Solution Techniques
Solving the geodesic equations means obtaining an exact solution, possibly even the general solution, of the geodesic equations. Most attacks secretly employ the point symmetry group of the system of geodesic equations. This often yields a result giving a family of solutions implicitly, but in many examples does yield the general solution in explicit form.
In general relativity, to obtain timelike geodesics it is often simplest to start from the spacetime metric, after dividing by to obtain the form
where the dot represents differentiation with respect to . Because timelike geodesics are maximal, one may apply the Euler-Lagrange equation directly, and thus obtain a set of equations equivalent to the geodesic equations. This method has the advantage of bypassing a tedious calculation of Christoffel symbols.
Read more about this topic: Solving The Geodesic Equations
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