theorem 29.29 Poinsot's construction

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:905 · p. 986

Rests on

depends_on definition 29.19 Inertia ellipsoid ¶
- depends_on definition 29.13 Inertia tensor, continuum form ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on definition 13.5 Tensor under orthogonal transformations ¶
    - depends_on definition 13.4 Orthogonal coordinate transformation ¶
      - depends_on definition 13.3 Orthogonal coordinate system ¶
  - depends_on definition 19.42 Inertia tensor ¶
    - depends_on definition 13.5 Tensor under orthogonal transformations ¶ ↺
    - depends_on definition 19.41 Moment of inertia about an axis ¶
      - depends_on definition 19.39 Body ¶
      - depends_on definition 19.3 Mass ¶
- depends_on proposition 29.15 Positivity ¶
  - depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
  - proves proof ch:12-rigid-body-rotating-frames@proof-9 ¶
- depends_on theorem 29.16 Principal axes ¶
  - depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
  - depends_on proposition 29.15 Positivity ¶ ↺
  - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      - proves proof ch:03-linear-algebra-representations@proof-27 ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - proves proof ch:03-linear-algebra-representations@proof-31 ¶
  - proves proof ch:12-rigid-body-rotating-frames@proof-10 ¶
depends_on proposition 29.26 The two integrals of torque-free motion ¶
- depends_on equation 29.12 eq:rigid-kinetic-energy ¶
- depends_on theorem 29.25 Euler's equations ¶
  - depends_on postulate 19.22 Newton's second law, rotational case ¶
    - depends_on definition 19.18 Angular momentum ¶
      - depends_on definition 18.3 Particle ¶
      - depends_on definition 19.5 Linear momentum ¶
    - depends_on postulate 19.8 Newton's second law ¶
      - depends_on definition 19.5 Linear momentum ¶ ↺
      - depends_on postulate 19.6 Newton's first law ¶
  - depends_on theorem 29.16 Principal axes ¶ ↺
  - depends_on theorem 29.7 Transport theorem ¶
    - 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 proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
    - depends_on proposition 29.5 The velocity field of a rigid body ¶
      - depends_on definition 29.4 Angular velocity ¶ ↺
      - depends_on equation 29.1 eq:rigid-constraint ¶
      - proves proof ch:12-rigid-body-rotating-frames@proof-3 ¶
    - proves proof ch:12-rigid-body-rotating-frames@proof-5 ¶
  - proves proof ch:12-rigid-body-rotating-frames@proof-16 ¶
- proves proof ch:12-rigid-body-rotating-frames@proof-17 ¶
proves proof ch:12-rigid-body-rotating-frames@proof-19 ¶

Supports

depends_on remark 29.32 What the linearization is entitled to conclude ¶
depends_on remark 29.30 Polhode and herpolhode ¶

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
`cites`	→	Théorie nouvelle de la rotation des corps	derived	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:910
`depends_on`	→	Inertia ellipsoid	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:911
`depends_on`	→	The two integrals of torque-free motion	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:911
`depends_on`	←	What the linearization is entitled to conclude	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1047
`depends_on`	←	Polhode and herpolhode	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:957
`proves`	←	ch:12-rigid-body-rotating-frames@proof-19	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:914