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The geodesic equation is obtained by applying the Euler-Lagrange equation to the Lagrangian $\mathscr{L} = g_{ij}\dot{x}^i\dot{x}^j$. The resulting differential equation for the curve components is $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$, where $\Gamma^k_{ij}$ are the connection coefficients.

Conditions: $\lambda$ is an affine parameter.; $\Gamma^k_{ij}$ are connection coefficients.; The equation is presented as the general solution of the Euler-Lagrange step for F.

The geodesic equation is obtained by applying the Euler-Lagrange equation to the Lagrangian $\mathscr{L} = g_{ij}\dot{x}^i\dot{x}^j$. The resulting differential equation for the curve components is $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$, where $\Gamma^k_{ij}$ are the connection coefficients.

Conditions: $\lambda$ is an affine parameter.; $\Gamma^k_{ij}$ are connection coefficients.; The equation is presented as the general solution of the Euler-Lagrange step for F.