docs: "Strong Angular System" chapter added

docs/users_guide/systems/strong_angular_system.md

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+# Strong Angular System
+
+## Some background information
+
+As per today's SI, both radian and steradian are dimensionless. This forces the convention to set the angle `1 radian`
+equal to the number `1` within equations (similar to what natural units system does for `c` constant).
+
+Following [Wikipedia](https://en.wikipedia.org/wiki/Radian#Dimensional_analysis):
+
+!!! quote "Wikipedia: Radian - Dimensional analysis"
+
+    Giacomo Prando says "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian
+    in the dimensional analysis of physical equations." For example, a mass hanging by a string from a pulley will rise or drop
+    by $y=rθ$ centimeters, where $r$ is the radius of the pulley in centimeters and $θ$ is the angle the pulley turns in radians.
+    When multiplying $r$ by $θ$ the unit of radians disappears from the result.
+    Similarly in the formula for the angular velocity of a rolling wheel, $ω=v/r$, radians appear in the units of
+    $ω$ but not on the right hand side.
+    Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical
+    advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention
+    and contextual knowledge is "pedagogically unsatisfying".
+
+At least a dozen scientists have made proposals to treat the radian as a base unit of measure defining its own dimension of "angle",
+as early as 1936 and as recently as 2022. This would bring the advantages of a physics-based, consistent, and logically-robust
+unit system, with unambiguous units for all physical quantities. At the same time the only notable changes for typical
+end-users would be: improved units for the quantities _torque_, _angular momentum_, and _moment of inertia_.
+
+Paul Quincey in his proposal [_"Angles in the SI: a detailed proposal for solving the problem"_](../../appendix/references.md#Quincey) states:
+
+!!! quote "Paul Quincey: Angles in the SI: a detailed proposal for solving the problem"
+
+    The familiar units assigned to some angular quantities are based on equations that have adopted the radian convention,
+    and so are missing `rad`s that would be present if the complete equation is used. The physically-correct units are
+    those with the `rad`s reinstated. Numerical values would not change, and the physical meanings of all quantities would
+    also be unaffected.
+
+He proposes the following changes:
+
+- The **radian** would become either a new base unit or a 'complementary' unit
+- The **steradian** would become a derived unit equal to $1\:rad^2$
+- The SI units for
+
+    - _Torque_ would change from $N\:m$ ($=J$) to $J\:rad^{-1}$
+    - _Angular momentum_ would change from $J\:s$ to $J\:s\:rad^{-1}$ (i.e. $J/(rad/s)$)
+    - _Moment of inertia_ would change from $kg\:m^2$ to $kg\:m^2\:rad^{-2}$ (i.e. $J/(rad/s)^2$)
+
+- The option to omit the radian from the SI units for _angle_, _angular velocity_, _angular frequency_,
+  _angular acceleration_, and _angular wavenumber_ would be removed, the only correct SI units being
+  $rad$, $rad/s$, $rad/s$, $rad/s^2$ and $rad/m$ respectively.
+
+Paul Quincey summarizes that with the above in action:
+
+!!! quote "Paul Quincey: Angles in the SI: a detailed proposal for solving the problem"
+
+    However, the physical clarity this would build into the SI should be recognised very quickly. The units would tell us that
+    $torque \times angle = energy$, and $angular\:momentum \times angle = action$, for example, in the same way that they do for
+    $force \times distance = energy$, $linear\:momentum \times distance = action$, and
+    $radiant\:intensity \times solid\:angle = radiant\:flux$.
+    Dimensional analysis could be used to its full extent. Software involving angular quantities would be rationalised.
+    Arguments about the correct units for frequency and angular frequency, and the meaning of the unit $Hz$, could be left behind.
+    The explanation of these changes would be considerably easier and more rewarding than explaining how a kilogram-sized mass
+    can be measured in terms of the Planck constant.
+
+
+## Angular quantities in the SI
+
+Even though the SI somehow ignores the dimensionality of angle:
+
+!!! quote "SI Brochure"
+
+    Plane and solid angles, when expressed in radians and steradians respectively, are in effect
+    also treated within the SI as quantities with the unit one. The symbols $rad$ and $sr$
+    are written explicitly where appropriate, in order to emphasize that, for radians or
+    steradians, the quantity being considered is, or involves the plane angle or solid angle
