperature $\beta \in \mathfrak{g}$ (a Lie-algebra element, not just a real number); and he derived the Bose-Fermi alternative for the spin coadjoint orbit from the geometric prequantisation condition on the spin orbit, giving a classical-symplectic origin for the spin-statistics distinction.

Distinctive contributions, in roughly the order SDS develops them (reconstructed from chapter listings cited in Marsden-Ratiu and Guillemin-Sternberg):

1. Pre-symplectic manifolds. Souriau works in the more general pre-symplectic category (closed 2-form, possibly with kernel) and recovers symplectic geometry as the quotient by the characteristic foliation. This is the natural setting for constrained systems and for the reduced phase space of a Lie-group action with a non-trivial stabiliser. Codex's symplectic chapter is exclusively symplectic; pre-symplectic is absent.
2. The moment map (introduced here under the name moment — French "moment cinétique" → English "momentum map"). Souriau's definition is the same as the modern one: a $G$-equivariant map $\mu : M \to \mathfrak{g}^*$with$d\mu^X = \iota_{X^\sharp}\omega$ for every $X\in \mathfrak{g}$. Codex 05.04.01 ships this.
3. The Souriau cocycle. When a Hamiltonian $G$-action does not admit an equivariant moment map, the failure-of-equivariance is measured by a Lie-algebra 2-cocycle $\sigma : \mathfrak{g} \times \mathfrak{g} \to \mathbb{R}$, the Souriau cocycle. It is a central extension of $\mathfrak{g}$ — the moment-map extension — and classifies the non-equivariant case. This is Souriau-original; it is the basis for Marsden-Ratiu's treatment of non-equivariant reduction in Introduction to Mechanics and Symmetry §12. Codex has no coverage of the cocycle or of non-equivariant moment maps.
4. Coadjoint orbits as the classical configuration space of an elementary system. Souriau classifies the connected, transitive $G$-Hamiltonian symplectic manifolds (for $G$ a connected Lie group) as covers of coadjoint orbits of the cocycle-extended central extension $\tilde{G}$. For $G$ the Galilei group, the elementary systems are parametrised by mass $m>0$ (the central charge) and spin $s\ge 0$ (a Casimir of the rotation subgroup); for $G$ the Poincaré group, mass and spin again. This is the symplectic-classical analogue of the Wigner classification of irreducible unitary representations. Codex 05.03.01 ships coadjoint orbits as symplectic manifolds (with the Kirillov-Kostant-Souriau form) but does not develop the classification-of-elementary-systems direction.
5. Geometric formulation of statistical mechanics on a symplectic $G$-space. Souriau replaces the canonical-ensemble Gibbs state $\rho \propto e^{-\beta H}$ with the generalised ${\rho }_{\beta }\propto e^{-⟨\beta ,\mu (x)⟩}$ where $\beta \in \mathfrak{g}$ is a Lie-algebra-valued generalised temperature and $\mu :M\to {\mathfrak{g}}^{\ast }$ is the moment map. The partition function $Z(\beta) = \int_M e^{-\langle \beta, \mu\rangle}, \omega^n / n!$isafunctionon$\mathfrak{g}$; its first and second derivatives recover the mean of the moment and its covariance. When $G=\mathbb{R}$ acting by time translations and $\mu =H$ the energy, the classical Gibbs ensemble is recovered. This is a uniquely-Souriau framework; it has been picked up in the modern information-geometric literature (Barbaresco, Marle) but the originator-statement is Souriau Chapter IV. Codex 11.04.01 ships the canonical-ensemble Gibbs state on phase space; the Lie-group / moment-map generalisation is absent.
6. Prequantisation and the integrality condition. Souriau states the prequantisation condition independently of Kostant (1970): a symplectic $(M,\omega )$ admits a prequantum line bundle with connection of curvature $\omega /\hslash$ iff $[\omega/(2\pi\hbar)] \in H^2(M, \mathbb{Z})$ is integral. Souriau Chapter V develops prequantisation, polarisations, and geometric quantisation in parallel with Kostant's Quantization and Unitary Representations (Springer LNM 170, 1970). The full geometric-quantisation programme has its own FT entry (Woodhouse, FT 3.09) and is deferred there; Souriau's contribution here is to give the prequantisation condition simultaneously with Kostant. Codex has no prequantisation unit.
7. The Bose-Fermi alternative from coadjoint-orbit topology. Souriau Chapter V applies geometric quantisation to the spin coadjoint orbit $S^2$ (with KKS form $s\cdot {\omega }_{S^2}$) and shows that the integrality / prequantisation condition forces the spin $s$ to be a half-integer. The connected component of the identity in the prequantisation group then has two connected components depending on whether $s$ is integer (single-valued representation = Bose statistics) or half-integer (double-valued = Fermi statistics). This is the classical-symplectic origin of the spin-statistics distinction, independent of the full relativistic-QFT spin-statistics theorem. Codex has no unit on this.
8. Galilei and Poincaré symmetry as the structural axiom. Souriau treats the Galilei group (for non-relativistic systems) and the Poincaré group (for relativistic systems) as the primary structural data — the phase space is built from the symmetry, not imposed on top of it. The Galilei group's Souriau-cocycle central charge is the mass; the Poincaré group has no analogous non-trivial cocycle (it is its own universal central extension), so mass appears instead as a Casimir of the orbit. This is the cleanest single statement of why Galilean and Poincaré mechanics look structurally different. Codex's 05.00.06 Galilean-Newtonian setup is a templated stub; no Poincaré-symplectic unit exists.

