Second variation formula and Jacobi fields Let $\bar{\alpha}:U\rightarrow \Omega$ be a two-parameter variation of a geodesic $\gamma$. For $i=1,2$ we define
$$W_{i}=\frac{\partial \bar{\alpha}}{\partial u_{i}} \in T_{\gamma}\Omega$$
the variation vector fields. We have:
$$-\frac{1}{2}\frac{\partial^{2}E}{\partial u_{1}\partial u_{2}}(0,0) = \sum_{t\in[0,1]} \langle W_{2},\Delta_t\frac{DW_{1}}{dt} \rangle +\int^{1}_{0} \langle W_{2},\frac{D^{2}W_{1}}{dt^{2}} + R(V,W_{1})V \rangle dt $$
where $V=\frac{d\gamma}{dt}$, $R$ is the curvature form and
$$ \Delta_t\frac{DW_{1}}{dt^{2}} := \frac{DW_{1}}{dt}(t^{+}) - \frac{DW_{1}}{dt}(t^{-}) $$
denotes the jump in $\frac{DW_{1}}{dt}$ at one of its finitely many points of discontinuity in the open unit interval.
We can define a Jacobi field along a geodesic $\gamma$ as the field that satisfies the differential equation
$$   \frac{D^{2}J}{dt^{2}} + R(V,J)V = 0 $$
that is the equation in the integral in the second variation formula. Is there a method to introduce Jacobi fields using the second variation formula? Does it happen that when a geodesic is not minimal, the form isn't defined?
 A: The second variation of energy is also called index form (along a geodesic).
It is indeed possible to introduce Jacobi fields using the index form.
Let $\gamma:[0,L]\to M$ be a geodesic and $I_\gamma$ the index form along it.
The index form is a quadratic form on the space of vector fields along $\gamma$.
Let us denote this space of vector fields by $H$ and the subspace that vanishes at both endpoints by $H_0$.
(Regularity issues are not very important, but you can take $W^{1,2}$ regularity for you vector fields and the index form behaves nicely. Piecewise $C^1$ vector fields make a subspace of this.)
Let me give you two theorems that illustrate what the index form can tell you.
Theorem 1:
Let $Y\in H$.
Then $Y$ is a Jacobi field along $\gamma$ if and only if $I_\gamma(Y,W)=0$ for all $W\in H_0$.
(This is essentially a weak formulation of the Jacobi equation.)
Theorem 2:
The index form is well-defined for any geodesic.
The index form $I_\gamma$ is positive semi-definite if and only if there are no conjugate points along $\gamma$ (exculding the endpoints).
If the index form takes a negative value, then $\gamma$ is not (locally) length minimizing.
Let me know if you have difficulties finding material about the index form and Jacobi fields, and I can find some lecture notes online.
