theorem 29.16 Principal axes

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:455 · p. 981

Rests on

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 18.4 Body ¶
      - depends_on definition 19.3 Mass ¶
    - 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 ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - 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 5.56 Endomorphism ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - proves proof ch:03-linear-algebra-representations@proof-25 ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on definition 5.56 Endomorphism ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - 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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-31 ¶
proves proof ch:12-rigid-body-rotating-frames@proof-10 ¶

Supports

depends_on definition 29.19 Inertia ellipsoid ¶
- 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 definition 29.35 The Lagrange top ¶
depends_on definition 29.18 Classification of tops ¶
- depends_on proposition 29.28 Free rotation of a symmetric body ¶
  - depends_on phenomenon 29.34 Free nutation of the Earth ¶
depends_on phenomenon 29.31 The intermediate-axis instability ¶
- depends_on remark 29.32 What the linearization is entitled to conclude ¶ ↺
depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
depends_on proposition 29.17 Triangle inequalities ¶
- depends_on proposition 29.22 Perpendicular-axis theorem ¶
depends_on theorem 29.25 Euler's equations ¶
- 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 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 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 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

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`	→	Specimen theoriae turbinum	derived	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:463
`depends_on`	→	Inertia tensor, continuum form	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:466
`depends_on`	→	Positivity	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:466
`depends_on`	→	Spectral theorem for a self-adjoint operator	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:466
`depends_on`	←	Inertia ellipsoid	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:535
`depends_on`	←	The Lagrange top	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1248
`depends_on`	←	Classification of tops	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:525
`depends_on`	←	The intermediate-axis instability	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:976
`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`	←	Triangle inequalities	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:504
`depends_on`	←	Euler's equations	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:752
`proves`	←	ch:12-rigid-body-rotating-frames@proof-10	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:470