Hamiltonian Mechanics
#Physics
Classical Dynamics
$\displaystyle H=\sum_{i = 1}^{n}p_{i}\dot{q}{i}-\mathcal{L}(q{i},\dot{q}{i},t),,p{i}=\frac{ \partial \mathcal{L} }{ \partial \dot{q}{i}},,\dot{q}{k}=\frac{ \partial H }{ \partial p_{k} },,\dot{p}{k}=-\frac{ \partial H }{ \partial q{k} }$
- Generalized coordinates version
- $\displaystyle H$ is the hamiltonian of the system
- $\displaystyle p_{i}$ is the canonical momentum of the
$U \text{ not dependent on }\dot{q}_{i} \text{ or }t\rightarrow H=E$
- $\displaystyle U$ would only be dependent on $\displaystyle q_{i}$ at this point then