X_H: ι_{X_H}ω=dH. Hamiltonian flow preserves ω. Liouville: phase vol preserved.
graph TD
D1["Def: X_H\nι_{X_H}ω = dH"]
D2["Def: Hamiltonian flow\nφ_t^H"]
D3["Def: Symplectic map\nφ*ω = ω"]
D4["Def: Poisson bracket\n{f,g} = ω(X_f,X_g)"]
T1["Thm: φ_t*ω = ω\nflow preserves ω"]
T2["Thm: Liouville\nvol preserved"]
T3["Thm: d/dt f∘φ_t = {H,f}"]
T4["Thm: Arnold–Liouville\nint sys action-angle"]
T5["Thm: Noether\nsymm⇒conserved"]
D1 --> D2
D2 --> T1
D3 --> T1
D4 --> T3
T1 --> T2
T3 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 14
- Edges: 11
Frontier: math.SG