From 726ec09882e97363dd910eda02f467b35d8c58e0 Mon Sep 17 00:00:00 2001
From: Astrid Giuliani <astrid.giuliani@etu.inp-n7.fr>
Date: Tue, 9 Jul 2024 18:10:56 +0200
Subject: [PATCH] fix pages names and add problem def

---
 docs/make.jl      |  8 ++++----
 docs/src/index.md | 29 ++++++++++++++++++++++++++++-
 2 files changed, 32 insertions(+), 5 deletions(-)

diff --git a/docs/make.jl b/docs/make.jl
index 76fca97..815b1be 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -4,11 +4,11 @@ makedocs(
     sitename = "AveragedKepler",
     format = Documenter.HTML(prettyurls = false),
     pages = [
-        "Home" => "index.md",
+        "Problem definition" => "index.md",
         "3D Visualization" => [
-                               "synthesis.md",
-                               "spheres.md",
-                               "geodesics.md",
+                               "Optimal Synthesis" => "synthesis.md",
+                               "Spheres" => "spheres.md",
+                               "Geodesics" => "geodesics.md",
                               ],
     ]
 )
diff --git a/docs/src/index.md b/docs/src/index.md
index 3359da5..ed2594a 100644
--- a/docs/src/index.md
+++ b/docs/src/index.md
@@ -1 +1,28 @@
-# Averaged Kepler problem
\ No newline at end of file
+# The averaged Kepler problem
+
+We consider the Hamiltonian
+
+$$
+    H(r, \theta, p_r, p_\theta) = v p_\theta + \Vert p \Vert_{g}
+$$
+
+where $v$ is a constant, $p = (p_r, p_\theta)$, and $\Vert \cdot \Vert_{g}$ is the norm induced by the metric
+
+$$
+    g = \mathrm{d}r^2 + m_\lambda^2(r)\, \mathrm{d}\theta^2, \quad 
+    m_\lambda^2(r) = \frac{\sin^2 r}{1 - \lambda \sin^2 r}
+$$
+
+with $\lambda = 4/5$.
+
+Along the geodesics, we have $H+p^0 = 0$. The parameter $p^0$ is constant equal to $-1$ (hyperbolic), $0$ (abnormal) or $1$ (elliptic).
+
+**Remark.** We can parameterize the geodesics by the norm of the initial convector, setting $\Vert{p_0}\Vert_g = 1$. 
+This amounts to parameterize by the initial angle $\alpha_0$:
+$$
+    p_r = \sin \alpha_0, \quad p_\theta = m_\lambda(r) \cos \alpha_0.
+$$
+In that case, the hyperbolic geodeics are given by
+$$
+    p_\theta\, v + 1 = v\, m_\lambda(r) \cos \alpha_0 + 1 > 0.  
+$$
\ No newline at end of file