SDS is not a first introduction to symplectic geometry or to classical mechanics; it assumes ease with manifolds, Lie groups, Lie-algebra cohomology, and Hamiltonian mechanics in the Arnold sense. It is the canonical originator-text once a reader is past Cannas / Arnold and wants the originator-flavour of the moment-map / coadjoint-orbit / prequantisation programme. Its modern downstream is Marsden-Ratiu (Introduction to Mechanics and Symmetry, 2nd ed. Springer 1999, the standard modern reference, which cites Souriau on essentially every chapter), Guillemin-Sternberg (Symplectic Techniques in Physics, Cambridge 1984, which builds on Souriau-Kostant geometric quantisation), Abraham-Marsden (Foundations of Mechanics 2nd ed. 1978, the historical companion volume), and Woodhouse (Geometric Quantization 2nd ed. 1991, FT 3.09 — the standard reference for the Souriau-Kostant programme on the quantum side).

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§2 Coverage table (Codex vs Souriau)

Cross-referenced against the 50 shipped units in 05-symplectic/ (per the v0.5 Strand-B production, plus the Cycle 4-7 deepenings of the moment map and reduction units), the partition-function unit 11.04.01 (canonical ensemble), and the Fock-space unit 12.03.01. ✓ = covered at Souriau-equivalent depth, △ = topic present but Codex unit shallower than SDS's chapter (typically a Cannas-anchored modern treatment lacking the Souriau-originator framing), ✗ = not covered. The symplectic chapter retains the templated-prose problem flagged in the Cannas plan: the Master "Key theorem with proof" on many units is a generic Cartan-formula calculation, so several △ entries below reflect Souriau-distinctive depth gaps on top of the modern-depth gaps the Cannas plan already tracks.

Chapter I — Differential geometry foundations (manifolds, forms, Lie groups)

Souriau topic	Codex unit(s)	Status	Note
Smooth manifold, tangent / cotangent bundle, differential forms	03.01.*, 03.02.*	✓	Standard foundations; covered fully in 03-modern-geometry/.
Lie group, Lie algebra, exponential map	03.03.*, 03.04.*	✓	Covered in 03-modern-geometry/03-lie/.
Lie-algebra cohomology (low degrees: $H^1,H^2$)	—	✗	Gap. Required for the Souriau cocycle. The cohomology infrastructure exists in the 03-modern-geometry/ chapter but the specifically Lie-algebraic case at $H^2$ has no dedicated unit. Souriau-distinctive prerequisite.

Chapter II — Symplectic geometry (pre-symplectic, symplectic, Hamiltonian actions)

Souriau topic	Codex unit(s)	Status	Note
Symplectic manifold, Darboux	05.01.02, 05.01.04	△	Topic present; Darboux proof is templated. Same gap as Cannas plan item 2.
Pre-symplectic manifold (closed 2-form, kernel = characteristic foliation, symplectic reduction $M/\ker \omega$)	—	✗	Gap (Souriau-distinctive). Souriau's natural framework for constrained systems and reduced phase spaces; modern symplectic books drop it. No Codex unit.
Hamiltonian vector field, Poisson bracket	05.02.01, 05.02.02	✓	Covered.
Symplectic action of a Lie group	05.04.01 (mention)	△	Setup-level fact appears in passing in the moment-map unit; not its own unit.
Moment map (Souriau-original definition)	05.04.01	△	Modern Cannas-anchored unit; Souriau's naming and historical priority not flagged. Add a sentence in the Master section + originator citation.
Souriau cocycle (non-equivariant moment map; $\sigma :\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ measuring failure of equivariance; central extension of $\mathfrak{g}$)	—	✗	Gap (high priority — Souriau-distinctive). The single most-load-bearing missing piece for any honest Souriau equivalence claim. Used as foundational machinery in Marsden-Ratiu §12.
Equivariant moment maps and the obstruction to equivariance	05.04.01 (partial)	△	Equivariance is assumed in the current Codex moment-map unit; the obstruction theorem (when can the cocycle be killed by adding a constant) is not stated.
Marsden-Weinstein-Meyer reduction (regular case)	05.04.02	△	Modern reduction unit; the Cycle 4-7 deepening fixed the proof template; Souriau-originator citation needed.
Coadjoint action, coadjoint orbit, Kirillov-Kostant-Souriau form	05.03.01	△	KKS form covered; Souriau's name in the eponym appears but the originator framing (every transitive Hamiltonian $G$-space is a cover of a coadjoint orbit) is not built out.
Classification of homogeneous symplectic $G$-spaces (Kirillov-Kostant-Souriau theorem: connected transitive Hamiltonian $G$-actions on symplectic manifolds correspond to coadjoint orbits of the cocycle-extension $\tilde{G}$)	—	✗	Gap (Souriau-distinctive). The classification half of the KKS triple is not stated. Codex covers what coadjoint orbits are but not that they are the classical objects.

