The Geodesic Equation
On an n-dimensional Riemannian manifold, the geodesic equation written in a coordinate chart with coordinates is:
where the coordinates xa(s) are regarded as the coordinates of a curve γ(s) in and are the Christoffel symbols. The Christoffel symbols are functions of the metric and are given by:
where the comma indicates a partial derivative with respect to the coordinates:
As the manifold has dimension, the geodesic equations are a system of ordinary differential equations for the coordinate variables. Thus, allied with initial conditions, the system can, according to the Picard–Lindelöf theorem, be solved. One can also use a Lagrangian approach to the problem: defining
and applying the Euler-Lagrange equation.
Read more about this topic: Solving The Geodesic Equations
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