Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...