remark 22.25 This is Liouville's theorem

Rests on

• depends_on corollary 22.24 Invariance of the phase-space volume ¶
  • depends_on lemma 22.20 The symplectic condition ¶
    • depends_on definition 5.135 The symplectic group ¶
      • depends_on definition 5.113 Non-degenerate form ¶
        • depends_on definition 5.110 Bilinear map ¶
      • depends_on equation 5.138 eq:lin-bilinear-expansion ¶
      • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
        • depends_on definition 5.110 Bilinear map ¶ ↺
        • depends_on definition 5.113 Non-degenerate form ¶ ↺
        • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
        • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
        • proves proof ch:03-linear-algebra-representations@proof-54 ¶
    • depends_on equation 22.18 eq:ham-generating ¶
    • depends_on proposition 7.105 Clairaut–Schwarz ¶
      • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.16 Limit ¶
        • 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 theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-64 ¶
    • proves proof ch:05-hamiltonian-mechanics@proof-7 ¶
  • depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
    • depends_on definition 5.135 The symplectic group ¶ ↺
    • depends_on lemma 5.137 The alternating top form is unique up to scale ¶
      • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
      • depends_on definition 5.110 Bilinear map ¶ ↺
      • depends_on equation 5.19 eq:lin-leibniz-det ¶
      • proves proof ch:03-linear-algebra-representations@proof-60 ¶
    • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-61 ¶
  • proves proof ch:05-hamiltonian-mechanics@proof-10 ¶
• depends_on theorem 24.21 Liouville ¶
  • depends_on definition 24.20 Liouville volume ¶
    • depends_on definition 13.102 Wedge product ¶
      • depends_on definition 13.98 $k$-form ¶
        • depends_on definition 13.83 Covector ¶
        • depends_on definition 13.84 Tensor ¶
      • depends_on equation 13.239 eq:mfd-dual-basis ¶
    • depends_on definition 24.8 Symplectic manifold ¶
      • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.103 Exterior derivative ¶
        • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on definition 24.2 Symplectic vector space ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
    • 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 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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