Why the Hamiltonian is constant along the integral curves of the hamiltonian vector field?

Question

Let $H$ the Hamiltonian of a system and $\gamma $ an integral curve of the Hamiltonian vector field, i.e. if $\gamma (t)=(q(t),p(t))$ and $H(p,q)$ is the Hamiltonian, then $$\begin{cases} \dot p=-H_q\\ \dot q= H_p\end{cases}.$$

In wikipedia the say that $H$ is constant along $\gamma $. Why is this true ? i.e. why $H(\gamma (t))$ is constant ?

Answer 1

For $H(p,q,t) = H(p(t),q(t))$ think that

$$ \frac{d}{dt}H(p(t),q(t)) = H_p \dot p + H_q \dot q $$