definition A.67 Symplectic manifold
open in the book ·
appendices/A-long-proofs.tex:4411
· p. 2826
- ground object -- no derivation owed
Rests on
-
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 13.81 Vector on a manifold ¶
-
depends_on
definition 13.84
Tensor
¶
- depends_on definition 13.83 Covector ¶ ↺
- depends_on definition 13.81 Vector on a manifold ¶ ↺
-
depends_on
definition 13.83
Covector
¶
-
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 13.103
Exterior derivative
¶
- depends_on definition 13.98 $k$-form ¶ ↺
-
depends_on
definition 13.48
Differentiable manifold
¶
-
depends_on
definition 13.47
Differentiable structure
¶
-
depends_on
definition 13.44
Atlas
¶
- depends_on definition 13.43 Coordinate system ¶
-
depends_on
definition 13.44
Atlas
¶
-
depends_on
definition 13.47
Differentiable structure
¶
Supports
-
depends_on
theorem A.68
Darboux
¶
-
depends_on
corollary A.69
No local invariants
¶
- depends_on remark 24.13 What Darboux's theorem forbids ¶
- depends_on example A.80 The phase space of one particle in space ¶
-
depends_on
theorem 24.12
Darboux
¶
-
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 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 theorem 24.24 Poincaré recurrence ¶ ↺
-
depends_on
proposition 26.25
The Liouville measure of a second-class surface
¶
- depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺
-
depends_on
definition 24.20
Liouville volume
¶
-
depends_on
corollary A.69
No local invariants
¶
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 |
→ | Closed form | declared | appendices/A-long-proofs.tex:4422 |
depends_on |
→ | $k$-form | declared | appendices/A-long-proofs.tex:4422 |
depends_on |
→ | Differentiable manifold | declared | appendices/A-long-proofs.tex:4422 |
depends_on |
← | Darboux | declared | appendices/A-long-proofs.tex:4436 |