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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.