Landau, Lifshitz — Mechanics (Fast Track 1.02) — Audit + Gap Plan

Book: L. D. Landau, E. M. Lifshitz, Mechanics (Volume 1 of the Course of Theoretical Physics).

Russian original: Mekhanika, Nauka, Moscow, 1958.
3rd English edition: translated by J. B. Sykes and J. S. Bell, Butterworth-Heinemann, 1976; reprinted continuously through 2000. Pagination and section references in this plan are to the 3rd English edition (vii + 169 pp.).

Fast Track entry: 1.02 — the classical-mechanics anchor of Section 1 of the booklist (after 1.01 Shilov Linear Algebra, before 1.03 Classical Theory of Fields). Catalog status: BUY (under active copyright; Pergamon / Butterworth-Heinemann commercial title).

Audit completeness: full. Local PDF resolved at reference/textbooks-extra/Vol 1 - Landau, Lifshitz - Mechanics (3rd ed, 1976).pdf (7.3 MB, the canonical Sykes-Bell translation). The catalog flags 1.02 as BUY-only, but a copy was already in reference/textbooks-extra/; full TOC and key sections (Ch. I-VII) inventoried directly. WebFetch fallback was not needed.

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite). Output is a punch-list of new units and existing-unit deepenings so that Mechanics is covered to the FT-equivalence threshold (≥95% effective coverage; see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Not a full P1 audit — Landau's 251 Problems are inventoried at chapter level only.

Substantial overlap with the Arnold plan (plans/fasttrack/arnold-mathematical-methods.md). Landau Vol 1 and Arnold cover the same canonical material — least action, Lagrangian, Hamiltonian, canonical transformations, Hamilton-Jacobi, action-angle, adiabatic invariants — with opposite editorial choices (Landau: physicist-coordinates-first, examples-driven, 169 pp; Arnold: geometer-manifolds-first, theorem-driven, 500+ pp). The Arnold batch (items 1-7, 11-13) has already shipped the Codex equivalents of Landau Chapters I-II and VII. This Landau plan does not double-count those units; it itemises Landau-distinctive deepenings and the worked-example layer Landau is canonically known for.

§1 What Landau's Mechanics is for

Landau-Lifshitz Volume 1 is the originating physics text the Fast Track uses to ground all later mechanics tracks (Arnold 1.11, Cannas 1.11, Souriau 2.11, Woodhouse 3.09). Its distinctive editorial choice: derive all of classical mechanics from the principle of least action. The book opens at §1 with generalised coordinates, reaches Lagrange's equations by §5, and the entire 169-page exposition is one unbroken chain of variational consequences — Lagrangian → conservation laws → integrable problems → small oscillations → rigid body → Hamiltonian → canonical transformations → Hamilton-Jacobi → adiabatic invariants. No Newtonian setup chapter; Newton's $F = ma$ first appears at eq. (5.4) as a corollary of the Euler-Lagrange equations for a specific Lagrangian. Galilean relativity (§3) is used to constrain the Lagrangian of a free particle and derive $L = \frac{1}{2} m v^{2}$ from symmetry alone — a derivation Arnold copies but most physicists do not.

Density: ~30 named results in 169 pp., compared to Arnold's ~120 in 500+ pp. Landau is extremely terse; arguments are often one paragraph, examples are problems with brief solutions, and what Arnold spends a chapter motivating, Landau states in three sentences. Reading Landau is a multiplier on prior mechanics knowledge; reading him as a first source is famously hard.

Distinctive contributions, in roughly the order the book develops them:

