  In the context of classical mechanics, we study the conditions under which
higher-order derivative theories can evade the so-called Ostrogradsky
instability. More precisely, we consider general Lagrangians with second order
time derivatives, of the form $L(\ddot\phi^a,\dot\phi^a,\phi^a;\dot q^i,q^i)$
with $a = 1,\cdots, n$ and $i = 1,\cdots, m$. For $n=1$, assuming that the
$q^i$'s form a nondegenerate subsystem, we confirm that the degeneracy of the
kinetic matrix eliminates the Ostrogradsky instability. The degeneracy implies,
in the Hamiltonian formulation of the theory, the existence of a primary
constraint, which generates a secondary constraint, thus eliminating the
Ostrogradsky ghost. For $n>1$, we show that, in addition to the degeneracy of
the kinetic matrix, one needs to impose extra conditions to ensure the presence
of a sufficient number of secondary constraints that can eliminate all the
Ostrogradsky ghosts. When these conditions that ensure the disappearance of the
Ostrogradsky instability are satisfied, we show that the Euler-Lagrange
equations, which involve a priori higher order derivatives, can be reduced to a
second order system.
