# Conditions for minima in calculus of variations

In the calculus of variations (unconstrained), one applies a first-order variation on a general functional of the form

$$\int_{a}^{b}F(x,y,y')\,dx$$

to obtain the first-order necessary condition for optimality as expressed by the Euler-Lagrange equality

$$\frac{d}{dx}F_{y'} -F_y = 0.$$

What is the necessary condition for a minima? How about a sufficient condition?

Coming from the world of optimal control, I tend to view the calculus of variations from a Pontryagin point of view. The conditions stated by Pontryagin are necessary, and sufficient under certain hypotheses (mainly related to the convexity of the function $$F$$). I used to refer to this article during my PhD:
For $$|\varepsilon|\ll1$$ and and a differentiable function $$h$$, we can write, using Taylor series,$$F(x,y+\varepsilon h,y'+\varepsilon h')=F(x,y,y')+\varepsilon\mathcal{I}(\Delta[y,h])+\frac{\varepsilon^2}2\mathcal I(\Delta_2[y,h])+\mathcal O(\varepsilon^3)$$ where $$\mathcal I$$ denotes the integrand, $$\Delta$$ the Gâteaux differential and $$\Delta_2[y,h]=\int_a^b\left(F_{y'y'}h'^2+\left(F_{yy}-\frac {dF_{yy'}}{dx}\right)h^2\right)\,dx$$ over the domain $$[a,b]$$. Analogous to functions, at a stationary point path, the Gâteaux differential is zero. Note that $$\Delta_2$$ is also analogous to the second derivative.
Therefore, it is sufficient that $$\Delta_2>0$$ for all non-zero $$h$$ for a minimum to occur, whereas the necessary condition is weaker with $$\Delta_2\ge0$$.