Principle of least action as foundational, not derived. §2: the action $S = \int_{t_{1}}^{t_{2}} L (q, \overset{q}{˙}, t) d t$ is the defining object of mechanics; Euler-Lagrange equations $\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = 0$ are derived in two pages.
Galilean-relativity derivation of the free-particle Lagrangian. §4: $L$ for a free particle must be a function of $v^{2}$ alone (isotropy + homogeneity of space + homogeneity of time); Galilean invariance under infinitesimal velocity boost forces $L \propto v^{2}$ ; so $L = \frac{1}{2} m v^{2}$ up to an additive total derivative. The mass appears as the proportionality constant, not as a primitive.
Mechanical similarity / virial theorem. §10: if $U (α r) = α^{k} U (r)$ then for similar trajectories $t^{'} / t = (l^{'} / l)^{1 - k /2}$ ; gives Kepler's third law $T^{2} \propto a^{3}$ (with $k = - 1$ ), gives oscillator period independent of amplitude ( $k = 2$ ), gives free-fall law (constant force, $k = 1$ ). Landau-distinctive — Arnold does not isolate similarity as a technique.
The reduced-mass / one-body reduction. §13: two-body central problem reduces to one body of reduced mass $\mu = m_1 m_2 / (m_1 + m_2)$ in central field. Standard, but Landau's treatment is the physics-canonical one.
Kepler problem — full solution including the Runge-Lenz vector. §15: orbit equation $p / r = 1 + e cos φ$ , time-period via Kepler III, the additional conserved vector $A = p \times M - m α \hat{r}$ named in the §15 problems but not as "Runge-Lenz". Closure of orbits as the signature of the $1/ r$ potential.
Scattering: Rutherford's formula and small-angle scattering. §17-§20: cross-section as Jacobian, Rutherford $d σ = (α /2 m v_{\infty}^{2})^{2} sin^{- 4} (θ /2) d Ω$ , small-angle scattering, scattering in the laboratory frame (boost from CM to lab — physicist-essential, Arnold-absent).
Small oscillations — full treatment. Ch. V (§§21-30, 36 pp.): 1D harmonic, forced oscillations including resonance, damped, forced+damped, parametric resonance, anharmonic oscillations and nonlinear resonance, normal modes of coupled systems, oscillations of molecules. Landau-canonical; reproduced in virtually every physics curriculum.
Rigid-body dynamics. Ch. VI (§§31-39, 35 pp.): angular velocity, inertia tensor, principal axes, angular momentum of a rigid body, Euler's equations $I_{1} \dot{Ω}_{1} = (I_{2} - I_{3}) Ω_{2} Ω_{3}$ etc., Eulerian angles, the symmetrical top, rigid bodies in contact (rolling, slipping, no-slip constraints), motion in a non-inertial frame (Coriolis, centrifugal). Landau-canonical.
Hamilton's equations / canonical equations. Ch. VII §40: Hamiltonian $H = \sum p_{i} \overset{q}{˙}_{i} - L$ as Legendre transform, Hamilton's equations $\overset{q}{˙} = \partial H / \partial p$ , $\overset{p}{˙} = - \partial H / \partial q$ .
Routhian. §41: hybrid Lagrangian-Hamiltonian formulation eliminating cyclic coordinates. Used in physics but rarely in geometric-mechanics texts.
Poisson brackets — full algebraic treatment. §42: definition, Jacobi identity (with explicit operator-commutator proof), Poisson's theorem ( $[f, g]$ of two integrals is itself an integral), Poisson brackets of angular momentum components.
Action as a function of the coordinates; Maupertuis' principle. §43-§44: $\partial S / \partial q_{i} = p_{i}$ , $\partial S / \partial t = -H $, abb r e v ia t e d a c t i o n$ S_0 = \int p, dq$ minimised at fixed energy. Maupertuis is Landau's bridge to geometric optics and Hamilton-Jacobi.
Canonical transformations and Liouville's theorem. §45-§46: four types of generating functions $F_{1}, F_{2}, F_{3}, F_{4}$ , invariance of $\sum p_{i} d q_{i} - H d t$ , phase-volume preservation via Jacobian = 1.
Hamilton-Jacobi equation and separation of variables. §47-§48: $\partial S / \partial t + H(q, \partial S / \partial q, t) = 0 $, co m pl e t e in t e g r a l$ S(q, t; \alpha_1, \ldots, \alpha_s)$, Jacobi's $β_{i} = \partial S / \partial α_{i}$ theorem, separation in spherical, parabolic, elliptic coordinates with worked Kepler and two-centre cases.
Adiabatic invariants and canonical action variables. §49-§51: $I = \frac{1}{2 π} \oint p d q$ is invariant under slow parameter variation; accuracy of adiabatic conservation (§51) — Landau shows $Δ I$ is exponentially small in $1/ ϵ$ via contour integration around complex singularities of $q (t)$ , a Landau-canonical small-divisor-free derivation. §52 conditionally periodic motion.
Worked Problems as primary pedagogical mechanism. Roughly 251 Problems with solutions, distributed §-by-§. The Problems are not exercises in the modern sense — they are mini-chapters. Examples: Lagrangian of double pendulum (§5 Pr. 1), Kepler in a parabolic orbit, anharmonic oscillator perturbation series, Lagrange's top (§35 Pr. 1), nutation of a symmetrical top.

The book ends before relativistic mechanics (deferred to Vol. 2) and before continuum mechanics (deferred to Vol. 6-7). It also stops short of KAM, perturbation theory beyond Birkhoff, and the modern symplectic-topology programme — those are mid-20th-century and later developments, not Landau's territory.

§2 Coverage table (Codex vs Landau Vol 1)

Cross-referenced against the current Codex corpus (49 units in 05-symplectic/, including the Arnold-MM substitute batch already shipped). ✓ = covered at Landau-equivalent depth, △ = topic present but Landau-distinctive examples / framing not captured, ✗ = not covered.

Chapter I — Equations of Motion (§§1-5, 12 pp.)

Landau topic	Codex unit	Status	Note
Generalised coordinates, configuration space	(touched in `05.00.01`)	△	Coordinate-language framing not isolated; Arnold-MM substitute uses $T M$ .
Principle of least action (§2)	`05.00.02-hamilton-principle`	△	Unit exists (Arnold-MM batch); Landau's two-page derivation is the canonical short proof — anchor citation needed.
Galilean relativity, derivation of $L = \frac{1}{2} m v^{2}$ (§3-§4)	`05.00.06-galilean-newtonian-setup`	△	Unit shipped; Landau's symmetry-derives-the-Lagrangian argument needs explicit anchor.
Lagrangian for system of particles (§5)	`05.00.01-lagrangian-on-tm`	△	Coordinate-form $L = T - U = \frac{1}{2} \sum m_{a} v_{a}^{2} - U (r_{a})$ not the master statement (which uses $L : T M \to R$ ).

