proposition 14.27 Generators of $\mathfrak{so}(3)$

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1046 · p. 553

Rests on

depends_on definition 14.26 The rotation group ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶

Supports

depends_on definition 14.28 Hermitian rotation generators ¶
depends_on definition 29.4 Angular velocity ¶
- depends_on proposition 29.11 Angular velocity in Euler angles ¶
  - depends_on definition 29.35 The Lagrange top ¶
  - depends_on remark 29.12 The coordinate singularity, and what it is not ¶
- depends_on proposition 29.5 The velocity field of a rigid body ¶
  - depends_on proposition 29.14 Angular momentum and kinetic energy ¶
  - depends_on proposition 29.8 Chasles: the general displacement is a screw ¶
  - depends_on proposition 29.6 The angular velocity does not depend on the base point ¶
  - depends_on theorem 29.7 Transport theorem ¶
    - depends_on proposition 29.47 The gyrocompass ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
    - depends_on theorem 29.51 Equation of motion in a rotating frame ¶
      - depends_on phenomenon 42.4 The lunar orbit is not polarized toward the Sun ¶
- depends_on theorem 29.7 Transport theorem ¶ ↺
depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
- depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
  - depends_on remark 29.12 The coordinate singularity, and what it is not ¶ ↺
- depends_on theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$ ¶
  - depends_on corollary 14.38 $\SU(2)$ is the universal cover ¶
    - depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶ ↺
    - depends_on proposition 14.45 Which representations descend to $\SO(3,\R)$ ¶
  - depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
- depends_on theorem 29.3 Euler's rotation theorem ¶
  - depends_on notation 29.9 The $z$–$x$–$z$ Euler angles used in this treatise ¶
    - depends_on proposition 29.11 Angular velocity in Euler angles ¶ ↺
    - depends_on remark 29.10 Other conventions, and why the choice must be stated ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on remark 125.3 What the theorem does and does not forbid ¶
  - depends_on proposition 29.8 Chasles: the general displacement is a screw ¶ ↺
depends_on remark 18.30 The angular velocity is an element of $\mathfrak{so}(3)$ ¶

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`	→	The rotation group	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1066
`depends_on`	←	Hermitian rotation generators	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1117
`depends_on`	←	Angular velocity	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:114
`depends_on`	←	Rodrigues formula; the exponential map is onto	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1153
`depends_on`	←	The angular velocity is an element of $\mathfrak{so}(3)$	declared	parts/03-classical-mechanics/01-kinematics.tex:901
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-8	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1069