Lagrangian & Hamiltonian

graph TD D1["Def: Lagrangian L\nL(q,q̇,t)"] D2["Def: Action S\nS = ∫L dt"] D3["Def: Hamiltonian H\nH = p·q̇ − L"] D4["Def: Poisson bracket\n{f,g} = ∂f/∂q ∂g/∂p − ..."] T1["Thm: Euler–Lagrange\n∂L/∂q = d/dt ∂L/∂q̇"] T2["Thm: Hamilton eqs\nq̇=∂H/∂p, ṗ=−∂H/∂q"] T3["Thm: Least action\nδS=0 ⇒ E-L"] T4["Thm: Noether\nsymmetry → conserved"] T5["Thm: Liouville\n{H,f}=0 ⇒ const"] D1 --> D2 D2 --> D3 D3 --> D4 D1 --> T1 T1 --> T3 T2 --> T4 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