Chapter II — Conservation Laws (§§6-10, 12 pp.)

Landau topic	Codex unit	Status	Note
Energy conservation from time-translation (§6)	(covered in `05.00.04` Noether)	△	Noether-anchored; Landau's coordinate-language derivation is one paragraph and worth quoting in the Beginner tier.
Momentum conservation from space-translation (§7)	`05.00.04` Noether	△	Same.
Centre of mass (§8)	—	✗	Gap. Trivially derivable but not isolated as a unit.
Angular momentum (§9)	(touched in `05.00.04`)	△	Noether-rotational-symmetry. Worked formula $M = \sum r_{a} \times p_{a}$ not isolated.
Mechanical similarity / virial theorem (§10)	—	✗	Gap (Landau-distinctive). No Codex unit. Kepler III, Bertrand's theorem flavour, isolated as own technique.

Chapter III — Integration of the Equations of Motion (§§11-15, 17 pp.)

Landau topic	Codex unit	Status	Note
Motion in 1D, quadrature by energy conservation (§11)	—	✗	Gap. Foundational physics example.
Period of oscillation from potential (§12)	—	✗	Gap. Worked example of inversion technique.
Reduced mass (§13)	—	✗	Gap. Two-body to one-body reduction.
Motion in a central field (§14)	—	✗	Gap. Conservation of $M$ , effective potential $U_{eff} (r) = U + M^{2} /2 m r^{2}$ , orbit equation.
Kepler problem with Runge-Lenz (§15)	—	✗	Gap (Landau-distinctive). Worked-example anchor.

Chapter IV — Collisions Between Particles (§§16-20, 14 pp.)

Landau topic	Codex unit	Status	Note
Disintegration / decay (§16)	—	✗	Gap. Physicist-required, math-curriculum-absent.
Elastic collisions (§17)	—	✗	Gap.
Scattering cross-section (§18)	—	✗	Gap. Definition of $d σ$ , lab-frame vs CM-frame.
Rutherford formula (§19)	—	✗	Gap (Landau-distinctive worked example).
Small-angle scattering (§20)	—	✗	Gap.

Chapter V — Small Oscillations (§§21-30, 32 pp.)

Landau topic	Codex unit	Status	Note
1D harmonic oscillator (§21)	scattered in Codex examples	△	Not isolated as own unit.
Forced oscillations, resonance (§22)	—	✗	Gap.
Coupled oscillators / normal modes (§23)	—	✗	Gap. Standard physics, Codex-absent.
Molecular vibrations (§24)	—	✗	Gap. Application example.
Damped oscillations (§25)	—	✗	Gap.
Forced + damped (§26)	—	✗	Gap.
Parametric resonance (§27)	—	✗	Gap (Landau-canonical). Mathieu equation territory.
Anharmonic oscillations (§28)	—	✗	Gap. Perturbation series.
Nonlinear resonance (§29)	—	✗	Gap.
Motion in rapidly oscillating field — Kapitza pendulum (§30)	—	✗	Gap (Landau-canonical, originator-prose).

Chapter VI — Motion of a Rigid Body (§§31-39, 33 pp.)

Landau topic	Codex unit	Status	Note
Angular velocity (§31)	—	✗	Gap.
Inertia tensor (§32)	—	✗	Gap. Physicist-canonical.
Angular momentum of a rigid body (§33)	—	✗	Gap.
Equations of motion of a rigid body (§34)	—	✗	Gap. Newton-Euler equations.
Eulerian angles (§35)	—	✗	Gap. $S O (3)$ parametrisation, used in QM, robotics.
Euler's equations (§36)	(touched in `05.09.05-euler-arnold-equations`)	△	Euler-Arnold unit exists at master tier; Landau's coordinate Euler equations on $S O (3)$ not the beginner-tier focus.
The symmetrical top (§37)	—	✗	Gap (Landau-canonical worked example). Lagrange's top by quadrature.
Rigid bodies in contact / constraints (§38)	—	✗	Gap. Holonomic vs. non-holonomic.
Motion in a non-inertial frame (§39)	—	✗	Gap. Centrifugal + Coriolis force derived from Lagrangian.

Chapter VII — The Canonical Equations (§§40-52, 32 pp.)

Bulk of Codex's existing symplectic chapter lives here. Most items shipped by the Arnold-MM batch.

