Semana 16

Questão 1

$$\frac{\partial L}{\partial q_{s}}=0$$
$$\Rightarrow \frac{d}{dt}\frac{\partial L}{\partial \dot{q_{s}}}=0$$
$$\therefore \frac{\partial L}{\partial \dot{q_{s}}}=C (constante\; no \;tempo)$$
$$H=\sum_{s=1}^{n}(p_{s}\dot{q_{s}})-L $$
$$\frac{\partial H}{\partial q_{s}}=\sum_{k=1}^{n}\frac{\partial }{\partial q_{s}}(p_{k}\dot{q_{k}})-\frac{\partial L}{\partial q_{s}}$$
$$\frac{\partial }{\partial q_{s}}(p_{k}\dot{q_{k}})=\frac{\partial p_{k}}{\partial q_{s}}\dot{q_{k}}+p_{k}\frac{\partial \dot{q_{k}}}{\partial q_{s}}$$
$$=\frac{\partial }{\partial q_{s}}\frac{\partial L}{\partial \dot{q_{k}}}\dot{q_{k}}+\frac{\partial L}{\partial \dot{q_{k}}}\frac{\partial \dot{q_{k}}}{\partial q_{s}}$$
para$ k\neq s$:
$$\frac{\partial q_{k}}{\partial q_{s}}=0$$
para $k = s$:
$$\frac{\partial q_{s}}{\partial q_{s}}=1$$
Portanto, o somatório fica:
$$\sum_{k=1}^{n}\frac{\partial }{\partial q_{s}}(p_{k}\dot{q_{k}})=\sum_{k=1}^{n}(\frac{\partial }{\partial t}\frac{\partial L}{\partial \dot{q_{s}}}+\frac{\partial L}{\partial q_{s}})$$
$$mas \frac{\partial L}{\partial \dot{q_{s}}}=C (constante\; no \;tempo)$$
$$\therefore \frac{\partial }{\partial t}\frac{\partial L}{\partial \dot{q_{s}}}=0$$
$$e \frac{\partial L}{\partial q_{s}}=0$$
$$\therefore \sum_{k=1}^{n}\frac{\partial }{\partial q_{s}}(p_{k}\dot{q_{k}})=0$$
$$Como \frac{\partial L}{\partial q_{s}}=0, tem-se \;que:$$
$$\Rightarrow \frac{\partial H}{\partial q_{s}}=0$$