Chapter III — Mechanics (Galilei / Poincaré symmetry, elementary systems)

Souriau topic	Codex unit(s)	Status	Note
Galilei group, Galilei algebra, central extension by mass (Bargmann extension)	05.00.06 (stub)	△	The Galilean-Newtonian setup unit exists but is templated; the Bargmann central extension and the Souriau-cocycle origin of mass are not covered.
Poincaré group, Poincaré algebra	—	✗	Gap. No symplectic-Poincaré unit. Some adjacent SR content in 10-em-sr/.
Elementary classical system = coadjoint orbit of the (extended) symmetry group	—	✗	Gap (high priority — Souriau-distinctive). The classification-of-particles story symplectically. Galilei elementary systems classified by $(m,s)$; Poincaré by $(m,s)$ + sign of energy.
Classical spin as the $S^2$ coadjoint orbit of $\mathrm{SO}(3)$ with the area form scaled by $s$	05.03.01 (general coadjoint orbits)	△	$S^2$ as the canonical coadjoint orbit appears as the headline example in 05.03.01 Intermediate section but not as a classical spin model.
Charged particle in an electromagnetic field as a Hamiltonian system on $T^{\ast }{\mathbb{R}}^3$ with twisted symplectic form $\omega +{\pi }^{\ast }F$	—	✗	Gap. Souriau's geometric model of magnetic coupling via the "minimally-coupled" symplectic form. Cross-link to 10-em-sr/ electromagnetism.
Time-dependent Hamiltonian systems via the extended pre-symplectic phase space $T^{\ast }Q\times {\mathbb{R}}_t$	—	✗	Gap. Souriau's natural setting for non-autonomous mechanics; modern texts split this off into contact geometry.

Chapter IV — Statistical mechanics on a symplectic $G$-space (Souriau-distinctive)

Souriau topic	Codex unit(s)	Status	Note
Canonical Gibbs ensemble on phase space, partition function $Z(\beta )$	11.04.01 canonical-ensemble	✓	Covered at the standard Landau-Lifshitz / Pathria level.
Souriau Gibbs state ${\rho }_{\beta }\propto e^{-⟨\beta ,\mu (x)⟩}$ on a symplectic $G$-space, generalised temperature $\beta \in \mathfrak{g}$	—	✗	Gap (high priority — Souriau-distinctive). The headline Chapter-IV result. The generalised-temperature framework.
Souriau partition function $Z:\mathfrak{g}\to \mathbb{R}$, mean moment $\bar{\mu }=-d\log Z/d\beta$, covariance / Fisher information $-d^2\log Z/d{\beta }^2$	—	✗	Gap. The Souriau-information-geometric framework picked up by Barbaresco and Marle in the 2010s.
Recovery of the canonical ensemble as the $G=\mathbb{R}$ time-translation case	—	✗	Gap (low priority — a remark in the new unit).
Recovery of relativistic Gibbs / Jüttner ensemble from the Poincaré-group case	—	✗	Gap (low priority — Master section in the new unit).

Chapter V — Quantum mechanics (prequantisation, geometric quantisation, Bose-Fermi)

Souriau topic	Codex unit(s)	Status	Note
Prequantum line bundle, connection of curvature $\omega /\hslash$, prequantisation map $f\mapsto -i\hslash {\nabla }_{X_f}+f$	—	✗	Gap. Pointer unit at FT-equivalence; full development is FT 3.09 Woodhouse.
Souriau-Kostant integrality condition $[\omega /(2\pi \hslash )]\in H^2(M,\mathbb{Z})$	—	✗	Gap (medium priority — Souriau-distinctive originator share with Kostant). Stub-level unit suffices for FT-equivalence; depth deferred to Woodhouse.
Polarisations (real, complex, Kähler), geometric quantisation	—	✗	Deferred to FT 3.09 Woodhouse.
Quantisation of the spin coadjoint orbit $S^2$, integrality forces $s\in \frac{1}{2}{\mathbb{Z}}_{\ge 0}$	—	✗	Gap (Souriau-distinctive). The classical-symplectic origin of half-integer spin.
Bose-Fermi alternative from the spin orbit prequantisation ($s$ integer ⇒ Bose; $s$ half-integer ⇒ Fermi; double-valued representation of the prequantum group)	—	✗	Gap (high priority — Souriau-distinctive headline result of Chapter V). No comparable Codex content.
Spin-statistics theorem at the QFT level	—	✗	Cross-strand to QFT; not Souriau's territory. Defer to Weinberg QFT (FT 2.17).
Bose / Fermi Fock spaces, second quantisation	12.03.01 bosonic-fock-space	△	Bosonic case shipped; Fermi case stub-only; the Souriau-cocycle origin of the Bose-Fermi distinction is not the framing.

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

Souriau topic	Status	Note
Quantisation of the harmonic oscillator and the hydrogen atom (Chapter V worked examples)	—	Pointer-only in Souriau (the headline examples in the Kostant programme); defer to Woodhouse FT 3.09.
Symplectic geometry of fluid mechanics (Arnold-Khesin programme)	—	Souriau touches it; defer to Arnold-Khesin Topological Methods in Hydrodynamics (not yet on FT).
Diffeology (post-1970 Souriau extension framework)	—	Not in Structure of Dynamical Systems; in Souriau's 1980s-90s papers. Defer entirely.

