Second Variation: Euler-Lagrange Equation & \delta^2 S

[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