Landau topic	Codex unit	Status	Note
Hamilton's equations (§40)	`05.02.01-hamiltonian-vector-field`	△	Shipped (Arnold-MM); Landau's coordinate-form derivation worth a Beginner-tier anchor.
The Routhian (§41)	—	✗	Gap (Landau-distinctive). Hybrid Lagrangian-Hamiltonian for cyclic coords.
Poisson brackets (§42)	`05.02.02-poisson-bracket`	△	Shipped (Arnold-MM); Landau's operator-commutator proof of Jacobi worth anchoring.
Action as function of coordinates (§43)	(touched in `05.05.04` Hamilton-Jacobi)	△	$\partial S / \partial q = p$ , $\partial S / \partial t = - H$ ; folded into HJ unit.
Maupertuis' principle (§44)	—	✗	Gap. Abbreviated action $S_{0} = \int p d q$ , Jacobi metric; bridge to geodesics.
Canonical transformations (§45)	`05.05.03-generating-functions`	△	Shipped; Landau's $F_{1}, F_{2}, F_{3}, F_{4}$ four-types table not fully in master tier.
Liouville's theorem (§46)	`05.02.07-liouville-volume`	✓	Shipped (Arnold-MM).
Hamilton-Jacobi equation (§47)	`05.05.04-hamilton-jacobi`	△	Shipped (Arnold-MM); separation-of-variables technique (§48) not its own unit.
Separation of variables (§48)	—	✗	Gap. Worked separations (spherical, parabolic, elliptic) including the two-centre Coulomb problem.
Adiabatic invariants (§49)	`05.09.02-adiabatic-invariants`	△	Shipped (Arnold-MM); Landau's exponential-accuracy result (§51) is the Landau contribution and is not in the shipped unit.
Canonical variables (action-angle) (§50)	`05.02.04-action-angle-coordinates`	△	Shipped (Arnold-MM).
Accuracy of conservation of the adiabatic invariant (§51)	—	✗	Gap (Landau-originator). Exponential $\sim e^{- c / ϵ}$ improvement via complex-contour argument.
Conditionally periodic motion (§52)	(touched in `05.02.03` integrable + `05.02.04` action-angle)	△	Shipped (Arnold-MM).

Aggregate coverage estimate

Theorem layer: ~40% topic-level, ~30% Landau-equivalent proof-depth. The Hamiltonian half (Ch. VII §§40, 42, 45-47, 49-50) is well-shipped via Arnold-MM substitute batch. The Lagrangian-side worked-example layer (Ch. III-VI: central-field, scattering, small oscillations, rigid body) is almost entirely absent.
Exercise / worked-example layer: ~10%. Landau's 251 Problems are the canonical physics problem corpus; Codex has the templated 7-Q pack. Dedicated exercise-pack pass is P3-priority-3 follow-up.
Notation layer: ~70%. Major crosswalk concerns:
- Landau writes $L$ for Lagrangian on coordinate space (no tangent-bundle language); Codex uses $L : T M \to R$ .
- Landau writes $M$ for angular momentum; Codex / modern geometric texts write $L$ or $J$ .
- Landau writes $H$ for Hamiltonian, ${f, g}$ or $[f, g]$ for Poisson bracket (the latter in §42); Codex uses ${f, g}$ .
- Landau's action symbol is $S$ for the on-shell action and $S_{0}$ for the abbreviated action; Codex uses $S$ for both contextually.
- Russian-physics convention puts $T - U$ rather than $K - V$ ; sign conventions on canonical transformations differ from Arnold/Cannas.
Sequencing layer: ~50%. Codex orders Lagrangian → Hamiltonian → canonical → HJ; Landau orders identically except he postpones Hamiltonian to Ch. VII (after small oscillations and rigid body). Codex's ordering matches Arnold's, not Landau's.
Intuition layer: ~25%. Landau's physicist-intuition (resonance, scattering, similarity, virial) is largely absent from Codex's geometer-flavoured symplectic chapter.
Application layer: ~15%. Kepler, Rutherford, Lagrange's top, Coriolis force, normal-mode molecular vibrations — all standard physics examples, absent from Codex.

After priority-1 batch: theorem layer ~60%, application ~50%. After priority-1+2: ~80%, ~75%. Priority-3 brings to ~92%, ~88%. Full ≥95% requires a dedicated Landau-style Problem set.

§3 Gap punch-list (priority-ordered)

The Codex symplectic chapter is mature on the abstract-Hamiltonian side (Arnold-MM-batch already shipped Hamilton-Jacobi, action-angle, adiabatic, KAM, Noether). Landau exposes gaps on (a) the coordinate- physics worked-example layer (Kepler, scattering, small oscillations, rigid body, Coriolis) and (b) a handful of Landau-distinctive items absent from Arnold (mechanical similarity, Routhian, Maupertuis, exponential-accuracy adiabatic, Kapitza pendulum, parametric resonance).

Recommended slot ranges:

Extend 05.00-lagrangian-mechanics/ with worked examples and Landau-distinctive sections.
Extend 05.02-hamiltonian/ with Routhian, Maupertuis, exponential-accuracy adiabatic.
New 09.* math-physics chapter (or extend 05.00-*) for Newtonian-mechanics worked examples (Kepler, Rutherford, rigid body, small oscillations, Coriolis). Landau's most canonical territory.

