definition 24.2 Symplectic vector space

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:88 · p. 849

ground object -- no derivation owed

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 24.8 Symplectic manifold ¶
- 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 24.14 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 proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - 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 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 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 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 proposition A.571 Invariant Hamiltonians descend with their flows ¶ ↺
    - 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 theorem 25.57 The Moyal equation, and its classical limit ¶
      - depends_on theorem 24.24 Poincaré recurrence ¶
- 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 remark 24.57 The half-integer, and the honest status of the construction ¶
- 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 24.33 Symplectic capacity ¶
      - depends_on proposition 24.34 Capacities exist if and only if non-squeezing holds ¶
    - depends_on example 24.29 A volume-preserving squeeze ¶ ↺
    - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶ ↺
  - 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 ¶ ↺
depends_on lemma A.564 Dimension and double orthogonal ¶
- depends_on proposition A.565 Freeness makes every value regular ¶ ↺
- depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
depends_on proposition 24.3 Even dimension and the canonical basis ¶
- depends_on definition 24.4 Symplectic group ¶
  - depends_on proposition 24.5 Properties of the symplectic group ¶
    - depends_on remark 24.6 Connectedness, and what it rests on ¶
    - depends_on remark 24.7 Contrast with the orthogonal group ¶
  - depends_on proposition 24.30 Linear non-squeezing ¶ ↺
- depends_on proposition 26.25 The Liouville measure of a second-class surface ¶ ↺

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`	→	Normal form of a non-degenerate antisymmetric form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:95
`depends_on`	←	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:255
`depends_on`	←	Dimension and double orthogonal	declared	appendices/A-long-proofs.tex:27153
`depends_on`	←	Even dimension and the canonical basis	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:112