From 610af37f442bc3b2efefe8ea883de0f654450813 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 21:20:39 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/report.tex b/report.tex
index 57f9ca1..99d139c 100644
--- a/report.tex
+++ b/report.tex
@@ -316,7 +316,15 @@
   \end{frame}
 
   \begin{frame}{Towards first order Z4c}{constraints during the reduction}
-    For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$, where $\gamma > 0$ is a constant. Then the evolution of the constraints becomes
+    For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$,
+    \begin{equation}
+      \begin{cases}
+        \Partial{t}{\phi} = \pi, \\
+        \Partial{t}{\pi} = \tensor{\delta}{^i^j} \Partial{i} \tensor{\psi}{_j},\\
+        \Partial{t}{\tensor{\psi}{_i}} = \Partial{i} \pi - \gamma \mathcal{C}_i,
+      \end{cases}
+    \end{equation}
+    where $\gamma > 0$ is a constant. Then the evolution of the constraints becomes
     \begin{equation}
       \Partial{t} \mathcal{C}_i = -\gamma \mathcal{C}_i,
     \end{equation}