Second order necessary condition for a strong minimum in the classical problem of calculus of variations (Q6574280)

The author presents the original result in formulating second order necessary conditions for a strong minimum of a functional \N\[\NJ(x(\cdot))=\int_0^T L(t,x(t),\dot x(t)\,dt,\ x(0)=a,\ x(T)=b\N\]\Non the set of \(C^1\) curves \(x(\cdot):[0,T]\to\mathbb{R}^n\). His proof is relatively simple compared with proofs of previous similar results. The idea of considering second order conditions for a strong minima has never appeared in the context of classical calculus of variations.