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Minimizing the length functional $L$ is described as tricky. When $\lambda$ is an affine parameter, minimizing the simpler functional $F = \int_A^B g_{ij}\dot{x}^i\dot{x}^j\,d\lambda$ is equivalent to minimizing $L$ and is much easier.

Conditions: $\lambda$ must be an affine parameter.; $F$ is used as an easier surrogate for $L$ under that condition.

Minimizing the length functional $L$ is described as tricky. When $\lambda$ is an affine parameter, minimizing the simpler functional $F = \int_A^B g_{ij}\dot{x}^i\dot{x}^j\,d\lambda$ is equivalent to minimizing $L$ and is much easier.

Conditions: $\lambda$ must be an affine parameter.; $F$ is used as an easier surrogate for $L$ under that condition.