Second variation formula and Jacobi fields

Question

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?