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The length functional $L$ between endpoints $A$ and $B$ is defined as the integral of the square root of the squared norm of the tangent vector with respect to the parameter $\lambda$: $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}\,d\lambda$. Minimizing this functional yields the geodesic curves.

Conditions: Endpoints are $A$ and $B$.; $\dot{x}^i$ and $\dot{x}^j$ denote derivatives with respect to $\lambda$.; $g_{ij}$ are metric components.

The length functional $L$ between endpoints $A$ and $B$ is defined as the integral of the square root of the squared norm of the tangent vector with respect to the parameter $\lambda$: $L = \int_A^B \sqrt{g_{ij}\dot{x}^i\dot{x}^j}\,d\lambda$. Minimizing this functional yields the geodesic curves.

Conditions: Endpoints are $A$ and $B$.; $\dot{x}^i$ and $\dot{x}^j$ denote derivatives with respect to $\lambda$.; $g_{ij}$ are metric components.

A geodesic is treated as a curve in space-time and is written componentwise as functions of a parameter $\lambda$. Every space-time component of the curve can be expressed as $x^\mu = x^\mu(\lambda)$, which converts the geometric object into coordinate functions suitable for differentiation.

Conditions: $\lambda$ is the chosen curve parameter.; Each space-time component $x^\mu$ is expressed as a function of $\lambda$.

A geodesic is treated as a curve in space-time and is written componentwise as functions of a parameter $\lambda$. Every space-time component of the curve can be expressed as $x^\mu = x^\mu(\lambda)$, which converts the geometric object into coordinate functions suitable for differentiation.

Conditions: $\lambda$ is the chosen curve parameter.; Each space-time component $x^\mu$ is expressed as a function of $\lambda$.