Hamiltonian Vector Fields

graph TD D1["D1 Hamiltonian vector field X_H\nι_{X_H} ω = dH"] D2["D2 Poisson bracket {f,g}\nω(X_f, X_g) = df(X_g)"] D3["D3 Hamiltonian flow φ_t\nGenerated by X_H, dφ_t/dt = X_H ∘ φ_t"] D4["D4 Liouville form\nα on cotangent bundle, dα = ω"] T1["T1 Flow preserves ω\nφ_t* ω = ω"] T2["T2 Liouville theorem\nHamiltonian flow preserves volume ω^n"] T3["T3 {f,g} = X_f(g)\nPoisson = Lie derivative of g along X_f"] T4["T4 Arnold-Liouville\nn integrals in involution ⇒ tori, action-angle"] T5["T5 Darboux on cotangent\nT*Q has canonical (q,p) with ω = dp∧dq"] D1 --> D2 D1 --> D3 D4 --> D1 D3 --> T1 D3 --> T2 D2 --> T3 D2 --> T4 D4 --> T5 T1 --> T2 D1 --> T3 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem