theorem 24.47 Marsden–Weinstein reduction

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1600 · p. 867

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 lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
- depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on definition 24.43 Momentum map ¶
  - depends_on definition 14.2 Lie group ¶
  - depends_on definition 24.14 Hamiltonian vector field ¶
    - depends_on definition 24.8 Symplectic manifold ¶ ↺
    - depends_on equation 22.4 eq:ham-pdot ¶
    - depends_on equation 22.3 eq:ham-qdot ¶
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition A.70 Pullback ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on theorem 13.59 Regular value theorem ¶
  - depends_on definition 13.52 Differential; pushforward ¶
    - depends_on definition 13.49 Smooth map between manifolds ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.58 Hypersurface ¶
    - depends_on definition 13.54 Embedded submanifold ¶
      - depends_on definition 13.53 Immersion, submersion, embedding ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.49 Smooth map between manifolds ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶
- proves proof ch:07-symplectic-geometry@proof-15 ¶
depends_on theorem 13.59 Regular value theorem ¶ ↺
proves proof ch:07-symplectic-geometry@prooflink-2 ¶

Supports

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 ¶

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:1613
`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:1613
`depends_on`	→	Regular value theorem	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1613
`depends_on`	←	Reduction is what physicists do without saying so	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1655
`proves`	←	ch:07-symplectic-geometry@prooflink-2	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1616