# reference request – Determination of the nature of stationary values in variational calculus

In variational calculus, when we solve the Euler-Lagrange equation $$frac{d}{dx}L_p(u’,u,x)-L_z(u’,u,x)$$, where $$L=L(p,z,x)$$, to find stationary inputs of the functional
$$I(u)=int_0^1 L(u’,u,x)dx,$$
we also need to determine whether this solution $$u$$ is a maximum, a minimum, or a saddle point. However, the second order variation
$$left.frac{d^2}{dt^2}right|_{t=0} I(u+tv)= int_0^1 v’^2L_{pp}+2vv’L_{pz}+L_{zz}v^2 dx$$
is very hard to work with since we need to prove, for instance, this is positive for all functions $$v$$ to show that $$u$$ is a minimum. I have kept everything in one dimension for simplicity, but of course the above applies to higher dimensions as well.

I can think of two ways to confirm that it is a minimum:

1. If the function $$L$$ satisfy certain convexity assumption, then a minimizer exists, and the minimizer is a solution to the differential equation. Therefore, if the differential equation has a unique solution, it is the minimizer.
2. In some situations, we can write an inequality satisfied by $$I$$, and then consider the condition for the equality to be achieved in the inequality. An example is the equation for geodesic on the sphere $$S^2$$.

However, both methods applies only to a limited range of functions.

What are some other methods to establish that the stationary solution is a minimum/maximum? I am looking for some other ideas.

Perhaps I need some reference on this, since parts of my variational calculus and PDE books that I have read does not focus on this question.

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