Scope of stationarity: stationary is not minimum, and no second-variation theory is developed here

Stationarity, as defined in A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations and characterised by Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression, is the vanishing of the first variation for all fixed-endpoint variations. It is not the statement that $q$ minimizes the action, nor that it is a local minimum. The companion page exhibits an explicit quadratic model in which a stationary curve is not a local minimum: for the Lagrangian $\mathcal{L}(q,v)=\frac{1}{2}(v^2-4q^2)$ on $[0,\pi ]$ and the stationary curve $q_{\ast }(t)=\sin 2t$, the exact identity $S(q_{\ast }+ϵ\sin )-S(q_{\ast })=-\frac{3\pi }{4}{ϵ}^2$ shows that every nearby element of the one-parameter family has strictly smaller action.

No second variation, and no convexity or coercivity hypothesis, is introduced on this page. The stationarity theorem is an equivalence between zero first variation and the Euler–Lagrange equation. For a local extremum on fixed-endpoint $C^2$ curves, use the norm $\mathrm{\parallel }q{\mathrm{\parallel }}_{C^2}:={\max }_{0\le j\le 2}{\sup }_{t\in [a,b]}\mathrm{\parallel }{q}^{(j)}(t){\mathrm{\parallel }}_2$. Each admissible family $q+s\eta$ tends to $q$ in this norm; a local extremum therefore gives an extremum at $s=0$ of its differentiable action, whose derivative is zero. Thus the Euler–Lagrange equation is necessary for such extrema, while stationarity alone need not give an extremum.

The finite-interval chart statements on this page concern $C^2$ data and a fixed interval $[a,b]$. They make no claim about minimizers in a Sobolev or absolutely continuous class, about broken extremals, or about boundary conditions other than fixed endpoints; those belong to the calculus-of-variations development of the mathematical library.