equation 22.3 eq:ham-qdot

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Supports

• depends_on definition 32.54 The kicked rotor and the standard map ¶
• depends_on definition 24.14 Hamiltonian vector field ¶
  • depends_on definition 24.43 Momentum map ¶
    • depends_on definition A.561 Coadjoint action and equivariance ¶
      • depends_on lemma A.567 The differential of the momentum map along the orbit ¶
        • depends_on theorem A.568 The radical is the isotropy orbit ¶
      • depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
        • depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      • depends_on theorem A.563 Marsden–Weinstein reduction ¶
        • depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
        • depends_on example A.572 Rotational reduction of the central-force problem ¶
    • depends_on example 24.44 Linear and angular momentum ¶
      • depends_on example A.572 Rotational reduction of the central-force problem ¶ ↺
    • depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      • depends_on proposition A.565 Freeness makes every value regular ¶
        • depends_on lemma A.567 The differential of the momentum map along the orbit ¶ ↺
        • depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶ ↺
      • depends_on theorem A.563 Marsden–Weinstein reduction ¶ ↺
      • depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
      • depends_on theorem 24.47 Marsden–Weinstein reduction ¶
        • depends_on remark 24.48 Reduction is what physicists do without saying so ¶
    • depends_on theorem 24.45 Noether, symplectic form ¶
      • depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
  • depends_on definition 24.50 Prequantum operator ¶
    • depends_on proposition 24.51 Prequantization is a Lie-algebra homomorphism ¶
      • depends_on example 24.52 The prequantum line and the Schrödinger representation ¶
    • depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
      • depends_on remark 24.55 What polarization costs ¶
  • depends_on proposition 25.4 Leibniz rule and derivations ¶
    • depends_on lemma A.616 The four identities ¶
      • depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
  • depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
    • depends_on definition 24.37 Casimir function ¶
      • depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
        • depends_on remark 24.48 Reduction is what physicists do without saying so ¶ ↺
        • depends_on remark 24.42 The intermediate axis ¶
      • depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
      • depends_on theorem 24.38 Symplectic foliation, quoted ¶
    • depends_on theorem 24.45 Noether, symplectic form ¶ ↺
  • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    • depends_on proposition A.570 Existence and uniqueness of the reduced form ¶ ↺
    • depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
    • depends_on theorem 24.21 Liouville ¶
      • depends_on example 24.29 A volume-preserving squeeze ¶
      • depends_on proposition 32.40 A Hamiltonian section preserves area ¶
      • depends_on remark 22.25 This is Liouville's theorem ¶
      • depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
      • depends_on theorem 23.31 Liouville–Arnold ¶
        • depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
        • depends_on remark 23.33 Integrability is exceptional ¶
        • depends_on remark 23.32 What that proof imports ¶
        • depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
        • depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
      • depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
        • depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
      • depends_on theorem 24.24 Poincaré recurrence ¶
        • depends_on remark 24.26 Recurrence and irreversibility ¶
        • depends_on remark 24.25 What the recurrence proof assumes about volume ¶
• depends_on proposition 22.9 Time derivative of the Hamiltonian ¶
  • depends_on theorem 22.10 Conservation of the Hamiltonian ¶
    • depends_on proposition 23.9 Separation of the time ¶
      • depends_on proposition 23.10 The characteristic function is the abbreviated action ¶
        • depends_on remark 23.11 Whose principle, and where it fails ¶
      • depends_on remark 23.12 Two functions, two roles ¶
• depends_on proposition 22.41 Symplectic form of Hamilton's equations ¶
  • depends_on remark 22.42 What the matrix form is for ¶
• depends_on theorem 22.34 Evolution of a phase-space function ¶
  • depends_on corollary 22.35 cor:ham-evolution-static ¶
    • depends_on proposition 22.37 Constants of motion ¶
  • depends_on remark 22.36 The whole dynamics in one operation ¶
  • depends_on theorem 22.38 Poisson's theorem ¶
    • depends_on proposition 25.12 The algebra of generators closes ¶
    • depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
• depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