Darboux Theorem and Canonical Coordinates

graph TD D1["D1 Standard symplectic ω_0\nω_0 = Σ dp_i ∧ dq_i on R^{2n}"] D2["D2 Darboux chart\nLocal coords (q,p) with ω = Σ dp_i ∧ dq_i"] D3["D3 Lagrangian submanifold L\ndim L = n, ω|_L = 0"] D4["D4 Isotopy\nSmooth 1-parameter family of diffeomorphisms"] T1["T1 Darboux theorem\n(M,ω) locally ≅ (R^{2n}, ω_0)"] T2["T2 Moser trick\nIsotopy of symplectic forms via Moser vector field"] T3["T3 Weinstein Lagrangian\nNhd of L symplectomorphic to T*L"] T4["T4 Symplectic embedding\nB^{2n}(r) → (M,ω) for small r"] T5["T5 Relative Darboux\nTwo forms agree on L ⇒ differ by exact near L"] D1 --> D2 D1 --> T1 D2 --> T1 D3 --> T3 D4 --> T2 D1 --> T2 T2 --> T1 D3 --> T4 T1 --> T3 T2 --> T5 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