Aggregate coverage estimate

Theorem layer: ~70% of SDS's named theorems map to Codex units at the topic level (modern Cannas-anchored versions); only ~45% are at Souriau-distinctive originator-framing depth. The remaining 30% (Souriau cocycle, non-equivariant moment maps, KKS classification theorem, pre-symplectic framework, elementary-system classification for Galilei / Poincaré, Souriau Gibbs state and partition function, prequantisation integrality, Bose-Fermi alternative) are absent at any depth and are concentrated in Chapters III-V.

After the priority-1 punch-list below, theorem-level coverage rises to ~90% and originator-framing depth to ~75%; after priority-1+2 to ~95% theorem-level and ~85% framing; priority-3 brings framing to ~92%.

Exercise layer: not separately audited. SDS has fewer formal "exercises" than a modern textbook; it has worked examples and remarks that function pedagogically as exercises. Defer to a dedicated pass after the priority-1 batch.

Worked-example layer: ~30% covered. SDS's lead examples ($S^2$ coadjoint orbit as classical spin, charged particle in $F$, Galilei / Poincaré elementary systems, Souriau Gibbs ensemble for rotation / translation actions) are mostly absent from the corresponding Codex units. The Cycle 4-7 deepening of 05.04.01 / 05.04.02 partially fixed the moment-map / reduction worked-example slot but not the Galilei-Poincaré / statistical-mechanics ones.

Notation layer: ~60% aligned. Souriau's notation is idiosyncratic (French-Bourbaki Cartan-calculus conventions, distinctive symbols for the moment $\mu$ vs. modern $J$, the cocycle $\sigma$ or $\theta$, the generalised temperature $\beta \in \mathfrak{g}$). A notation/souriau.md crosswalk is recommended for the new Chapter-III / IV units.

Sequencing layer: ~70%. Souriau's "symmetry-first" DAG (Lie group → coadjoint orbit → mechanics) is the reverse of Codex's current "manifold-first → symplectic-form-first → Hamiltonian → moment map → orbit" DAG. The DAG itself does not need to change; the exposition in Master sections should add a Souriau-style symmetry-first recapitulation paragraph for the new Chapter-III / IV units.

Intuition layer: ~45%. Souriau's distinctive "every elementary system is a coadjoint orbit" intuition is the headline missing intuition; needs to land in 05.03.01 (deepening) and in the new Galilei / Poincaré elementary-system units.

Application layer: ~60%. Souriau's downstream applications (classification of relativistic particles, classical spin, magnetic coupling, Souriau-Gibbs statistical mechanics, Bose-Fermi from prequantisation) are the primary differentiator from Cannas. Closing the punch-list closes most of the application gap.

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§3 Gap punch-list (P3-lite — units to write or deepen, priority-ordered)

The Codex symplectic chapter is mature in modern topic coverage (50 shipped units; the Cycle 4-7 reduction-batch deepenings already fixed the load-bearing-technique gaps Cannas was tracking). Most of the Souriau-specific work below is new units in Chapters III-V territory (Souriau cocycle, KKS classification, Galilei / Poincaré elementary systems, Souriau Gibbs state, prequantisation, Bose-Fermi) plus deepenings of existing Cannas-anchored units to add Souriau-originator framing. The recommended slot ranges are:

• New Souriau-cocycle / non-equivariant-moment-map content: 05.04.07-08 (extending the moment-reduction sub-chapter).
• New KKS classification and elementary-system content: 05.03.02-04 (extending the coadjoint sub-chapter).
• New Galilei / Poincaré symplectic-mechanics content: 05.00.07, 05.00.09-10 (extending the lagrangian-mechanics sub-chapter, with a Galilei / Poincaré symplectic-mechanics sub-block).
• New Souriau-Gibbs / symplectic-statistical-mechanics content: 11.04.02-04 (extending the partition-functions sub-chapter of 11-stat-mech-physics/).
• New prequantisation / Bose-Fermi content: a small 05.11-geometric-quantisation/ stub sub-chapter (05.11.01-03), explicitly marked as pointer-level with depth deferred to Woodhouse FT 3.09.

Priority 1 — Souriau-distinctive load-bearing content

These items are the Souriau-originator content that the Cannas / Marsden-Weinstein modern-symplectic batches do not cover. Without them the symplectic chapter cannot honestly claim Souriau-equivalence.

1. 05.04.07 Souriau cocycle and non-equivariant moment maps. Definition of the cocycle $\sigma(X, Y) = {\mu^X, \mu^Y} - \mu^{[X,Y]}$, its closedness (Jacobi identity), the cohomology class in $H^2(\mathfrak{g},\mathbb{R})$, the central extension $\tilde{\mathfrak{g}}=\mathfrak{g}{\oplus }_{\sigma }\mathbb{R}$, the equivariance obstruction theorem (the cocycle is a coboundary iff there exists a constant shift of $\mu$ making the action equivariant). Souriau §11 anchor; Marsden-Ratiu §12.3-12.6 as modern reference. Three-tier; ~1800 words. Highest priority — load-bearing for items 2, 3, 5 below.

