Geodesic Equations

Differential Geometry · Axiom Academy

Understanding the differential equations that govern the shortest paths on curved surfaces A curve γ(t) parameterized by coordinates u k (t) is a geodesic if it satisfies the geodesic equation for each coordinate k: Here, Γ k ij are the Christoffel symbols encoding how the coordinate basis vectors change across the manifold. The equation states that the acceleration of the curve (second derivative) is balanced by the "pseudo-forces" arising from the curved geometry. 2. Derivation from Variational Principle Geodesics can be derived by minimizing the arc length functional. For a curve γ(t), we seek to minimize: where g ij is the metric tensor. Applying the Euler-Lagrange equations to this variational problem with Lagrangian L = √(g ij u̇ i u̇ j ) yields the geodesic equation. This connects the geometric concept of "shortest path" to the differential equation framework. For an n-dimensional manifold, the geodesic equation represents n coupled second-order ordinary differential equations (ODEs). We can rewrite this as a 2n-dimensional first-order system by introducing velocities v k = du k /dt: This formulation is well-suited for numerical integration and reveals that geodesics are determined by specifying both initial position u k (0) and initial velocity v k (0). 4. Initial Conditions Determine Geodesic The geodesic equation is a second-order ODE system, so by the existence and uniqueness theorem, a geodesic is uniquely determined by:

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