Yes, you can do it componentwise. As you have three momenta, you have to write Hamilton's equations for all of them.

$\dot{x} = \partial H/\partial p_x$

$\dot{y} = \partial H/\partial p_y$

$\dot{z} = \partial H/\partial p_z$

Similarly for the momenta

$\dot{p_x} = -\partial H/\partial x$

$\dot{p_y} = -\partial H/\partial y$

$\dot{p_z} = -\partial H/\partial z$

Yes, you can do it component-wise. As you have three momenta, you have to write Hamilton's equations for all of them.

\begin{align}\dot{x} &= \partial H/\partial p_x\\ \dot{y} &= \partial H/\partial p_y \\ \dot{z} &= \partial H/\partial p_z\end{align}

Similarly for the momenta

\begin{align}\dot{p_x} &= -~\partial H/\partial x\\ \dot{p_y} &= -~\partial H/\partial y\\ \dot{p_z} &= -~\partial H/\partial z\end{align}

Yes, you can do it componentwise. As you have three momenta, you have to write Hamilton's equations for all of them.

$\dot{x} = \partial H/\partial p_x$

$\dot{y} = \partial H/\partial p_y$

$\dot{z} = \partial H/\partial p_z$

Similarly for the momenta

$\dot{p_x} = \partial H/\partial x$

$\dot{p_y} = \partial H/\partial y$

$\dot{p_z} = \partial H/\partial z$

Yes, you can do it componentwise. As you have three momenta, you have to write Hamilton's equations for all of them.

$\dot{x} = \partial H/\partial p_x$

$\dot{y} = \partial H/\partial p_y$

$\dot{z} = \partial H/\partial p_z$

Similarly for the momenta

$\dot{p_x} = -\partial H/\partial x$

$\dot{p_y} = -\partial H/\partial y$

$\dot{p_z} = -\partial H/\partial z$