How is the second variation δ²S defined in the calculus of variations?

[Karlisbad]

Let be a functional S so $$\delta S =0$$ give the Euler-Lagrange equation where:

$$S= \int_{a}^{b}dtL(q,\dot q, t)$$

My question is ..How the "second variation" $$\delta ( \delta S )=0= \delta ^{2} S$$ defined??.. in order we could decide if a function q=q(t) is either a maximum or a minimum point of function space.. thanks.

 

[dextercioby]

How's the "first variation" defined ?

Daniel.

 

[Karlisbad]

The first variation of S (are you physicist..i say so because i use to see you in the QM forum ) are the Euler-Lagrange equation defined via the functional derivative