Priority 1 — Landau-distinctive load-bearing additions (5 items)

05.00.08 Mechanical similarity / virial theorem. Landau §10 anchor; Clausius 1870 (virial), Bertrand 1873 (closure of orbits for $1/ r$ and $r^{2}$ potentials only). Three-tier; ~1500 words. Statement: $U(\alpha r) = \alpha^k U(r) \Rightarrow t'/t = (l'/l)^{1-k/2} $. W or k e d e x am pl es :$ k = -1$ recovers Kepler III, $k = 2$ recovers period-independent oscillator. Master section: virial theorem $⟨ T ⟩ = (k /2) ⟨ U ⟩$ for homogeneous potentials. Landau-canonical, absent from Arnold-MM batch.
05.02.10 The Routhian. Landau §41 anchor; Routh 1877 Treatise on the Dynamics of a System of Rigid Bodies originator. Three-tier; ~1400 words. Definition: $R = \sum p_\alpha \dot q_\alpha
- L$ over cyclic coordinates only, leaving the non-cyclic ones in Lagrangian form. Use case: motion of symmetric top with cyclic precession angle. Landau-distinctive; absent from Arnold-MM batch; appears in physics curricula but not in modern geometric- mechanics texts.
05.02.11 Maupertuis' principle / abbreviated action. Landau §44 anchor; Maupertuis 1744 Accord de différentes lois de la nature qui avaient jusqu'ici paru incompatibles originator, Euler 1744, Jacobi 1837. Three-tier; ~1600 words. Statement: at fixed energy $E$ , $δ S_{0} = δ \int p d q = 0$ over paths with $H = E$ . Master section: Jacobi metric $d s^{2} = 2 (E - U) a_{ik} d q^{i} d q^{k}$ ; reduction of mechanics at fixed energy to geodesic problem in conformally-rescaled metric. Bridge to 05.02.06 geodesic-flow-hamiltonian; complements Arnold-MM batch.
05.09.07 Exponential accuracy of the adiabatic invariant. Landau §51 anchor; Landau is the originator (the complex-contour argument is his signature contribution to perturbation theory). Three-tier; ~1800 words. Statement: for $H (q, p; λ (ϵ t))$ with $λ$ analytic in $t$ , $Δ I \sim e^{- c / ϵ}$ where $c$ is the imaginary part of the nearest complex singularity of $q (t)$ from the real axis. Master section: contour-integration proof; comparison with the power-series-only result $Δ I = O (ϵ^{N})$ from averaging. Originator-prose mandatory — paraphrase Landau's §51 directly. Deepens 05.09.02-adiabatic-invariants (which currently has only the leading-order result).
05.00.09 Worked Lagrangian examples — coordinate-physics layer. Landau Ch. III §§11-15 anchor. Three-tier; ~2200 words. Beginner: 1D motion via energy quadrature (§11), period from potential (§12). Intermediate: reduced-mass two-body (§13), motion in central field with effective potential (§14). Master: full Kepler problem including the Runge-Lenz vector $\mathbf{A} = \mathbf{p} \times \mathbf{M} - m\alpha \hat{\mathbf{r}}$ and orbit closure as $S O (4)$ -degeneracy signature (§15 Problems). Provides the coordinate-physics worked-example layer Codex currently lacks.

Priority 2 — Landau-canonical physics chapters (5 items)

05.00.10 Scattering and Rutherford formula. Landau Ch. IV §§16-20 anchor; Rutherford 1911 (Phil. Mag. 21) originator of the formula. Three-tier; ~2000 words. Beginner: cross-section definition as flux/density ratio. Intermediate: elastic-collision kinematics in CM and lab frames; small-angle scattering. Master: Rutherford formula $d σ = (α /4 E)^{2} sin^{- 4} (θ /2) d Ω$ from hyperbolic orbits in central field, with CM-to-lab boost. First collision-theory unit in Codex; downstream prereq for 02.* Section-2 QM scattering.
05.00.11 Small oscillations and normal modes. Landau Ch. V §§21-23 anchor + §27 parametric resonance. Three-tier; ~2400 words. Beginner: 1D harmonic + forced + damped. Intermediate: coupled oscillators, normal-mode decomposition via eigenvalue problem $(ω^{2} m_{ij} - k_{ij}) A_{j} = 0$ . Master: parametric resonance from Mathieu equation $\overset{x}{¨} + ω^{2} (1 + h cos 2 ω_{0} t) x = 0$ , instability tongues. Foundational across physics; Codex-absent.
05.00.12 Kapitza pendulum / motion in rapidly oscillating field. Landau §30 anchor; Kapitza 1951 (Sov. Phys. Usp.) originator. Three-tier; ~1500 words. Master section: effective-potential method, $U_\text{eff} = U + \langle f^2 \rangle / 4 m \omega^2$, inverted-pendulum stabilisation. Landau-Kapitza originator-prose mandatory; canonical physics-pedagogy example absent from Arnold and Codex.
05.00.13 Rigid-body dynamics — inertia tensor and Eulerian angles. Landau Ch. VI §§31-35 anchor; Euler 1765 Theoria motus corporum solidorum originator. Three-tier; ~2200 words. Beginner: angular velocity vector $Ω$ , inertia tensor $I_{ik} = \sum m (r^{2} δ_{ik} - x_{i} x_{k})$ . Intermediate: principal axes, angular momentum $M_{i} = I_{ik} Ω_{k}$ , equations of motion. Master: Eulerian angles $(φ, θ, ψ)$ as $S O (3)$ chart, symmetrical top by quadrature. Complements 05.09.05 Euler-Arnold (which is geometric / coadjoint-orbit framing) with the coordinate-physics counterpart.
05.00.14 Motion in a non-inertial frame / Coriolis force. Landau §39 anchor; Coriolis 1835 (J. École Polytech.) originator. Three-tier; ~1400 words. Beginner: rotating-frame setup, pseudo-forces. Intermediate: Lagrangian-derivation of centrifugal + Coriolis terms; energy is not conserved in rotating frame, but $E_0 = E - \boldsymbol{\Omega} \cdot \mathbf{M}$ is. Master: Foucault pendulum, weather-system application. Universal physics topic; bridge to 1.03 classical fields and 2.* non-inertial QM.

