theorem 32.47 Poincaré's non-integrability, quoted

Rests on

• depends_on remark 32.44 Topological entropy, and where horseshoes come from ¶
• depends_on theorem 23.31 Liouville–Arnold ¶
  • depends_on equation 22.41 eq:ham-poisson-bracket ¶
  • depends_on theorem 23.30 Motion in action–angle variables ¶
    • depends_on definition 23.28 Action variables ¶
      • depends_on definition 16.99 Abbreviated action ¶
        • depends_on definition 16.96 Action; Hamilton's principle ¶
        • depends_on equation 16.86 eq:calcvar-hamilton-action ¶
      • depends_on definition 23.13 Additive separability ¶
        • depends_on definition 23.5 Complete integral ¶
        • depends_on equation 23.15 eq:hj-characteristic-equation ¶
    • depends_on proposition 22.18 The mixed generating functions ¶
      • depends_on definition 22.15 Canonical transformation ¶
        • depends_on definition 22.14 Phase-space coordinate transformation ¶
        • depends_on definition 21.45 Action ¶
        • depends_on theorem 22.6 Hamilton's equations ¶
      • depends_on equation 22.18 eq:ham-generating ¶
      • proves proof ch:05-hamiltonian-mechanics@proof-6 ¶
    • depends_on theorem 22.6 Hamilton's equations ¶ ↺
    • proves proof ch:06-hamilton-jacobi@proof-8 ¶
  • 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 equation 13.239 eq:mfd-dual-basis ¶
      • depends_on definition 24.8 Symplectic manifold ¶
        • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.98 $k$-form ¶ ↺
        • depends_on definition 24.2 Symplectic vector space ¶
      • depends_on theorem 24.12 Darboux ¶
        • depends_on definition 24.8 Symplectic manifold ¶ ↺
        • depends_on theorem A.68 Darboux ¶
        • 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 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 ¶
    • 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 ¶
        • 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 ¶
  • proves proof ch:06-hamilton-jacobi@prooflink-1 ¶

Supports

Nothing declares a dependency on this node yet.