Let $y$ be a broken extremal for $J[y] = \int_a^b L(x, y, y')\, dx$ with a corner at $c \in (a,b)$. Denote the left and right limits at $c$ by a superscript ${}^-$ and ${}^+$ respectively. Then the following two conditions must hold at $c$: 1. **First Weierstrass–Erdmann condition** (continuity of momentum): \begin{align*} \frac{\partial L}{\partial y'}(c, y(c), y'^-) = \frac{\partial L}{\partial y'}(c, y(c), y'^+). \end{align*} 2. **Second Weierstrass–Erdmann condition** (continuity of Hamiltonian): \begin{align*} \left[L - y' \frac{\partial L}{\partial y'}\right]^- = \left[L - y' \frac{\partial L}{\partial y'}\right]^+, \end{align*} where both sides are evaluated at $(c, y(c))$ with the appropriate one-sided derivative.