2. 05.03.02 Kirillov-Kostant-Souriau classification of homogeneous symplectic $G$-spaces. Statement and proof: every connected simply-connected transitive Hamiltonian $G$-action on a symplectic manifold is equivariantly symplectomorphic to a coadjoint orbit of the cocycle-extended group $\tilde{G}$, equipped with the KKS symplectic form. Souriau Chapter II (§13-§14) anchor; Kirillov 1962 Unitary representations of nilpotent Lie groups, Kostant 1970 Quantization and unitary representations, Souriau 1970 as the three independent originators. Three-tier; ~2000 words. Prerequisite: 05.04.07 (cocycle) and 05.03.01 (coadjoint orbits — deepened).

3. 05.03.03 Elementary classical system (Souriau definition): a connected simply-connected transitive Hamiltonian $G$-space; by item 2 = a coadjoint orbit of $\tilde{G}$; for the Galilei group parametrised by $(m,s)$, mass and spin; for the Poincaré group by $(m,s,ϵ)$ with energy sign. Souriau Chapter III headline result. Three-tier; ~1800 words. Worked examples: massive Galilei particle, massless Galilei (no analogue at $m=0$), Souriau classical spin from the $S^2$ orbit, Poincaré massive spin-$\frac{1}{2}$ particle, Poincaré massless helicity states.

4. 05.00.07 Galilei group and Bargmann central extension. Definition of the Galilei group $G$, the algebra $\mathfrak{g}$ (generators: $H,P_i,K_i,J_i$ with the Galilei commutators), the Bargmann cocycle $\sigma([K_i, P_j]) = m, \delta_{ij}$ giving the mass-shifted bracket $[K_i,P_j]=m{\delta }_{ij}$ in the extension. Souriau §12 anchor; Bargmann 1954 On unitary ray representations of continuous groups as parallel originator on the quantum side. Three-tier; ~1500 words. Foundational for items 3 and 5.

5. 11.04.02 Souriau Gibbs state on a symplectic $G$-space. Definition: ${\rho }_{\beta }=e^{-⟨\beta ,\mu ⟩}/Z(\beta )$ with $\beta \in \mathfrak{g}$ a generalised inverse temperature. Partition function $Z(\beta) = \int_M e^{-\langle \beta, \mu\rangle}, \omega^n / n!$;firstmoments$\bar\mu = -d\log Z / d\beta$; covariance / Fisher information $-d^2\log Z/d{\beta }^2⪰0$. Souriau Chapter IV anchor; Marsden-Ratiu §12.7 (modern reference); Marle 2014-2016 and Barbaresco 2014-2020 as modern information- geometric follow-ups. Three-tier; ~2000 words. Master section: recovery of canonical Gibbs for $G={\mathbb{R}}_t$; recovery of Jüttner relativistic Gibbs for $G=\mathrm{Poincar}\acute{\mathrm{e}}$. Souriau's headline Chapter-IV result; uniquely-Souriau framework.

Priority 2 — Souriau-distinctive depth deepenings on existing units

6. Deepen 05.04.01 moment-map. Add Souriau-originator framing in the Master section: the historical note that Souriau (1969-70) introduced the moment map independently and earlier than the Marsden-Weinstein 1974 modern formulation; the moment terminology; the connection to the cocycle (forward-reference to 05.04.07). No new unit ID; rewrite of the Master "Originator note" paragraph.

7. Deepen 05.03.01 coadjoint-orbit. Add the Souriau-originator half of the KKS triple in the Master section: every coadjoint orbit is symplectic with the KKS form, and every transitive Hamiltonian $G$-space is (a cover of) a coadjoint orbit of the extended group (forward-reference to 05.03.02). The current Cycle-7 deepening gave the KKS-form-is-symplectic half; this adds the classification half. No new unit ID.

8. 05.04.08 Equivariance obstruction and reduction in the non-equivariant case. Souriau §11; Marsden-Ratiu §12.6 modern reference. When the cocycle is non-trivial, the reduced space is the quotient by the affine action of $G$ on ${\mathfrak{g}}^{\ast }$ (shifted by the cocycle). Three-tier; ~1500 words. Cross-link to 05.04.02 symplectic-reduction.

9. 05.00.09 Poincaré group and the relativistic symplectic particle. Three-tier; ~1500 words. The relativistic-mechanics companion of 05.00.07. Mass-shell constraint as a pre-symplectic manifold; reduction to the symplectic mass-shell. Souriau Chapter III §13. Worked example: massive spinless Poincaré particle = $T^{\ast }{\mathbb{R}}^3$ with the modified symplectic form.

10. 05.00.10 Charged particle in an electromagnetic field (Souriau twisted symplectic form). Three-tier; ~1500 words. Twisted form $\omega +{\pi }^{\ast }F$ on $T^{\ast }{\mathbb{R}}^3$ for $F$ the field 2-form; Hamiltonian flow recovers the Lorentz force. Souriau §13 worked example. Cross-link to 10-em-sr/.

Priority 3 — Geometric-quantisation stub block (pointer-only, FT 3.09 Woodhouse handoff)

These items are pointer-level units that close the Souriau-Chapter-V coverage gap without duplicating the FT 3.09 Woodhouse equivalence work. Each is Master-only, ~800-1200 words, with a pointer to the upcoming Woodhouse audit for the full development.

