  As the first step to extend our understanding of higher-derivative theories,
within the framework of analytic mechanics of point particles, we construct a
ghost-free theory involving third-order time derivatives in Lagrangian. While
eliminating linear momentum terms in the Hamiltonian is necessary and
sufficient to kill the ghosts associated with higher derivatives for Lagrangian
with at most second-order derivatives, we find that this is necessary but not
sufficient for the Lagrangian with higher than second-order derivatives. We
clarify a set of ghost-free conditions under which we show that the Hamiltonian
is bounded, and that equations of motion are reducible into a second-order
system.
