Starting from the Hamiltonian system
$$ \dot{z}(t) = H(t,z(t)) $$
we deduce the variational system
$$ \dot{\delta z} = dH(t,z(t)).\delta z $$
$\delta z$ means we consider curves close to $z(t)$ for a certain topology. Is there a sense to consider the equation for $\ddot{z}(t)$ and how is it related to the variational system ?
Thanks !