11. 05.11.01 Prequantum line bundle and the Souriau-Kostant integrality condition. Souriau Chapter V §18; Kostant 1970 (joint originator). Statement of the integrality condition; pointer to Woodhouse FT 3.09 for the proof and the full prequantisation construction.

12. 05.11.02 Prequantisation of the spin coadjoint orbit. Souriau Chapter V §19. Half-integer spin from integrality. Pointer to Woodhouse FT 3.09 for the full quantisation.

13. 05.11.03 Bose-Fermi alternative from the spin orbit prequantisation. Souriau Chapter V §20 — the headline classical- symplectic origin of spin statistics. Pointer to QFT (FT 2.17 Weinberg) for the full spin-statistics theorem and to FT 3.09 Woodhouse for the geometric-quantisation depth.

Priority 4 — Optional pre-symplectic and Lie-algebra-cohomology stubs

These items are completist; they round out Souriau's preferred framework but are not load-bearing for FT-equivalence.

14. 05.01.06 Pre-symplectic manifold and its characteristic foliation. Souriau §1-§2 (foundational chapter). Three-tier; ~1200 words. Definition: closed 2-form, possibly with kernel; characteristic foliation = leaves tangent to $\ker \omega$; quotient by the foliation is symplectic when smooth. Cross-link to 05.10.01 contact-manifold (the odd-dim companion).

15. 03.04.05 Lie-algebra cohomology in low degrees ($H^1,H^2$, central extensions). Tucked into 03-modern-geometry/03-lie/ rather than 05-symplectic/. Three-tier; ~1500 words. Standard Chevalley-Eilenberg construction; the $H^2$-classifies-central- extensions theorem. Foundational for 05.04.07 Souriau cocycle. Cite Knapp / Weibel as anchors; Chevalley-Eilenberg 1948 as originator.

16. 11.04.03 Souriau partition function as a moment-generating function and Fisher-information geometry. Master-only deepening of item 5; the information-geometric reformulation (Barbaresco-Marle programme). ~1000 words. Optional unless the information-geometry / Riemannian-stat-mech direction is being pursued separately.

17. 11.04.04 Recovery of the canonical and relativistic Jüttner ensembles from the Souriau framework. Master-only worked-example deepening; expands the recovery argument from item 5's Master section into its own short unit. ~800 words. Optional.

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§4 Implementation sketch (P3 → P4)

Minimum Souriau-equivalence batch = priority 1 only (items 1-5): 5 new units (05.04.07, 05.03.02, 05.03.03, 05.00.07, 11.04.02). Realistic production estimate (mirroring earlier Cannas / Marsden-Weinstein / Bott-Tu batches):

• ~3-4 hours per new unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis prose). Souriau units skew higher because the originator-prose framing requires citing the 1970 Dunod text and the parallel Kostant / Kirillov / Bargmann literature.
• Priority 1 totals: 5 new × ~3.5 h = ~17-18 hours.
• Priority 1+2 totals: 5 priority-1 new + 2 deepenings (in-place rewrites) + 3 priority-2 new = ~30 hours. Fits a focused 3-4 day window with 2-3 production agents in parallel.
• Priority 1+2+3: add 3 pointer-only units = ~33 hours total.
• Priority 1-4 (full Souriau-equivalence): add 4 completist units = ~42 hours. Optional; not required for sign-off.

Originator-prose targets (per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10). Souriau is himself the originator of the modern moment-map / coadjoint-orbit / symplectic-statistical-mechanics package; the priority-1 units must carry originator-prose treatment citing:

• J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris 1970 (and the 1997 Birkhäuser English translation Structure of Dynamical Systems) — the book itself, the originator-text for items 1, 2, 5.
• A. A. Kirillov, "Unitary representations of nilpotent Lie groups," Russian Math. Surveys 17 (1962) 53-104 — independent originator of the coadjoint-orbit method on the representation-theory side (parallel originator citation for item 2).
• B. Kostant, "Quantization and unitary representations," Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics 170 (Springer 1970) 87-208 — independent originator of geometric quantisation and the prequantisation integrality condition (parallel originator citation for items 2, 11).
• V. Bargmann, "On unitary ray representations of continuous groups," Ann. Math. 59 (1954) 1-46 — parallel originator of the Galilei central extension on the quantum side (citation for item 4).
• J. E. Marsden and A. Weinstein, "Reduction of symplectic manifolds with symmetry," Rep. Math. Phys. 5 (1974) 121-130 — later systematic modern formulation, cited as the modern reference rather than as the originator (Souriau predates).
• J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Applied Mathematics 17, Springer 1999 — modern textbook reference, anchor for items 1, 5, 8.
• V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge UP 1984 — modern textbook reference for the representation-theory side, anchor for items 2, 3.
• R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings 1978 — historical companion reference.
• N. M. J. Woodhouse, Geometric Quantization, 2nd ed., Oxford 1991 — FT 3.09 forward-reference for items 11-13.

Each priority-1 unit's Master section should cite Souriau 1970 (the originator) and the modern textbook anchor (Marsden-Ratiu or Guillemin-Sternberg) explicitly.

