  This paper deals with necessary and sufficient conditions for weak and strong
minimizers of functionals $\Phi(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\in
C^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler than
the known ones, and then apply them to several particular problems, including
stability problems in the elasticity theory. In particular, we solve some open
problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and
buckling solutions in three dimensions, Nonlinearity 27 (2014), 2841--2867] by
finding optimal conditions for the stability of a naturally straight Kirchhoff
rod under various types of endpoint constraints.
