definition 22.2 Hamiltonian

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:67 · p. 805

ground object -- no derivation owed

Rests on

depends_on definition 21.34 Generalized momentum ¶
depends_on definition 21.31 Lagrangian ¶

Supports

depends_on definition 22.14 Phase-space coordinate transformation ¶
- 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 theorem 23.31 Liouville–Arnold ¶
  - depends_on remark 22.16 Why the class is larger than the point transformations ¶
- depends_on definition 22.26 Lagrange bracket ¶
- depends_on definition 22.29 Poisson bracket ¶
  - depends_on definition 26.20 Dirac bracket ¶
    - depends_on definition 26.23 Gauge conditions ¶
      - depends_on proposition 26.24 The Faddeev–Popov determinant ¶
    - depends_on example 26.22 Holonomic constraints ¶
    - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
    - depends_on theorem 26.21 Properties of the Dirac bracket ¶
      - depends_on postulate 26.47 Dirac quantization ¶
  - depends_on definition 26.12 First and second class ¶
    - depends_on proposition 26.13 The first-class functions close ¶
      - depends_on proposition 26.43 The hypersurface-deformation algebra ¶
      - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
    - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on definition 26.20 Dirac bracket ¶ ↺
      - depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
    - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 113.3 Degrees of freedom before and after ¶
  - depends_on definition 26.8 Total Hamiltonian ¶
    - depends_on definition 26.16 Extended Hamiltonian ¶
    - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
    - depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
      - depends_on definition 26.16 Extended Hamiltonian ¶ ↺
      - depends_on postulate 26.47 Dirac quantization ¶ ↺
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶ ↺
  - depends_on definition 25.11 Generators of the Galilei transformations ¶
    - depends_on proposition 25.12 The algebra of generators closes ¶
  - depends_on definition 25.6 Infinitesimal canonical transformation ¶
    - depends_on definition 25.11 Generators of the Galilei transformations ¶ ↺
  - depends_on definition 25.2 Poisson algebra ¶
  - depends_on definition 24.36 Poisson manifold ¶
    - depends_on definition 25.2 Poisson algebra ¶ ↺
    - depends_on definition 24.37 Casimir function ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
      - depends_on theorem 24.38 Symplectic foliation, quoted ¶
    - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶ ↺
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶ ↺
    - depends_on theorem 24.38 Symplectic foliation, quoted ¶ ↺
  - depends_on notation 26.28 Field systems ¶
    - depends_on definition 26.40 The $3+1$ decomposition ¶
      - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
      - depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
    - depends_on notation A.636 Densities and smearing ¶
      - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
    - depends_on notation 26.36 Gauge fields and their SI dimensions ¶
      - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶ ↺
  - depends_on notation 26.1 Weak and strong equality ¶
    - depends_on notation 26.28 Field systems ¶ ↺
  - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
    - depends_on definition 24.37 Casimir function ¶ ↺
    - depends_on theorem 24.45 Noether, symplectic form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
depends_on definition 23.15 Orthogonal Hamiltonian ¶
depends_on definition 25.18 Phase-space action ¶
depends_on theorem 22.6 Hamilton's equations ¶
- depends_on definition 22.15 Canonical transformation ¶ ↺
- 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`	→	Generalized momentum	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:75
`depends_on`	→	Lagrangian	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:75
`depends_on`	←	Phase-space coordinate transformation	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:378
`depends_on`	←	Orthogonal Hamiltonian	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:472
`depends_on`	←	Phase-space action	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:739
`depends_on`	←	Hamilton's equations	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:165