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.