Simplify initial guess in tutorial-mam by inlining helper functions

Author: ocots

docs/src/tutorial-mam.md

@@ -1,5 +1,9 @@
 # [Minimal Action Method using Optimal Control](@id tutorial-mam)
 
+```@meta
+Draft = false
+```
+
 The Minimal Action Method (MAM) is a numerical technique for finding the most probable transition pathway between stable states in stochastic dynamical systems. It achieves this by minimizing an action functional that represents the path's deviation from the deterministic dynamics, effectively identifying the path of least resistance through the system's landscape.
 
 This tutorial demonstrates how to implement MAM as an optimal control problem, using the classical Maier-Stein model as a benchmark example.
@@ -17,7 +21,7 @@ using Plots, Printf
 We aim to find the most probable transition path between two stable states of a stochastic dynamical system. For a system with deterministic dynamics $f(x)$ and small noise, the transition path minimizes the action functional:
 
 ```math
-S[x(\cdot), u(\cdot)] = \int_0^T \|u(t) - f(x(t))\|^2 \, dt
+S[x(\cdot), u(\cdot)] = \int_0^T \|u(t) - f(x(t))\|^2 \, \mathrm{d}t
 ```
 
 subject to the path dynamics:
@@ -72,17 +76,10 @@ We provide an initial guess for the path using a simple interpolation with the `
 # Time horizon
 T = 50
 
-# Helper functions for initial state guess
-L(t) = -(1 - t/T) + t/T      # Linear interpolation from -1 to 1
-P(t) = 0.3*(-L(t)^2 + 1)     # Parabolic arc (approximates saddle crossing)
-
 init = @init ocp(T) begin
-    # Linear interpolation for x₁
-    x₁(t) := L(t)
-    # Parabolic guess for x₂
-    x₂(t) := P(t)
-    # Control from vector field
-    u(t) := f(L(t), P(t))
+    x₁(t) := -(1 - t/T) + t/T       # Linear interpolation from -1 to 1
+    x₂(t) := 0.3*(-x₁(t)^2 + 1)     # Parabolic arc (approximates saddle crossing)
+    u(t) := f(x₁(t), x₂(t))         # Control from vector field
 end
 nothing # hide
 ```