A variational approach to deriving Hamilton's equations

I am a math grad student learning mechanics from Landau-Lifshitz's textbook, and I have a question about their derivation of Hamilton's equations from Lagrange's. In short, they use a variational principle with respect to $p$ and $q$ , where they allow arbitrary variations $p\to p+\delta p$ and $q\to q+\delta q$ . However, $\delta q$ determines $\delta p$ , so this is not quite rigorous. Let me summarize the derivation. Since $L=p\dot q-H$ , the action integral is $S=\int_{t_0}^{t_1}(pdq-Hdt).$ Now, to apply a variational principle, consider a variation $q\to q+\delta q$ and $p\to p+\delta p$ where $\delta q(t_0)=\delta q(t_1)=0$ . The change is $S$ is $\delta S=\int_{t_0}^{t_1}\bigg(\delta p\cdot dq+pd(\delta q)-\big(\frac{\partial H}{\partial p}dp+\frac{\partial H}{\partial q}dq\big)\bigg).$ By integration by parts this can be re-written as $\delta S=\int\delta p\bigg(dq-\frac{\partial H}{\partial p}dt\bigg)-\int\delta q\bigg(dp+\frac{\partial H}{\partial q}dt\bigg).$ For this to always be zero for all $\delta p$ and $\delta q$ , we require Hamilton's equations $\frac{\partial H}{\partial p}=\dot q$ and $\frac{\partial H}{\partial q}=-\dot p$ . Let me say why I'm unhappy with this derivation. In the usual derivation of the Euler-Lagrange equations, you only consider a variation $q\to q+\delta q$ . This determines a variation $\dot q\to \dot q+\dot{(\delta q)}$ and we solve for $\delta S=0$ . However, suddenly Landau-Lifshitz is telling us to take a variation of both $q$ and $p$ freely. This is problematic, since the change in $p:=\frac{\partial L}{\partial\dot q}$ is determined by $\delta q$ . That would be analogus to, if, in the Lagrangian formalism, you took a variation $q\to q+\delta q$ and $\dot q\to\dot q+\delta\dot q$ , where $\delta\dot q$ was an arbitrary function unrelated to the derivative $\dot{(\delta q)}$ of $\delta q$ . So, my question is: What justifies taking an arbitrary variation $q\to q+\delta q$ and $p\to p+\delta p$ in the derivation of Hamilton's equations, while you are not allowed to take arbitrary variations $q\to q+\delta q$ and $\dot q\to\dot q+\delta\dot q$ in the derivation of the Euler-Lagrange equations?