the geodesic equation
A geodesic is the curved-space analogue of a straight line: the path a particle follows when no force pushes it sideways, or the path that locally minimizes length. The geodesic equation is the precise differential equation that captures 'going as straight as the space allows' — a curve that never turns relative to the connection, so its own velocity is parallel-transported along itself.
Intrinsically, a curve gamma(t) is a geodesic if its acceleration vanishes covariantly: nabla_(gamma') gamma' = 0, i.e. the velocity is parallel along the curve. In coordinates this is the system d^2 x^k/dt^2 + Gamma^k_ij (dx^i/dt)(dx^j/dt) = 0 for each k, a second-order ODE driven entirely by the Christoffel symbols. Given a starting point p and an initial velocity v in T_p M, the standard existence-and-uniqueness theorem for ODEs produces a unique geodesic with gamma(0) = p, gamma'(0) = v, at least for small time. Note the parametrization matters: this equation gives geodesics with constant speed (affine parameter); a curve can trace a geodesic's image yet fail the equation if reparametrized non-affinely.
Where it lives: geodesics are the autoparallel curves of the Levi-Civita connection and, on a complete manifold, the locally length-minimizing curves. This entry is about the autoparallel ODE itself; the variational story — that geodesics are critical points of the length or energy functional, and the analysis of when they actually minimize — belongs to the calculus of variations, and the global question of whether geodesics extend forever belongs to comparison geometry (Hopf-Rinow). Honesty: a geodesic is only LOCALLY minimizing — a great-circle arc longer than half the sphere is a geodesic but not the shortest path between its endpoints.
On the round sphere the geodesics are exactly the great circles. The flight path from Tokyo to New York looks curved on a flat map but is a great-circle geodesic: locally the shortest route. Continue past the antipode, though, and the same great circle is no longer minimizing even though it still solves the geodesic equation.
Great circles solve the geodesic equation everywhere, but only minimize length up to the antipode.
Solving the geodesic equation (an autoparallel ODE) is different from being a length-minimizer: every minimizer is a geodesic, but a geodesic minimizes only locally. The minimization/variational characterization is a separate, global story.