definition 24.14 Hamiltonian vector field

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:391 · p. 853

ground object -- no derivation owed

Rests on

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 13.83 Covector ¶
      - depends_on definition 13.84 Tensor ¶
    - 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 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 definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-54 ¶
depends_on equation 22.4 eq:ham-pdot ¶
depends_on equation 22.3 eq:ham-qdot ¶

Supports

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.570 Existence and uniqueness of the reduced form ¶
  - 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 example A.573 The abelian case, and eliminating a cyclic coordinate ¶ ↺
- 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 24.57 The half-integer, and the honest status of the construction ¶
    - 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 phenomenon 32.58 Regular and chaotic motion coexist ¶
      - depends_on phenomenon 32.59 The solar system is chaotic ¶
      - depends_on remark 23.33 Integrability is exceptional ¶ ↺
    - 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 ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:404
`depends_on`	→	eq:ham-pdot	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:404
`depends_on`	→	eq:ham-qdot	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:404
`depends_on`	←	Momentum map	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1464
`depends_on`	←	Prequantum operator	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1713
`depends_on`	←	Leibniz rule and derivations	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:178
`depends_on`	←	Poisson bracket from the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:442
`depends_on`	←	The Hamiltonian flow preserves the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:499