From 716157f80d383ea963df6cb2b6da2540a06f4b4f Mon Sep 17 00:00:00 2001
From: LM <hi@leima.is>
Date: Mon, 15 Apr 2024 23:31:27 +0200
Subject: [PATCH] docs(harmonic-oscillator): adjust tutorials to include ations
 or equation of motion

#49
---
 docs/tutorials/harmonic_oscillator.py   | 20 ++++++++++++++++++++
 hamilflow/models/harmonic_oscillator.py |  2 +-
 2 files changed, 21 insertions(+), 1 deletion(-)

diff --git a/docs/tutorials/harmonic_oscillator.py b/docs/tutorials/harmonic_oscillator.py
index 3ee81af..9bf2db7 100644
--- a/docs/tutorials/harmonic_oscillator.py
+++ b/docs/tutorials/harmonic_oscillator.py
@@ -33,6 +33,18 @@
 
 # %% [markdown]
 # ## Simple Harmonic Oscillator
+#
+# For an simple harmonic oscillator, the action of a simple harmonic oscillator is
+#
+# $$S_L[x] = \int_{t_0}^{t_1} \mathbb{d}t \left\{\frac{1}{2} m \dot x^2 - \frac{1}{2} m \omega^2 x^2 \right\}\,,$$
+#
+# where the least action principle leads to the following equation of motion,
+#
+# $$
+# \ddot x + \omega^2 x = 0\,.
+# $$
+#
+# A simple harmonic oscillator is a periodic motion.
 
 # %%
 sho_omega = 0.5
@@ -57,6 +69,14 @@
 
 # %% [markdown]
 # ## Damped Harmonic Oscillator
+#
+# A damped harmonic oscillator is a simple harmonic oscillator with damping force that is proportional to its velocity,
+#
+# $$
+# \ddot x + \omega^2 x = - 2\xi\omega \dot x\,.
+# $$
+#
+# In this section, we demonstrate three scenarios of a damped harmonic oscillator.
 
 # %%
 dho_systems = {
diff --git a/hamilflow/models/harmonic_oscillator.py b/hamilflow/models/harmonic_oscillator.py
index bda660d..072d869 100644
--- a/hamilflow/models/harmonic_oscillator.py
+++ b/hamilflow/models/harmonic_oscillator.py
@@ -174,7 +174,7 @@ def _x(self, t: ArrayLike) -> ArrayLike:
 
 
 class DampedHarmonicOscillator(HarmonicOscillatorBase):
-    r"""Generate time series data for a [simple harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator).
+    r"""Generate time series data for a [damped harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator).
 
     The equation for a general un-driven harmonic oscillator is[^wiki_ho][^libretext_ho]