Priority 3 — depth on existing units, examples, applications (4 items)

Deepen 05.00.02-hamilton-principle. Add a Master-tier section anchored to Landau §2: the two-page canonical derivation of Euler-Lagrange from $δ S = 0$ , with explicit endpoint- fixity and integration-by-parts. Currently the unit cites Arnold; add Landau as joint originator-anchor. No new ID.
Deepen 05.00.06-galilean-newtonian-setup. Add Master-tier section anchored to Landau §3-§4: derive $L = \frac{1}{2} m v^{2}$ from Galilean invariance + homogeneity + isotropy. Landau- distinctive symmetry-derives-the-Lagrangian argument absent from Arnold. No new ID.
Deepen 05.02.02-poisson-bracket. Add Master-tier proof of Jacobi identity via Landau's §42 operator-commutator argument ( $D_{1} D_{2} - D_{2} D_{1}$ is a first-order operator) — the cleanest elementary proof. No new ID.
Deepen 05.05.03-generating-functions. Tabulate the four types $F_{1} (q, Q)$ , $F_{2} (q, P)$ , $F_{3} (p, Q)$ , $F_{4} (p, P)$ in the master tier per Landau §45 (eqs. (45.7)-(45.8)) — physicist- canonical reference table. Already on Arnold plan priority-3 item 15; no double-count.

Priority 4 — survey / advanced (1 item)

05.00.15 Separation of variables in Hamilton-Jacobi — elliptic and parabolic. Landau §48 anchor; Jacobi 1837 / Stäckel 1891 originator. Master-only; ~1600 words. Worked separations: central field in spherical, two-centre Coulomb in elliptic, parabolic-cylinder Kepler. Bridge to integrable-systems classics; deferable unless curriculum expands into integrable systems beyond Liouville-Arnold.

§4 Implementation sketch

Minimum Landau-equivalence batch = priority 1 only (items 1-5): 4 new units + 1 deepening. Estimates:

~3 hours per typical new unit; item 5 (worked Lagrangian examples) is ~4.5 h given the worked-example density. Item 4 (exponential- accuracy adiabatic) is ~4 h for originator research.
Priority 1: 4 typical × 3 h + 1 worked-example × 4.5 h + 1 originator-prose × 4 h = ~20 hours. Effectively the same envelope as Arnold priority-1.
Priority 1+2: ~20 + 5 new × 3.5 h = ~38 hours.
Priority 1+2+3 (excl. duplicate items 14): ~38 + 3 deepenings × 1.5 h = ~43 hours.

Originator-prose targets. Landau is partly originator — exponential adiabatic accuracy (item 4) is the Landau contribution. Otherwise he is a synthesizer in the Lagrange-Hamilton-Jacobi-Poisson- Noether tradition. Joint originator-citations:

Item 1 (similarity / virial): Clausius 1870 (virial); Bertrand 1873 (closure of orbits).
Item 2 (Routhian): Routh 1877.
Item 3 (Maupertuis): Maupertuis 1744 / Euler 1744 / Jacobi 1837.
Item 4 (exponential adiabatic): Landau §51 — originator, mandatory originator-prose.
Item 5 (Kepler / Runge-Lenz): Newton 1687 (Kepler problem); Runge 1919 / Lenz 1924 (the vector).
Item 6 (Rutherford): Rutherford 1911.
Item 7 (parametric resonance): Mathieu 1868; Faraday 1831 (parametric instability of pendula).
Item 8 (Kapitza pendulum): Kapitza 1951 — originator, mandatory originator-prose.
Item 9 (rigid body): Euler 1765.
Item 10 (Coriolis): Coriolis 1835.

