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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.