How Do You Derive the Geodesic Equation?

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SUMMARY

The geodesic equation can be derived from the condition that the tangent vector \( U^u \) satisfies the equation \( (U^v)(d/dx^v)(U^u)=0 \), where \( U^u = dx^u/ds \). To make the equation covariant, one must incorporate the Levi-Civita connection, which ensures that the derivative of the tangent vector is parallel transported along the curve. This approach clarifies the relationship between the geometry of the space and the motion of particles within it.

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1. A straight line in flat space may be defined by the equation:
(when I use the ^ symbol in this case it means like upper subscript not to the power)


(U^v)(d/dx^v)(U^u)=0


(U^u=dx^u/ds)

derive the geodesic equation.

Please help I'm completely clueless all I can really see to do is mix the two equations but that doesn't really show me anything more clearly, any help would be greatly appreciated.


 
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also it says to make the equation covariant if that helps
 

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