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.