proposition 24.3 Even dimension and the canonical basis
open in the book ·
parts/03-classical-mechanics/07-symplectic-geometry.tex:98
· p. 850
Rests on
-
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 4.33 Vector space ¶
-
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
definition 5.110
Bilinear map
¶
-
depends_on
proposition 5.119
Normal form of a non-degenerate antisymmetric form
¶
- depends_on equation 22.56 eq:ham-symplectic-matrix ¶
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- proves proof ch:07-symplectic-geometry@proof-1 ¶
Supports
-
depends_on
definition 24.4
Symplectic group
¶
-
depends_on
proposition 24.5
Properties of the symplectic group
¶
- depends_on remark 24.6 Connectedness, and what it rests on ¶
- depends_on remark 24.7 Contrast with the orthogonal group ¶
-
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
proposition 24.5
Properties of the symplectic group
¶
-
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 ¶
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 vector space | declared | parts/03-classical-mechanics/07-symplectic-geometry.tex:112 |
depends_on |
→ | eq:ham-symplectic-matrix | declared | parts/03-classical-mechanics/07-symplectic-geometry.tex:112 |
depends_on |
→ | Normal form of a non-degenerate antisymmetric form | declared | parts/03-classical-mechanics/07-symplectic-geometry.tex:112 |
depends_on |
← | Symplectic group | declared | parts/03-classical-mechanics/07-symplectic-geometry.tex:155 |
depends_on |
← | The Liouville measure of a second-class surface | declared | parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:876 |
proves |
← | ch:07-symplectic-geometry@proof-1 | declared | parts/03-classical-mechanics/07-symplectic-geometry.tex:116 |