theorem 22.6 Hamilton's equations

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:153 · p. 806

Rests on

depends_on definition 22.2 Hamiltonian ¶
- depends_on definition 21.34 Generalized momentum ¶
- depends_on definition 21.31 Lagrangian ¶
depends_on equation 21.39 eq:lag-el-conservative ¶
depends_on equation 21.43 eq:lag-generalized-momentum ¶
proves proof ch:05-hamiltonian-mechanics@proof-2 ¶

Supports

depends_on definition 22.15 Canonical transformation ¶
- depends_on proposition 22.17 Transformation conditions ¶
  - depends_on remark 22.21 The count ¶
- depends_on proposition 22.18 The mixed generating functions ¶
  - depends_on definition 23.2 Hamilton's principal function and the Hamilton–Jacobi equation ¶
    - depends_on remark 23.12 Two functions, two roles ¶
  - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
    - depends_on example A.557 The check that fixes the convention ¶
  - depends_on remark 22.19 Which form exists ¶
  - depends_on theorem 23.30 Motion in action–angle variables ¶
    - depends_on example 23.24 Harmonic oscillator ¶
    - depends_on theorem 23.34 Adiabatic invariance of the action ¶
      - depends_on example 23.35 The slowly stiffened oscillator ¶
      - depends_on example 23.27 Charged particle in a uniform magnetic field ¶
      - depends_on remark 23.36 How good the invariance really is ¶
      - depends_on remark 23.37 Why this matters observationally ¶
    - depends_on theorem 23.31 Liouville–Arnold ¶
      - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 23.33 Integrability is exceptional ¶
      - depends_on remark 23.32 What that proof imports ¶
      - depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
      - depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
- depends_on remark 22.16 Why the class is larger than the point transformations ¶
depends_on remark 22.8 Non-conservative forces ¶
depends_on remark 22.7 Which half is kinematics and which is dynamics ¶
depends_on remark 23.4 What the equation costs and what it buys ¶
depends_on theorem 23.30 Motion in action–angle variables ¶ ↺
depends_on theorem 23.6 Jacobi ¶
- depends_on example 23.25 Central force in plane polar coordinates ¶
- depends_on example 23.22 Free particle ¶
  - depends_on remark 23.42 Phase speed and particle speed ¶
- depends_on example 23.23 Uniform gravity, and the parabola ¶
- depends_on proposition 23.7 The principal function is the action ¶
  - depends_on remark 23.8 The action as a function, not a functional ¶

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`	→	Hamiltonian	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:165
`depends_on`	→	eq:lag-el-conservative	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:165
`depends_on`	→	eq:lag-generalized-momentum	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:165
`depends_on`	←	Canonical transformation	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:393
`depends_on`	←	Non-conservative forces	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:252
`depends_on`	←	Which half is kinematics and which is dynamics	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:236
`depends_on`	←	What the equation costs and what it buys	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:141
`depends_on`	←	Motion in action–angle variables	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:940
`depends_on`	←	Jacobi	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:173
`proves`	←	ch:05-hamiltonian-mechanics@proof-2	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:169