+    respectively. For steradians it emphasizes the distinction between units of flux and intensity
+    in radiometry and photometry for example. However, it is a long-established practice in
+    mathematics and across all areas of science to make use of $rad = 1$ and $sr = 1$.
+    For historical reasons the radian and steradian are treated as derived units.
+
+It also explicitly states:
+
+!!! quote "SI Brochure"
+
+    The SI unit of frequency is hertz, the SI unit of angular velocity and angular frequency is
+    radian per second, and the SI unit of activity is becquerel, implying counts per second.
+    Although it is formally correct to write all three of these units as the reciprocal second, the
+    use of the different names emphasizes the different nature of the quantities concerned. It is
+    especially important to carefully distinguish frequencies from angular frequencies, because
+    by definition their numerical values differ by a factor of $2\pi$. Ignoring this fact may cause
+    an error of $2\pi$. Note that in some countries, frequency values are conventionally expressed
+    using “cycle/s” or “cps” instead of the SI unit $Hz$, although “cycle” and “cps” are not units
+    in the SI. Note also that it is common, although not recommended, to use the term
+    frequency for quantities expressed in $rad/s$. Because of this, it is recommended that
+    quantities called “frequency”, “angular frequency”, and “angular velocity” always be given
+    explicit units of $Hz$ or $rad/s$ and not $s^{-1}$.
+
+
+## Strong Angular extensions in the library
+
+The mp-units library strives to define physically-correct quantities and their units to provide maximum help
+to its users. As treating angle as a dimensional quantity can lead to many "trivial" mistakes in dimensional
+analysis and calculation, it was decided to provide additional experimental systems of quantities and units
+that follow the above approach and treat _angle_ as a base quantity with a base unit of radian and
+_solid angle_ as its derived quantity.
+
+As those (at least for now) are not a part of SI, the _plain angle_ and _solid angle_ definitions can be found
+in a dedicated `angular` system. Those definitions are also used in the `isq_angle` system of quantities
+to make the recipes for angle-based quantities like _torque_ or _angular velocity_ physically correct:
+
+```cpp
+using namespace mp_units;
+using namespace mp_units::si::unit_symbols;
+using mp_units::angular::unit_symbols::deg;
+using mp_units::angular::unit_symbols::rad;
+
+const quantity lever = isq_angle::position_vector(20 * cm);
+const quantity force = isq_angle::force(500 * N);
+const quantity angle = isq_angle::angular_measure(90. * deg);
+const quantity torque = isq_angle::torque(lever * force * angular::sin(angle) / (1 * isq_angle::cotes_angle));
+
+std::cout << "Applying a perpendicular force of " << force << " to a " << lever << " long lever results in "
+          << torque.in(N * m / rad) << " of torque.\n";
+```
+
+The above program prints:
+
+```text
+Applying a perpendicular force of 500 N to a 20 cm long lever results in 100 N m/rad of torque.
+```
+
+!!! note
+
+    `cotes_angle` is a constant which represents an angle with the value of exactly `1 radian`.
+    You can find more information about this constant in [Quincey](../../appendix/references.md#Quincey).
+
+!!! example "[Try it on Compiler Explorer](https://godbolt.org/z/qfxosv9YM)"

mkdocs.yml

@@ -151,12 +151,12 @@ nav:
           - Obtaining Metadata: users_guide/framework_basics/obtaining_metadata.md
           - Concepts: users_guide/framework_basics/concepts.md
           - Text Output: users_guide/framework_basics/text_output.md
-      - Defining Systems:
-          - Introduction: users_guide/defining_systems/introduction.md
-          - International System of Quantities (ISQ): users_guide/defining_systems/isq.md
-          - International System of Units (SI): users_guide/defining_systems/si.md
-          - Strong Angular System: users_guide/defining_systems/strong_angular_system.md
-          - Natural Units: users_guide/defining_systems/natural_units.md
+      - Systems:
+          - Introduction: users_guide/systems/introduction.md
+          - International System of Quantities (ISQ): users_guide/systems/isq.md
+          - International System of Units (SI): users_guide/systems/si.md
+          - Strong Angular System: users_guide/systems/strong_angular_system.md
+          - Natural Units: users_guide/systems/natural_units.md
       - Use Cases:
           - Pure Dimensional Analysis: users_guide/use_cases/pure_dimensional_analysis.md
           - Working with Legacy Interfaces: users_guide/use_cases/working_with_legacy_interfaces.md