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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.

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.

An affine parameter is the special choice of the curve parameter $\lambda$ for which the scalar proportionality function $f(\lambda)$ in the tangent evolution equation vanishes identically. Specifically, the tangent vector $\tilde{T}$ satisfies $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$, and the parameter is affine if $f(\lambda)=0$ for all $\lambda$.

Conditions: $\tilde{T}$ is the tangent to the geodesic.; $f(\lambda)$ is a scalar function of the parameter.; The affine case is the subcase $f(\lambda)=0$ for all $\lambda$.

An affine parameter is the special choice of the curve parameter $\lambda$ for which the scalar proportionality function $f(\lambda)$ in the tangent evolution equation vanishes identically. Specifically, the tangent vector $\tilde{T}$ satisfies $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$, and the parameter is affine if $f(\lambda)=0$ for all $\lambda$.

Conditions: $\tilde{T}$ is the tangent to the geodesic.; $f(\lambda)$ is a scalar function of the parameter.; The affine case is the subcase $f(\lambda)=0$ for all $\lambda$.

The Lagrangian $\mathscr{L}$ used to minimize the integral $F$ via the Euler-Lagrange equation is $\mathscr{L} = g_{ij}\dot{x}^i\dot{x}^j$. This is the integrand of the functional $F$.

Conditions: Applied to the integral $F$.; $\dot{x}^i$ and $\dot{x}^j$ are derivatives with respect to $\lambda$.; $g_{ij}$ are metric components.

The Lagrangian $\mathscr{L}$ used to minimize the integral $F$ via the Euler-Lagrange equation is $\mathscr{L} = g_{ij}\dot{x}^i\dot{x}^j$. This is the integrand of the functional $F$.

Conditions: Applied to the integral $F$.; $\dot{x}^i$ and $\dot{x}^j$ are 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.

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.