theorem 24.18 Symplectic form of the transformation condition

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:532 · p. 855

Rests on

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 7.16 Limit ¶
      - 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 5.14 Linear independence ¶
  - 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 13.45 Differentiable map on a topological space ¶
      - 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 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 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
      - proves proof ch:03-linear-algebra-representations@proof-54 ¶
depends_on equation 22.42 eq:ham-poisson-matrix ¶
depends_on equation 22.57 eq:ham-symplectic-eom ¶
proves proof ch:07-symplectic-geometry@proof-6 ¶

Supports

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`	→	Symplectomorphism	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:549
`depends_on`	→	eq:ham-poisson-matrix	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:549
`depends_on`	→	eq:ham-symplectic-eom	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:549
`depends_on`	←	Canonical transformations preserve phase volume	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:622
`proves`	←	ch:07-symplectic-geometry@proof-6	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:553