Notation crosswalk concerns (Landau-specific). Recorded inline in the Master sections of each new unit:

Lagrangian: Landau writes $L (q, \overset{q}{˙}, t)$ as a function on coordinate patches; Codex Master-tier writes $L : T M \to R$ . Pin the equivalence $L_\text{Landau}(q, \dot q, t) = L_\text{Codex}|_{\text{chart}}(q, v, t)$.
Angular momentum symbol: Landau $M$ , Codex $J$ / $L$ . Resolution: keep $M$ in Landau-anchored Master sections, write $L$ in coordinate-free statements.
Sign convention on Hamiltonian: Landau uses $H = \sum p \overset{q}{˙} - L$ (eq. 40.2); Arnold and Cannas agree. No conflict.
Sign convention on symplectic form: Landau does not introduce $ω$ explicitly. Codex's choice $\omega = dp \wedge dq = -d(p, dq)$ matches Arnold; Landau's $\sum d p_{i} \land d q_{i}$ (implicit in §45) is the same.
Russian physicist conventions: Landau writes $T = \frac{1}{2} a_{ik} (q) \overset{q}{˙}_{i} \overset{q}{˙}_{k}$ (kinetic-energy metric); modern math-physics writes $\frac{1}{2}g_{ij}\dot q^i \dot q^j $. P in e q u i v a l e n ce :$ a_{ik} = g_{ik}$ for the configuration metric.
Cross-volume notation alert: Landau Vol 1's $S$ for action reappears in Vol 2 as the relativistic action $S = - m c \int d s$ . When citing across volumes, disambiguate. (Plan applies to 1.03 audit also.)

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

05.00.08 (similarity) ← 05.00.02 (Hamilton's principle)
05.02.10 (Routhian) ← {05.02.01, 05.00.01}
05.02.11 (Maupertuis) ← {05.02.05 cotangent-bundle, 05.05.04 Hamilton-Jacobi}; → 05.02.06 geodesic-flow
05.09.07 (exponential adiabatic) ← {05.09.02 adiabatic, 06.01.13 argument-principle / contour integration}
05.00.09 (Lagrangian worked examples) ← 05.00.01-05 Lagrangian block
05.00.10 (scattering) ← 05.00.09 (central field)
05.00.11 (small oscillations) ← 05.00.02
05.00.12 (Kapitza) ← 05.00.11 (parametric resonance)
05.00.13 (rigid body) ← {05.00.01, 03.03.02 group-action, 05.09.05 Euler-Arnold}
05.00.14 (Coriolis) ← 05.00.13

Chapter structure. No new chapter folder needed. Extensions:

Extend 05.00-lagrangian-mechanics/: items 1, 5, 6, 7, 8, 9, 10 (plus deepenings 11, 12).
Extend 05.02-hamiltonian/: items 2, 3 (plus deepening 13, 14).
Extend 05.09-perturbation/ (alias integrable/): item 4.
Optional priority-4 item 15 either extends 05.05-lagrangian/ or opens a Master-tier sub-chapter on separation-of-variables.

Composite Arnold + Landau batch. Most efficient production order: because the Arnold-MM substitute batch (priority-1+2 of arnold-mathematical-methods.md) has already shipped 13 of the 17 core units that Landau and Arnold share, this Landau batch is purely Landau-distinctive additions. No re-production of Hamilton's equations, Poisson brackets, action-angle, KAM, Liouville, Hamilton- Jacobi. The Landau plan adds:

4 new Landau-distinctive Hamiltonian-side units (items 1, 2, 3, 4),
6 new physics-canonical Lagrangian/Newtonian-side worked-example units (items 5, 6, 7, 8, 9, 10) — these are the bulk of the shipped Landau pedagogy and have no Arnold-MM substitute because Arnold dispatches them as Problems in 2 pages.

Apex unit designation. Item 4 (05.09.07 exponential-accuracy adiabatic) is a near-apex unit — originator-prose, requires complex- analysis contour-integration argument that bridges to 06-riemann-surfaces/, paraphrases Landau §51 directly. Not as research-heavy as Arnold-KAM but warrants extended budget.

§5 What this plan does NOT cover

Relativistic mechanics. Deferred to Landau Vol 2 Classical Theory of Fields (FT 1.03); separate plan.
Continuum / fluid mechanics. Deferred to Vol 6 Fluid Mechanics (not on FT) and Arnold-Khesin (FT 1.12).
Statistical mechanics. Deferred to Vol 5 / Souriau (FT 2.10, 2.11).
Quantum mechanics. Deferred to Vol 3 (FT 2.01).
Line-number Problem inventory. Landau's ~251 Problems form a separate per-section problem corpus. Inventory deferred unless the punch-list expands into a dedicated Landau-exercise-pack pass (analogous to the proposed Arnold-exercise-pack — would be its own ~20-hour pass).
The remaining ~30% of Landau worked Problems outside priority-1 items: nutation of symmetrical top, Lagrange points in restricted three-body, oscillations of polyatomic molecules, anharmonic perturbation series, motion of charged particle in EM field as Lagrangian, motion of relativistic particle (deferred to Vol 2).
Goldstein / Marion-Thornton parallel-track coverage. Goldstein is cited as a peer source but not separately audited; if a future decision adds Goldstein to FT, dedicated plan. José-Saletan Classical Dynamics same.
Russian-language original. Translation differences (Sykes-Bell vs. Russian) not catalogued; physicist-conventions track the English edition.

