Add augment=true documentation and improve mathematical notation

- Add comprehensive augment=true section in manual-flow-ocp.md
- Replace simple example with harmonic oscillator (parameter in dynamics)
- Document mathematical background, usage, advantages, and error handling
- Use \mathrm{d} for all differentials (mathematical rigor)
- Update example-control-free.md with proper differential notation
- Update manual-differential-geometry.md with \mathrm{d} notation
- Remove @meta Draft blocks (use global draft setting)
- Restore private API documentation file

Author: ocots

docs/src/api/private.md

@@ -0,0 +1,244 @@
+```@meta
+EditURL = nothing
+```
+
+# Private API
+
+This page lists **non-exported** (internal) symbols of `OptimalControl`.
+
+
+---
+
+### From `OptimalControl`
+
+
+## `DescriptiveMode`
+
+```@docs
+OptimalControl.DescriptiveMode
+```
+
+
+## `ExplicitMode`
+
+```@docs
+OptimalControl.ExplicitMode
+```
+
+
+## `SolveMode`
+
+```@docs
+OptimalControl.SolveMode
+```
+
+
+## `_DEFAULT_DISPLAY`
+
+```@docs
+OptimalControl._DEFAULT_DISPLAY
+```
+
+
+## `_DEFAULT_INITIAL_GUESS`
+
+```@docs
+OptimalControl._DEFAULT_INITIAL_GUESS
+```
+
+
+## `_INITIAL_GUESS_ALIASES`
+
+```@docs
+OptimalControl._INITIAL_GUESS_ALIASES
+```
+
+
+## `_INITIAL_GUESS_ALIASES_ONLY`
+
+```@docs
+OptimalControl._INITIAL_GUESS_ALIASES_ONLY
+```
+
+
+## `_ansi_color`
+
+```@docs
+OptimalControl._ansi_color
+```
+
+
+## `_ansi_reset`
+
+```@docs
+OptimalControl._ansi_reset
+```
+
+
+## `_build_components_from_routed`
+
+```@docs
+OptimalControl._build_components_from_routed
+```
+
+
+## `_build_or_use_strategy`
+
+```@docs
+OptimalControl._build_or_use_strategy
+```
+
+
+## `_build_partial_description`
+
+```@docs
+OptimalControl._build_partial_description
+```
+
+
+## `_build_partial_tuple`
+
+```@docs
+OptimalControl._build_partial_tuple
+```
+
+
+## `_build_source_tag`
+
+```@docs
+OptimalControl._build_source_tag
+```
+
+
+## `_complete_components`
+
+```@docs
+OptimalControl._complete_components
+```
+
+
+## `_complete_description`
+
+```@docs
+OptimalControl._complete_description
+```
+
+
+## `_descriptive_action_defs`
+
+```@docs
+OptimalControl._descriptive_action_defs
+```
+
+
+## `_descriptive_families`
+
+```@docs
+OptimalControl._descriptive_families
+```
+
+
+## `_determine_parameter_display_strategy`
+
+```@docs
+OptimalControl._determine_parameter_display_strategy
+```
+
+
+## `_explicit_or_descriptive`
+
+```@docs
+OptimalControl._explicit_or_descriptive
+```
+
+
+## `_extract_action_kwarg`
+
+```@docs
+OptimalControl._extract_action_kwarg
+```
+
+
+## `_extract_kwarg`
+
+```@docs
+OptimalControl._extract_kwarg
+```
+
+
+## `_extract_strategy_parameters`
+
+```@docs
+OptimalControl._extract_strategy_parameters
+```
+
+
+## `_has_complete_components`
+
+```@docs
+OptimalControl._has_complete_components
+```
+
+
+## `_print_ansi_styled`
+
+```@docs
+OptimalControl._print_ansi_styled
+```
+
+
+## `_print_component_with_param`
+
+```@docs
+OptimalControl._print_component_with_param
+```
+
+
+## `_route_descriptive_options`
+
+```@docs
+OptimalControl._route_descriptive_options
+```
+
+
+## `_unwrap_option`
+
+```@docs
+OptimalControl._unwrap_option
+```
+
+
+## `display_ocp_configuration`
+
+```@docs
+OptimalControl.display_ocp_configuration
+```
+
+
+## `get_strategy_registry`
+
+```@docs
+OptimalControl.get_strategy_registry
+```
+
+
+## `solve_descriptive`
+
+```@docs
+OptimalControl.solve_descriptive
+```
+
+
+## `solve_explicit`
+
+```@docs
+OptimalControl.solve_explicit
+```
+
+
+## `will_solver_print`
+
+```@docs
+OptimalControl.will_solver_print
+```
+

docs/src/example-control-free.md

@@ -26,7 +26,7 @@ Consider a system with exponential growth:
 where $\lambda$ is an unknown growth rate parameter. We have observed data with some perturbations and want to estimate $\lambda$ by minimizing the squared error:
 
 ```math
-\min_{\lambda} \int_0^{2} (x(t) - x_{\text{obs}}(t))^2 \, dt
+\min_{\lambda} \int_0^{2} (x(t) - x_{\text{obs}}(t))^2 \, \mathrm{d}t
 ```
 
