+
+Thus the $\LAM$-dependent constraint equation is
+\beq
+g_k = |\ve{r}_{k}| - \left(\LL d_{k}^A + \LAM d_k^B\right).
+\eeq
+
+The (zero) contribution $G$ to the Hamiltonian from the constraints
+(using Lagrange multipliers $\lambda_k$, which are logically distinct
+from the free-energy $\LAM$) is