remark 24.22 Liouville's theorem is the foundation of statistical mechanics

Rests on

• depends_on remark 24.11 The SI dimension of every object in this chapter ¶
  • depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
    • depends_on definition 13.52 Differential; pushforward ¶
      • depends_on definition 13.49 Smooth map between manifolds ¶
        • depends_on definition 13.44 Atlas ¶
        • depends_on definition 13.45 Differentiable map on a topological space ¶
        • depends_on definition 13.48 Differentiable manifold ¶
      • depends_on definition 13.81 Vector on a manifold ¶
        • depends_on definition 13.71 Differentiable curve ¶
        • depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
        • depends_on definition 13.48 Differentiable manifold ¶ ↺
    • depends_on definition 24.9 The canonical form on a cotangent bundle ¶
      • depends_on definition 13.103 Exterior derivative ¶
        • depends_on definition 13.98 $k$-form ¶
        • depends_on definition 13.102 Wedge product ¶
      • depends_on definition 24.8 Symplectic manifold ¶
        • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.98 $k$-form ¶ ↺
        • depends_on definition 24.2 Symplectic vector space ¶
    • depends_on lemma 13.107 Poincaré lemma ¶
      • depends_on definition 13.105 Closed form ¶ ↺
      • depends_on definition 13.106 Exact form ¶
        • depends_on definition 13.103 Exterior derivative ¶ ↺
        • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on equation 13.245 eq:mfd-nilpotency ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-21 ¶
    • proves proof ch:07-symplectic-geometry@proof-3 ¶
  • depends_on remark 22.5 Units ¶
    • depends_on equation 22.1 eq:ham-hamiltonian ¶
    • depends_on equation 21.43 eq:lag-generalized-momentum ¶
• depends_on theorem 24.21 Liouville ¶
  • depends_on definition 24.20 Liouville volume ¶
    • depends_on definition 13.102 Wedge product ¶ ↺
    • depends_on definition 24.8 Symplectic manifold ¶ ↺
    • depends_on theorem 24.12 Darboux ¶
      • depends_on definition 24.8 Symplectic manifold ¶ ↺
      • depends_on theorem A.68 Darboux ¶
        • depends_on definition A.67 Symplectic manifold ¶
        • depends_on equation 5.144 eq:lin-symplectic-normal-form ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
        • proves proof app:A-long-proofs@proof-57 ¶
      • proves proof ch:07-symplectic-geometry@prooflink-1 ¶
  • depends_on proposition 7.105 Clairaut–Schwarz ¶
    • depends_on definition 7.20 Continuity at a point ¶
      • depends_on definition 7.16 Limit ¶
        • depends_on definition 7.2 Absolute value ¶
        • depends_on definition 7.9 Real function ¶
      • depends_on equation 7.7 eq:ana-limit-left ¶
      • depends_on equation 7.5 eq:ana-limit-right ¶
    • depends_on theorem 7.35 Mean value theorem ¶
      • depends_on proposition 7.29 Linearity ¶
        • depends_on definition 7.11 Sum of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-13 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-18 ¶
      • proves proof ch:05-real-analysis@proof-19 ¶
    • proves proof ch:05-real-analysis@proof-64 ¶
  • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    • depends_on definition 24.14 Hamiltonian vector field ¶
      • depends_on definition 24.8 Symplectic manifold ¶ ↺
      • depends_on equation 22.4 eq:ham-pdot ¶
      • depends_on equation 22.3 eq:ham-qdot ¶
    • depends_on definition 24.8 Symplectic manifold ¶ ↺
    • depends_on proposition 13.128 Cartan's magic formula ¶
      • depends_on definition 13.103 Exterior derivative ¶ ↺
      • depends_on equation 13.246 eq:mfd-leibniz-forms ¶
      • depends_on equation 13.245 eq:mfd-nilpotency ¶ ↺
      • depends_on proposition 13.127 Component formulas ¶
        • depends_on definition 13.89 Contravariant and covariant tensors ¶
        • depends_on definition 13.90 Mixed tensor ¶
        • depends_on equation 13.267 eq:mfd-lie-def ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
    • proves proof ch:07-symplectic-geometry@proof-5 ¶
  • proves proof ch:07-symplectic-geometry@proof-8 ¶

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