definition A.561 Coadjoint action and equivariance

open in the book · appendices/A-long-proofs.tex:27062 · p. 3063

ground object -- no derivation owed

Rests on

depends_on definition 4.21 Group ¶
- depends_on definition 4.11 Inverse element ¶
  - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
- depends_on definition 4.20 Monoid ¶
  - depends_on definition 4.9 Neutral element ¶ ↺
  - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
      - depends_on definition 4.2 Binary operation ¶ ↺
depends_on definition 14.2 Lie group ¶
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 definition 13.105 Closed form ¶
      - depends_on definition 13.103 Exterior derivative ¶
      - 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 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 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on definition 24.8 Symplectic manifold ¶ ↺

Supports

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.571 Invariant Hamiltonians descend with their flows ¶
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 ¶

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`	→	Group	declared	appendices/A-long-proofs.tex:27083
`depends_on`	→	Lie group	declared	appendices/A-long-proofs.tex:27083
`depends_on`	→	Momentum map	declared	appendices/A-long-proofs.tex:27083
`depends_on`	←	The differential of the momentum map along the orbit	declared	appendices/A-long-proofs.tex:27253
`depends_on`	←	The isotropy group acts, and the quotient is smooth	declared	appendices/A-long-proofs.tex:27215
`depends_on`	←	Marsden–Weinstein reduction	declared	appendices/A-long-proofs.tex:27134