proposition 29.11 Angular velocity in Euler angles

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:283 · p. 980

Rests on

depends_on definition 29.4 Angular velocity ¶
- depends_on definition 14.26 The rotation group ¶
- depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
  - depends_on equation 13.223 eq:mfd-epsilon-density ¶
  - depends_on example 13.94 The two densities this chapter uses ¶
    - depends_on definition 13.92 Tensor density ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on proposition 13.93 Algebra of densities ¶
      - depends_on definition 13.92 Tensor density ¶ ↺
      - proves proof ch:11-manifolds-tensors-curvature@proof-17 ¶
  - depends_on notation 14.1 not:lie-indices ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-29 ¶
- depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
  - depends_on definition 14.26 The rotation group ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
depends_on notation 29.9 The $z$–$x$–$z$ Euler angles used in this treatise ¶
- depends_on definition 14.26 The rotation group ¶ ↺
- depends_on theorem 29.3 Euler's rotation theorem ¶
  - depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
    - depends_on equation 14.28 eq:lie-expso2 ¶
    - depends_on equation 14.25 eq:lie-so2-matrix ¶
    - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
  - depends_on proposition 29.2 Six degrees of freedom ¶
    - depends_on definition 21.9 Degrees of freedom ¶
    - depends_on definition 29.1 Rigid body ¶
      - depends_on definition 21.2 Holonomic constraint ¶
      - depends_on definition 19.39 Body ¶
    - proves proof ch:12-rigid-body-rotating-frames@proof-1 ¶
  - proves proof ch:12-rigid-body-rotating-frames@proof-2 ¶
proves proof ch:12-rigid-body-rotating-frames@proof-7 ¶

Supports

depends_on definition 29.35 The Lagrange top ¶
depends_on remark 29.12 The coordinate singularity, and what it is not ¶

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`	→	Angular velocity	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:295
`depends_on`	→	The $z$–$x$–$z$ Euler angles used in this treatise	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:295
`depends_on`	←	The Lagrange top	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1248
`depends_on`	←	The coordinate singularity, and what it is not	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:345
`proves`	←	ch:12-rigid-body-rotating-frames@proof-7	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:298