Second variation of functionals

Second variation turns out to be

With

If is a stationary point of and for all nonzero ,
then is a local minimiser of .

Legendre condition

If is a local minimiser of , then .

This is not a sufficient condition.

Integrating by parts, we find

This is a Sturm-Liouville operator ,
so if has a solution for some real ,
we find .
Note that here still HAS to satisfy the 0 boundary conditions.

Example

So find , with .

Jacobi condition

Let be a differentiable function.
Then

Add this to the expression for second variation to get

Complete the square:

Hence if ,
we just need to find solution to
and we guarantee the positivity of the integrand.

Ricatti equation

Set to get

Jacobi equaiton

Related to the kernel problem for where solution never vanishes.

Ashton alternative

Let be a Sturm-Liouville operator
Find it’s eigenvalues with eigenfunctions .
Then for any satisfying boundary conditions we can write:

Now note that

As we can choose freely.
This is always positive if and only if for all .
So we only need to find eigenvalues of in order to figure out the second variation.