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Trajectory By Christoffel Symbols

Physics Asked by user285989 on March 5, 2021

So, basically Christoffel Symbols gives us the components of the vector that represents the rate of change of the basis vectors in a manifold (surface).

And this equation below tells the rate of change of the vector components of a velocity vector summarized by a given object in a manifold where the Christoffel Symbols != 0,

$$frac{dv^a}{dt} = -Gamma_{mn}^a v^m v^n$$

Let me be clear with simple terms, $v^m$ and $v^n$ are vector components respectively such that $v^1$ is the x-component, $v^2$ is the y-component of the velocity vector simultaneously.
And $Gamma$ with indices is the Christoffel Symbol, if you are interested with its derivation I will leave a link below for a video that explains step by step derivation and explanation.

My question is that will this equation hold true for in 3 spatial dimensions where the velocity vector has three components in each direction respectively?

Or it will take the form,

$$frac{dv^a}{dt} = -Gamma_{m n}^a v^m v^nv^{i}$$

And $a, m,n,i$ can take value $1,2,3$.
The link to the video:

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