theorem 21.43 Conservation of energy

open in the book · parts/03-classical-mechanics/04-lagrangian-mechanics.tex:1157 · p. 788

Rests on

depends_on equation 21.51 eq:lag-energy-dot-cons ¶
depends_on equation 21.46 eq:lag-energy-h ¶
depends_on theorem 19.32 Conservation of energy ¶
- depends_on definition 19.31 Energy ¶
  - depends_on definition 19.30 Conservative force and potential energy ¶
    - depends_on equation 19.2 eq:newt-second-law ¶
    - 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 19.29 Kinetic energy ¶
    - 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 equation 19.19 eq:newt-conservative ¶
- depends_on equation 19.4 eq:newt-fma ¶
- proves proof ch:02-newtonian-dynamics@proof-9 ¶
proves proof ch:04-lagrangian-mechanics@proof-11 ¶

Supports

depends_on proposition 29.36 Three first integrals ¶
- depends_on phenomenon 29.38 Nutation of a released top ¶
- 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 proposition 29.40 The sleeping top ¶
- depends_on theorem 29.37 Reduction to one degree of freedom ¶
  - depends_on phenomenon 29.38 Nutation of a released top ¶ ↺
  - depends_on proposition 29.40 The sleeping top ¶ ↺
  - depends_on remark 29.42 The closed form, and the mathematics it needs ¶
depends_on theorem 27.10 Lagrange–Dirichlet ¶

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`	→	eq:lag-energy-dot-cons	declared	parts/03-classical-mechanics/04-lagrangian-mechanics.tex:1175
`depends_on`	→	eq:lag-energy-h	declared	parts/03-classical-mechanics/04-lagrangian-mechanics.tex:1175
`depends_on`	→	Conservation of energy	declared	parts/03-classical-mechanics/04-lagrangian-mechanics.tex:1175
`depends_on`	←	Three first integrals	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1276
`depends_on`	←	Lagrange–Dirichlet	declared	parts/03-classical-mechanics/10-central-forces-statics.tex:304
`proves`	←	ch:04-lagrangian-mechanics@proof-11	declared	parts/03-classical-mechanics/04-lagrangian-mechanics.tex:1179