proposition 29.26 The two integrals of torque-free motion

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:781 · p. 985

Rests on

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 definition 18.10 Velocity ¶
      - depends_on definition 19.3 Mass ¶
  - 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 definition 18.8 Inertial frame of reference ¶
      - depends_on definition 19.5 Linear momentum ¶ ↺
- depends_on theorem 29.16 Principal axes ¶
  - 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 definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - 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 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 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 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 equation 13.223 eq:mfd-epsilon-density ¶
      - depends_on example 13.94 The two densities this chapter uses ¶
      - 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 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 ¶

Supports

depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
depends_on proposition 29.43 The three integrable heavy tops ¶
- depends_on remark 29.45 What integrability buys, and where it stops ¶
- depends_on remark 29.44 The Kovalevskaya integral ¶
depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
depends_on remark 29.30 Polhode and herpolhode ¶
depends_on remark 24.42 The intermediate axis ¶
depends_on theorem 29.29 Poinsot's construction ¶
- depends_on remark 29.32 What the linearization is entitled to conclude ¶
- depends_on remark 29.30 Polhode and herpolhode ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:rigid-kinetic-energy	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:794
`depends_on`	→	Euler's equations	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:794
`depends_on`	←	Dissipation drives a free body to its largest moment	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1062
`depends_on`	←	The three integrable heavy tops	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1591
`depends_on`	←	Euler's equations as a Lie–Poisson system	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:828
`depends_on`	←	Polhode and herpolhode	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:957
`depends_on`	←	The intermediate axis	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1448
`depends_on`	←	Poinsot's construction	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:911
`proves`	←	ch:12-rigid-body-rotating-frames@proof-17	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:797