Notation crosswalk. Souriau's notation is the largest single notation-decision the symplectic chapter has had to make:

• Moment map: Souriau writes $\mu$ (modern Codex matches). Cannas also uses $\mu$. ✓ already aligned.
• Souriau cocycle: Souriau writes $\Theta$ or $\sigma$; Marsden-Ratiu uses $\sigma$. Recommend $\sigma$ for 05.04.07.
• Generalised temperature: Souriau writes $\beta \in \mathfrak{g}$ (Lie-algebra-valued, distinct from the scalar $\beta =1/(kT)$ in 11.04.01). Recommend keeping $\beta$ in 11.04.02 but explicitly flagging the Lie-algebra-valued nature in the Notation paragraph and cross-referencing the scalar case in 11.04.01.
• Coadjoint action: Souriau writes ${\mathrm{Ad}}^{\ast }$ (modern Codex matches). ✓ already aligned.
• Elementary system: Souriau-specific terminology with no modern equivalent; introduce the term and flag the Wigner-classification analogue in 05.03.03.

Record these decisions in a short notation/souriau.md file alongside the existing notation crosswalks; mirror in the Master "Notation" paragraphs of items 1, 3, 5.

DAG edges to add. New prerequisites arrows for the priority-1+2 batch:

• 05.04.07 (Souriau cocycle) ← {03.04.05 Lie-algebra cohomology (item 15, priority 4 — block this with a TODO stub in the prerequisites field if item 15 ships late), 05.04.01 moment-map, 03.04.01 Lie-algebra (shipped)}
• 05.03.02 (KKS classification) ← {05.04.07, 05.03.01 coadjoint-orbit, 05.04.01 moment-map}
• 05.03.03 (elementary system) ← {05.03.02, 05.00.07 Galilei group / 05.00.09 Poincaré group}
• 05.00.07 (Galilei group) ← {03.03.01 Lie group, 05.04.07 for the Bargmann-cocycle framing}
• 05.00.09 (Poincaré group) ← {03.03.01, 10-em-sr/ cross-link for the SR adjacency}
• 11.04.02 (Souriau Gibbs state) ← {05.04.01 moment-map, 11.04.01 canonical-ensemble (modern Gibbs), 03.03.01 Lie group}
• 05.11.01-03 (prequantisation stubs) → forward-reference Woodhouse FT 3.09 (will be encoded once that audit ships).

Composite Souriau + Marsden-Ratiu batch recommendation. Because Marsden-Ratiu §12 (Hamiltonian actions and momentum maps) is the canonical modern statement of Souriau's Chapter II-III content, producing the Souriau priority-1+2 batch alongside the (yet-to-be- audited) Marsden-Ratiu per-book plan would yield a composite of ~10-12 units that closes both books' moment-map / cocycle / reduction gaps simultaneously. Recommended execution path once Marsden-Ratiu is audited.

Composite Souriau + Landau-Lifshitz 2.10 + Baxter 2.12 batch. The three §2 stat-physics endgame books overlap on the canonical Gibbs ensemble. The Souriau-Gibbs unit (11.04.02, item 5) is the unique Souriau-contribution; Landau-Lifshitz 2.10 and Baxter 2.12 cover the standard canonical / grand-canonical / lattice-model territory. Sequencing recommendation: produce Souriau item 5 after the Landau-Lifshitz per-book audit ships its canonical-ensemble baseline.

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§5 What this plan does NOT cover

• A line-number-level inventory of every named theorem in Souriau's ~400 pages. (Would require a copy of the book; deferred unless the priority-1+2 punch-list expands. REDUCED-PASS caveat applies.)
• The full Souriau exercise / worked-example pack. Souriau has fewer formal exercises than a modern textbook and many worked examples are embedded in chapter prose; the dedicated example-densification pass is deferred.
• The full Woodhouse FT 3.09 geometric-quantisation programme. Items 11-13 are pointer-only stubs; the Souriau-Kostant prequantisation proof, polarisations, BKS pairing, metaplectic correction, and the Borel-Weil construction are all deferred to the Woodhouse per-book plan. Souriau's Chapter V is treated at headline-statement depth only; the full development is Woodhouse territory.
• The relativistic spin-statistics theorem. Souriau Chapter V gives the classical-symplectic origin of the Bose-Fermi distinction; the QFT-level Pauli-Lüders spin-statistics theorem is Weinberg Quantum Theory of Fields Vol. 1 (FT 2.17) territory.
• Souriau's post-1970 diffeology programme (his 1980s-90s generalisation of differential geometry). Not in Structure of Dynamical Systems; defer entirely.
• The Arnold-Khesin Topological Methods in Hydrodynamics symplectic-fluid-mechanics direction. Souriau touches it in remarks; the full programme is its own book (not currently on the FT booklist).
• Information-geometry depth beyond the basic Fisher-information identity in item 5. The Barbaresco-Marle 2010s programme is a modern Souriau-descendant but its own line of research; treat as optional priority-4 (item 16) unless a dedicated information-geometry audit is opened.
• Modern moduli-space and Donaldson-Thomas applications of the moment-map framework. Cannas / McDuff-Salamon / Donaldson-Kronheimer territory; not Souriau's.