 The underlying model has $\lambda = 0.5$, but the observed data includes perturbations.
@@ -94,17 +94,17 @@ To handle the variable parameter $\lambda$, we treat it as an additional state w
 
 ```math
 \begin{aligned}
-\frac{dx}{dt} &= \frac{\partial H}{\partial p} = \lambda x \\
-\frac{d\lambda}{dt} &= 0 \quad \text{(constant parameter)} \\
-\frac{dp}{dt} &= -\frac{\partial H}{\partial x} = -p\lambda + 2(x - x_{\text{obs}}(t)) \\
-\frac{dp_\lambda}{dt} &= -\frac{\partial H}{\partial \lambda} = -px
+\frac{\mathrm{d}x}{\mathrm{d}t} &= \frac{\partial H}{\partial p} = \lambda x \\
+\frac{\mathrm{d}\lambda}{\mathrm{d}t} &= 0 \quad \text{(constant parameter)} \\
+\frac{\mathrm{d}p}{\mathrm{d}t} &= -\frac{\partial H}{\partial x} = -p\lambda + 2(x - x_{\text{obs}}(t)) \\
+\frac{\mathrm{d}p_\lambda}{\mathrm{d}t} &= -\frac{\partial H}{\partial \lambda} = -px
 \end{aligned}
 ```
 
 The transversality condition for the variable parameter requires $p_\lambda(t_f) - p_\lambda(t_0) = 0$. Assuming $p_\lambda(t_0) = 0$, we have to satisfy:
 
 ```math
-p_\lambda(t_f) = -\int_{t_0}^{t_f} \frac{\partial H}{\partial \lambda}(t, x(t), p(t), \lambda) \, dt = 0
+p_\lambda(t_f) = -\int_{t_0}^{t_f} \frac{\partial H}{\partial \lambda}(t, x(t), p(t), \lambda) \, \mathrm{d}t = 0
 ```
 
 We use CTFlows' `augment=true` feature to automatically compute $p_\lambda(t_f)$ without manually constructing the augmented system.
@@ -271,12 +271,12 @@ To handle the variable parameter $\omega$, we treat it as an additional state wi
 
 ```math
 \begin{aligned}
-\frac{dq}{dt} &= \frac{\partial H}{\partial p_1} = v \\
-\frac{dv}{dt} &= \frac{\partial H}{\partial p_2} = -\omega^2 q \\
-\frac{d\omega}{dt} &= 0 \quad \text{(constant parameter)} \\
-\frac{dp_1}{dt} &= -\frac{\partial H}{\partial q} = \omega^2 p_2 \\
-\frac{dp_2}{dt} &= -\frac{\partial H}{\partial v} = -p_1 \\
-\frac{dp_\omega}{dt} &= -\frac{\partial H}{\partial \omega} = 2\omega q p_2
+\frac{\mathrm{d}q}{\mathrm{d}t} &= \frac{\partial H}{\partial p_1} = v \\
+\frac{\mathrm{d}v}{\mathrm{d}t} &= \frac{\partial H}{\partial p_2} = -\omega^2 q \\
+\frac{\mathrm{d}\omega}{\mathrm{d}t} &= 0 \quad \text{(constant parameter)} \\
+\frac{\mathrm{d}p_1}{\mathrm{d}t} &= -\frac{\partial H}{\partial q} = \omega^2 p_2 \\
+\frac{\mathrm{d}p_2}{\mathrm{d}t} &= -\frac{\partial H}{\partial v} = -p_1 \\
+\frac{\mathrm{d}p_\omega}{\mathrm{d}t} &= -\frac{\partial H}{\partial \omega} = 2\omega q p_2
 \end{aligned}
 ```
 
@@ -289,7 +289,7 @@ p_\omega(t_f) - p_\omega(t_0)= -\frac{\partial g}{\partial \omega} = -2\omega
 Assuming $p_\omega(t_0) = 0$, we have:
 
 ```math
-p_\omega(t_f) = -\int_{t_0}^{t_f} \frac{\partial H}{\partial \omega}(t, x(t), p(t), \omega) \, dt = -2\omega
+p_\omega(t_f) = -\int_{t_0}^{t_f} \frac{\partial H}{\partial \omega}(t, x(t), p(t), \omega) \, \mathrm{d}t = -2\omega
 ```
 
 We use CTFlows' `augment=true` feature to automatically compute $p_\omega(t_f)$ without manually constructing the augmented system:

docs/src/manual-differential-geometry.md

@@ -607,7 +607,7 @@ dg(3, [1, 2], [4, 5])
 For a non-autonomous Hamiltonian $H(t, x, p)$ and a function $G(t, x, p)$, the **total time derivative** along the Hamiltonian flow is:
 
 ```math
-\frac{d}{dt} G(t, x(t), p(t)) = \partial_t G + \{H, G\}.
+\frac{\mathrm{d}}{\mathrm{d}t} G(t, x(t), p(t)) = \partial_t G + \{H, G\}.
 ```
 
 This is the sum of: