Suppose that we are given an extremal problem such that:
$$J(x) = \int_{t_0}^{t_1} L(t,x,x')dt \rightarrow inf$$
Where $L$ is known.
Find all conjugate points on the interval: $(t_0,t_1]$.
Provided the Jacobi condition:
$$- \frac{d}{dt}(L_{x'x'}h' + L_{x'x}h) + L_{x'x}h' + L_{xx}h = 0 $$
Suppose that we were able to find a solution: $h(t)$.
How can I go about finding the conjugate points?
Can I just check where $h(t)=0$?
I know that points are considered to be conjugate if $h(t_0)=h(T)=0$ where $t_0 \leq T \leq t_1$, but does that mean that I should set up the system:$$\begin{bmatrix} h(t_0)\\ h(T)\end{bmatrix} = \vec{0}$$
and solve for the coefficients of $h(t)$?
I'm having a hard time conceptually grasping what I am required to do for this problem. It also doesn't help that the endpoint $t_0$ is not included. If someone could better explain the concept of conjugate points to me that would be great. Thanks.