definition 24.17 Symplectomorphism

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:526 · p. 855

ground object -- no derivation owed

Rests on

depends_on definition A.70 Pullback ¶
- depends_on definition 7.97 Partial derivative; gradient ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
- depends_on definition 13.83 Covector ¶
  - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 13.98 $k$-form ¶
  - depends_on definition 13.83 Covector ¶ ↺
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶ ↺
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
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 13.102 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 ¶

Supports

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 proposition 24.34 Capacities exist if and only if non-squeezing holds ¶ ↺
    - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
- depends_on theorem 24.31 Gromov's non-squeezing theorem ¶ ↺
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 theorem 24.18 Symplectic form of the transformation condition ¶
- depends_on corollary 24.19 Canonical transformations preserve phase 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`	→	Pullback	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:529
`depends_on`	→	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:529
`depends_on`	←	Infinitesimal canonical transformation	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:262
`depends_on`	←	Ball and cylinder	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:960
`depends_on`	←	Momentum map	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1464
`depends_on`	←	Symplectic form of the transformation condition	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:549