PX285 - F1 - hamilton's equations

considering variations in the lagrangian:

\begin{aligned} d L (q_{i}, {\dot{q}}_{i}) & = \sum_{i} \frac{\partial L}{\partial {\dot{q}}_{i}} d {\dot{q}}_{i} + \sum_{i} \frac{\partial L}{\partial q_{i}} d q_{i} \\ (1) & ∴ d L & = \sum_{i} p_{i} d {\dot{q}}_{i} + \sum_{i} {\dot{p}}_{i} d q_{i} \end{aligned}

where, $p_{i} = \frac{\partial L}{\partial {\dot{q}}_{i}}$ , and ${\dot{p}}_{i} = \frac{\partial L}{\partial q_{i}}$

for $a = f g :$

\frac{d a}{d t} = f \frac{d g}{d t} + g \frac{d f}{d t} ⟹ d a = f d g + g d f

similarly, for $\sum_{i} p_{i} {\dot{q}}_{i} :$

\begin{array}{r} d (\sum_{i} p_{i} {\dot{q}}_{i}) = \sum_{i} p_{i} d {\dot{q}}_{i} + \sum_{i} {\dot{q}}_{i} d p_{i} \\ \sum_{i} p_{i} d {\dot{q}}_{i} = d (\sum_{i} p_{i} {\dot{q}}_{i}) - \sum_{i} {\dot{q}}_{i} d p_{i} \end{array}

substituting into equation $(1)$ and rearranging:

\begin{matrix} (2) & d (\sum_{i} p_{i} {\dot{q}}_{i} - L) = - \sum_{i} {\dot{p}}_{i} d q_{i} + \sum {\dot{q}}_{i} d p_{i} \end{matrix}

the argument of the differential on the $L H S$ of equation $(2)$ is defined to be the hamiltonian:

H (p, q, t) = \sum p_{i} {\dot{q}}_{i} - L

therefore, equation $(2)$ can be re written as:

\begin{matrix} (3) & d H = - \sum_{i} {\dot{p}}_{i} d q_{i} + \sum {\dot{q}}_{i} d p_{i} \end{matrix}

also:

\begin{matrix} (4) & d H = \sum_{i} \frac{\partial H}{\partial q_{i}} d q_{i} + \sum_{i} \frac{\partial H}{\partial p_{i}} d p_{i} \end{matrix}

comparing equations $(3)$ and $(4)$ , the canonical hamilton's equations are obtained:

\begin{matrix} {\dot{q}}_{i} = \frac{\partial H}{\partial p_{i}} \\ {\dot{p}}_{i} = - \frac{\partial H}{\partial q_{i}} \end{matrix}