Hamiltonian Mechanics

Classical Dynamics

$H=\sum _{i=1}^n{p}_i{\dot{q}}_i-\mathcal{L}(q_i,{\dot{q}}_i,t),p_i=\frac{\partial \mathcal{L}}{\partial {\dot{q}}_i},{\dot{q}}_k=\frac{\partial H}{\partial p_k},{\dot{p}}_k=-\frac{\partial H}{\partial q_k}$

• Generalized coordinates version
• $H$ is the hamiltonian of the system
• $p_i$ is the canonical momentum of the

$U\text{ not dependent on }{\dot{q}}_i\text{ or }t\to H=E$

• $U$ would only be dependent on $q_i$ at this point then

Equations of Motion

$\{\begin{array}{l}-{\dot{p}}_k=\frac{\partial H}{\partial q_k}\\ {\dot{q}}_k=\frac{\partial H}{\partial p_k}\end{array}$

Quantum Physics

• Hamiltonian Operator