  Given a functional for a one-dimensional physical system, a classical problem
is to minimize it by finding stationary solutions and then checking the
positive definiteness of the second variation. Establishing the positive
definiteness is, in general, analytically untractable. However, we show here
that a global geometric analysis of the phase-plane trajectories associated
with the stationary solutions leads to generic conditions for minimality. These
results provide a straightforward and direct proof of positive definiteness, or
lack thereof, in many important cases. In particular, when applied to
mechanical systems, the stability or instability of entire classes of solutions
can be obtained effortlessly from their geometry in phase-plane, as illustrated
on a problem of a mass hanging from an elastic rod with intrinsic curvature.
