  A different approach will be presented that aims to scrutinize the
phase-space trajectories of a general class of Hamiltonian systems with regard
to their regular or irregular behavior. The approach is based on the
`energy-second-moment map' that can be constructed for all Hamiltonian systems
of the generic form $H=p^{2}/2+V(q,t)$. With a three-component vector $s$
consisting of the system's energy $H$ and second moments $qp$, $q^{2}$, this
map linearly relates the vector $s(t)$ at time $t$ with the vector's initial
state $s(0)$ at $t=0$. It will turn out that this map is directly obtained from
the solution of a linear third-order equation that establishes an extension of
the set of canonical equations. The Lyapunov functions of the
energy-second-moment map will be shown to have simple analytical
representations in terms of the solutions of this linear third-order equation.
Applying Lyapunov's regularity analysis for linear systems, we will show that
the Lyapunov functions of the energy-second-moment map yields information on
the irregularity of the particular phase-space trajectory. Our results will be
illustrated by means of numerical examples.
