definition 24.8 Symplectic manifold

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:245 · p. 851

ground object -- no derivation owed

Rests on

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 13.81 Vector on a manifold ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - 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 4.33 Vector space ¶
  - 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 ¶

Supports

depends_on definition 24.9 The canonical form on a cotangent bundle ¶
- depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
  - depends_on remark 24.11 The SI dimension of every object in this chapter ¶
    - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
    - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
    - depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
      - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
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 definition 24.20 Liouville volume ¶
- depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
  - depends_on proposition A.619 The Dirac bracket is a projected Hamiltonian flow ¶
  - depends_on remark 26.26 What the factor is doing ¶
- depends_on theorem 24.21 Liouville ¶ ↺
- depends_on theorem 24.24 Poincaré recurrence ¶ ↺
depends_on definition 24.53 Real polarization and polarized sections ¶
- depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶ ↺
depends_on definition 24.49 Prequantum datum ¶
- depends_on definition 24.53 Real polarization and polarized sections ¶ ↺
- depends_on definition 24.50 Prequantum operator ¶ ↺
- depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶ ↺
depends_on definition 24.17 Symplectomorphism ¶
- depends_on definition 25.6 Infinitesimal canonical transformation ¶
  - depends_on definition 25.11 Generators of the Galilei transformations ¶
    - depends_on proposition 25.12 The algebra of generators closes ¶
- depends_on definition 24.28 Ball and cylinder ¶ ↺
- depends_on definition 24.43 Momentum map ¶ ↺
- depends_on theorem 24.18 Symplectic form of the transformation condition ¶
  - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
depends_on theorem 24.12 Darboux ¶
- depends_on definition 24.20 Liouville volume ¶ ↺
- depends_on remark 24.13 What Darboux's theorem forbids ¶
depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
depends_on theorem 24.38 Symplectic foliation, quoted ¶ ↺
depends_on theorem 24.47 Marsden–Weinstein reduction ¶ ↺

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`	→	Closed form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:255
`depends_on`	→	$k$-form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:255
`depends_on`	→	Symplectic vector space	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:255
`depends_on`	←	The canonical form on a cotangent bundle	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:269
`depends_on`	←	Hamiltonian vector field	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:404
`depends_on`	←	Liouville volume	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:653
`depends_on`	←	Real polarization and polarized sections	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1815
`depends_on`	←	Prequantum datum	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1701
`depends_on`	←	Symplectomorphism	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:529
`depends_on`	←	The level set of the momentum map is the symplectic orthogonal of the orbit	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1568
`depends_on`	←	Darboux	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:355
`depends_on`	←	The Hamiltonian flow preserves the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:499
`depends_on`	←	Symplectic foliation, quoted	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1270
`depends_on`	←	Marsden–Weinstein reduction	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1613