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§6 Acceptance criteria for FT equivalence (Souriau)

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

• ≥95% of Souriau's named theorems map to Codex units at Souriau-equivalent depth (currently ~45%; after priority-1 this rises to ~80%; after priority-1+2 to ~92%; after priority-3 pointer stubs to ~95%).
• ≥90% of Souriau's headline worked examples (Galilei elementary systems, Poincaré elementary systems, classical spin from $S^2$ orbit, charged particle in $F$, Souriau Gibbs ensemble for rotation / translation, $S^2$-prequantisation half-integer-spin derivation) are reproduced in some Codex unit (currently ~10%; the priority-1+2 batch brings this to ~80%).
• The notation alignment is recorded in notation/souriau.md and mirrored in the new units' Notation paragraphs.
• The DAG chain 03.04.05 → 05.04.07 → 05.03.02 → 05.03.03 is unbroken (priority-1 dependency chain).
• The DAG chain 05.04.01 → 11.04.02 (Souriau-Gibbs state depends on moment map) crosses the symplectic ↔ stat-physics chapter boundary cleanly.
• Pass-W weaving connects the new units (05.04.07-08, 05.03.02-03, 05.00.07/09/10, 11.04.02, 05.11.01-03) to the existing 05.02-hamiltonian/, 05.03-coadjoint/, 05.04-moment-reduction/, 05.10-contact/, 11-stat-mech-physics/04-partition-functions/, and 12-quantum/03-fock-spaces/ units via lateral connections.

The 5 priority-1 items close the Souriau-distinctive load-bearing content gap. The 5 priority-2 items close the deepening / equivariance- obstruction / Galilei-Poincaré-charged-particle gaps. The 3 priority-3 items close the Chapter-V Bose-Fermi gap at pointer-depth (full depth deferred to FT 3.09 Woodhouse). The 4 priority-4 items are completist; they bring originator-framing coverage from ~92% to ~95%+ but are not strictly required for sign-off.

Honest scope. Souriau's Structure of Dynamical Systems is the originator-text for the moment-map / coadjoint-orbit / symplectic- statistical-mechanics package that the modern Codex symplectic chapter already covers in Cannas-anchored modern form. The work in this plan is dominated by originator-framing additions (Souriau cocycle, KKS classification theorem, elementary-system classification, Souriau- Gibbs state) and small numbers of new units in the Galilei / Poincaré symplectic-mechanics adjacency, not by wholesale new-chapter production. The modern-symplectic and reduction infrastructure shipped via the Cannas and Marsden-Weinstein audits (Cycle 4-7) already provides ~70% topic-level coverage; the punch-list closes the remaining 30% concentrated in Souriau-distinctive Chapters III-V.

Reduced-pass caveat. This plan is produced without a local copy of the Birkhäuser 1997 PDF. The chapter listing, theorem set, and notation conventions cited above are reconstructed from canonical secondary sources (Marsden-Ratiu, Guillemin-Sternberg, Abraham-Marsden, Woodhouse) and from the existing Codex symplectic-chapter inventory. A full P1 line-number audit and an exercise / worked-example inventory pass require acquiring the book (BUY on the FT booklist). The priority-1 punch-list above is robust under any reasonable revision once a copy is in hand; the priority-3+4 items may shift slightly in scope. Re-audit at full depth when the book is acquired.

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§7 Sourcing

• Status: BUY on docs/catalogs/FASTTRACK_BOOKLIST.md line 71. Not freely available; the 1997 Birkhäuser English translation is the canonical edition for citation.
• Original. Jean-Marie Souriau, Structure des systèmes dynamiques, Dunod, Paris 1970 (in French; the original publication). Reissued by Éditions Jacques Gabay, Paris 2008.
• English translation. Structure of Dynamical Systems: A Symplectic View of Physics, translated by C.-H. Cushman-de Vries, Progress in Mathematics 149, Birkhäuser, Boston 1997, ISBN 978-0-8176-3695-1. The canonical English citation.
• Local copy. Not present in reference/fasttrack-texts/ or reference/book-collection/free-downloads/ or ~/Downloads. When acquired, file under reference/fasttrack-texts/02-quantum-stat/ as Souriau-StructureOfDynamicalSystems.pdf to mirror the pattern of other §2 stat-physics texts.
• Free secondary sources used for this reduced pass. Marsden-Ratiu Introduction to Mechanics and Symmetry (2nd ed., Springer 1999); Guillemin-Sternberg Symplectic Techniques in Physics (Cambridge UP 1984); Abraham-Marsden Foundations of Mechanics (2nd ed., Benjamin/Cummings 1978, currently free at CaltechAUTHORS); Woodhouse Geometric Quantization (2nd ed., Oxford 1991, partial free preview); Marle "From tools in symplectic and Poisson geometry to J.-M. Souriau's theories of statistical mechanics and thermodynamics" (Entropy 18 (2016) 370, free); the Souriau Wikipedia article and the Marsden archive at Caltech.
• Re-audit trigger. Re-run this plan at full P1 depth once the Birkhäuser 1997 PDF is acquired and filed under reference/fasttrack-texts/02-quantum-stat/. Expect priority-1 punch-list to remain stable; priority-3+4 items may shift.