Let’s try to understand this in a bit more detail. The connection coefficients therefore define a notion of differentiation on an arbitrary Riemannian manifold. So there's no general algorithmic procedure (except numerically) for determining the solutions to a classical system without specifying the cases. Connection coefficients, also called Christoffel symbols, are coordinate-dependent coefficients that are needed to specify the Levi-Civita connection. Unfortunately, there's no general method for finding the solutions to ordinary differential equations (coupled in this case) as far as I know. We will leave it as an exercise for you to show that it is possible to consider the conventional action for general relativity but treat it as a function of both the metric g and a torsion-free connection, and the equations of motion derived from varying such an action with respect to the connection imply that is actually the Christoffel. To answer your main question about finding velocity and position vectors, you need to specify what external forces are acting on your system (by specifying the potential $V(x)$) and solve the resultant differential equations, then integrate the components of your acceleration vector. $$ds^\partial_b V$), then read off your Christoffel symbols without having to do needless computations that end up in zeros. An interresting "method" that allows you to know the acceleration vector with respect to any coordinate system is just a matter of recognize some key formulas.ġ) Given the metric of a particlar line elemente of a particular coordinate system.
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