§6 Acceptance criteria for FT equivalence (Landau Vol 1)

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

≥95% of Landau's named results (and worked formulae) in Chapters I-VII map to Codex units at Landau-equivalent pedagogical depth (currently ~40%; after priority-1 ~60%; after priority-1+2 ~80%; after priority-1+2+3 ~92%; after priority-4 ~95%). Items 4 (exponential adiabatic), 8 (Kapitza), and 9 (rigid body) are the highest-leverage individual gaps.
≥80% of Landau's Problems have a Codex equivalent worked example (currently ~10%; closing requires the Landau-exercise-pack pass per §5).
≥90% of Landau's worked examples appear in a unit (currently ~15%; priority-1+2 brings to ~70%; remainder needs dedicated worked- example densification).
Notation crosswalk pinned in Master sections per §4. No standalone notation/landau.md needed.
DAG: for every Lagrangian → conservation-law → integrable chain in Landau Chs. I-III (least action → Noether → reduced mass → central field → Kepler), an unbroken prerequisite chain in Codex's DAG. Currently breaks at 05.00.09 (worked Lagrangian examples) — item 5 closes it.
Pass-W weaving connects new units to existing 05.00-*, 05.02-*, 05.05-*, 05.09-* and laterally to 06.01.13 (contour integration, for item 4) and 03.03.* (group actions, for items 9-10).

The 4 priority-1 new units (items 1-4) + 1 worked-example unit (item 5) close the Landau-distinctive gap given the Arnold-MM batch is shipped. Priority-2 closes the physics-canonical worked-example layer (items 6-10). Priority-3+4 are deepenings.

Prerequisite alert. Landau Vol 1 sits at FT 1.02 — after the Apostol Vols I (single-variable calculus) and II (multivariable + linear algebra) and Lang Basic Mathematics (FT 0.1, 0.2, 0.3). Codex must keep those prerequisite chains intact: Apostol Vol II calculus-of- variations (Ch. 7) and linear algebra (Ch. 3-5) are load-bearing for items 5, 7, 9, 11. Audit plans tom-m-apostol-calculus-vol-single.md and tom-m-apostol-calculus-vol-multi.md should be checked for any incomplete-coverage gaps that would block Landau units; this Landau plan does not re-audit Apostol.

Honest scope. Mid-sized equivalence gap. The Codex symplectic chapter is mature on the Arnold-geometric side (49 units shipped, of which ~13 already cite Landau-Arnold-equivalent material), but the Landau-canonical physics-pedagogy layer (worked examples, scattering, oscillations, rigid body, Coriolis, similarity, virial, Kapitza, Routhian, Maupertuis, exponential-accuracy adiabatic) is almost entirely uncaptured. Priority-1+2 batch is 10 new units + 4 deepenings — comparable to Arnold priority-1+2 in scope but with a different center of mass (Arnold added KAM and Lagrangian-on- $T M$ infrastructure; Landau adds the physicist worked-example corpus).

§7 Sourcing

Status: BUY per docs/catalogs/FASTTRACK_BOOKLIST.md.
Local copy resolved: reference/textbooks-extra/Vol 1 - Landau, Lifshitz - Mechanics (3rd ed, 1976).pdf (7.3 MB; also .djvu). Full text accessed for this audit.
Citation form: L. D. Landau, E. M. Lifshitz, Mechanics, 3rd ed., translated from the Russian by J. B. Sykes and J. S. Bell, Course of Theoretical Physics Volume 1 (Butterworth-Heinemann, Oxford, 1976; reprinted 2000). ISBN 0 7506 2896 0.
Russian original: Mekhanika (Mekhanika), Nauka, Moscow, 1958 (and 1965 / 1973 revisions). Pre-1989 Soviet text — copyright questions are non-trivial; English Pergamon / Butterworth edition remains commercial.
Peer sources cited in this audit (peer-anchor for §1):
- H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, 2002) — physicist-coordinate companion; canonical American counterpart to Landau.
- J. Marion, S. Thornton, Classical Dynamics of Particles and Systems, 5th ed. (Brooks/Cole, 2003) — undergraduate parallel.
- V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer GTM 60, 2nd ed. (1989) — geometric counterpart, audit plan at plans/fasttrack/arnold-mathematical-methods.md.
- J. V. José, E. J. Saletan, Classical Dynamics: A Contemporary Approach (Cambridge UP, 1998) — hybrid coordinate-geometric text; bridge between Landau and Arnold.
Additional originator references (cited in §3 punch-list): Lagrange 1788 Mécanique analytique; Hamilton 1834-35 (Phil. Trans. Roy. Soc. Lond. 124, 125); Jacobi 1843 Vorlesungen über Dynamik; Poisson 1809 (J. École Polytech. 8); Noether 1918 (Nachr. König. Ges. Wiss. Göttingen); Rutherford 1911 (Phil. Mag. 21); Kapitza 1951 (Sov. Phys. Usp.); Coriolis 1835 (J. École Polytech.); Euler 1765 Theoria motus corporum solidorum; Maupertuis 1744; Clausius 1870; Bertrand 1873; Routh 1877; Mathieu 1868.
Local copy retention. Keep at reference/textbooks-extra/ (the existing path); not promoted into reference/fasttrack-texts/01-fundamentals/ because the fasttrack-texts directory is for free-license / public-domain texts only. Landau remains commercial-copyright; the personal copy stays in the textbooks-extra/ parking area.

Landau, Lifshitz — *Mechanics* (Fast Track 1.02) — Audit + Gap Plan