theorem 29.25 Euler's equations

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:736 · p. 984

Rests on

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 7.26 Derivative of a function at a point ¶
      - depends_on definition 18.9 Position ¶
      - depends_on definition 13.9 Tangent vector to a curve ¶
    - depends_on definition 19.3 Mass ¶
      - depends_on definition 18.3 Particle ¶ ↺
      - depends_on postulate 19.1 Inertia ¶
- 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 18.7 Frame of reference ¶
      - depends_on definition 18.6 Free particle ¶
    - 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 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 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 8.18 Liouville ¶
      - proves proof ch:06-complex-analysis@proof-13 ¶
    - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - proves proof ch:03-linear-algebra-representations@proof-26 ¶
    - 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 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - 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 definition 13.92 Tensor density ¶
      - depends_on proposition 13.93 Algebra of densities ¶
    - 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 ¶

Supports

depends_on phenomenon 29.46 A fast spinning top does not fall ¶
- depends_on phenomenon 29.49 Precession of the equinoxes ¶
  - depends_on proposition 29.50 The principal nutation ¶
depends_on phenomenon 29.31 The intermediate-axis instability ¶
- depends_on remark 29.32 What the linearization is entitled to conclude ¶
depends_on proposition 29.26 The two integrals of torque-free motion ¶
- 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 ¶ ↺
depends_on proposition 29.28 Free rotation of a symmetric body ¶
- depends_on phenomenon 29.34 Free nutation of the Earth ¶
depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
- depends_on remark 24.48 Reduction is what physicists do without saying so ¶
  - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
- depends_on remark 24.42 The intermediate axis ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`cites`	→	Theoria motus corporum solidorum seu rigidorum	derived	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:751
`depends_on`	→	Newton's second law, rotational case	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:752
`depends_on`	→	Principal axes	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:752
`depends_on`	→	Transport theorem	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:752
`depends_on`	←	A fast spinning top does not fall	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1729
`depends_on`	←	The intermediate-axis instability	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:976
`depends_on`	←	The two integrals of torque-free motion	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:794
`depends_on`	←	Free rotation of a symmetric body	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:852
`depends_on`	←	The free rigid body is a Lie–Poisson system	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1353
`proves`	←	ch:12-rigid-body-rotating-frames@proof-16	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:756