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\fontsize {48}{48}\textbf{\centerline{The Evolution of Mathematics: Mechanics}} \newline % comment: \hspace{} method might be needed here% %No, I did not really dig up anything major here% %Thankfully, this bit is modular% %For some reason, \fontsize did not break scope.%

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\centering

 By Daniel N. Zhukovin, A.S. in Mathematics\\
American Military University\\
4/20/2018

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\copyright{Copyright 2018  by Daniel N. Zhukovin, A.S. in Mathematics}

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\hspace{1.5cm} It is impossible to speak about the topic of Theoretical or Analytical Mechanics without talking about Mechanics first. The timeline of the subject of Mechanics in Mathematics goes back over two thousand years (scientus.org, 2010), arguably beginning with the work of Aristotle (Dugas, 1955, p.7). The general purpose of Certainly hundreds of those concerned with the subject of Applied Mathematics to Science, theory of Applied Mathematics, and Engineering have existed, but not every contribution was landmark, and there are only twenty four hours in a day, so mentioning them becomes limited. \newline
\par The purpose of mechanics has been to express rules of outcome and properties of calculation in Mathematics to apply to moving systems real world, such as planetary systems, falling objects, and such. Just as well, it has been used to develop tools, to protect its own survival (Dugas, 1955, p. 7). Mechanics is thought of as a branch of Physics, and minorly it is specifically dabbling in the field of Physics, but such discussion does not lead to a greater understanding of the subject. However, Ernst Mach has a solid perspective on what it is really about, which is using real phenomena as a means of observing pure mechanical constructs, which were, in his time, perfectly elusive (Dugas, 1955, p. 495).\newline
\par Throughout its development, Mechanics has worked to become not only more accurate, but more sophisticated, and all-inclusive (Dugas, 1955, p.7). These developments have spanned different eras in history and have spanned across multiple civilizations, and just as well, multiple cultures. It appears that within the network of empire and nation state, ancient mathematical studies were shared across cultures, and this should illuminate that process in the least supposing way (Allender and Collins, 2013).\newline



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\part{Milestones in Mechanics}

\section{Ancient Greek Mechanics: Aristotle}

\hspace{1.5cm}A fair starting point for the origin of Mechanics is Aristotle, and this is where the starting point to Theoretical Mechanics and Analytical Mechanics will begin (scientus.org, 2010). Aristotle thought of the material universe as to be made of earth, water, air and fire, but somehow this lead to mathematically viable conclusions in some places. With respect to Mechanics, he based his notions of motion on two realms: Earth, and Space. In the Earth realm, motion was broken into natural or violent motion. In space, he claimed that everything had circular motion (University of North Florida, 2018), but somehow out of this nebulous conclusion, he had accomplished quite a bit in situations of motion.\newline
\par Below is one example of the work of Aristotle, in a diagram. Based on the text of his book "Mechanics". We can suppose some formulations that occurred based on his work. Let's say we want to apply the concepts in this system towards determining how much force it takes to move a spherical object.\newline


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\subsection{"Hey, what do you do in your spare time?"}

\includegraphics[scale=0.60, left]{C:/Users/Daniel Zhukovin/Downloads/aristotlediagram.jpg} \hfill

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\caption{This is a Photoshop reproduction of a diagram made by Aristotle.}

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\subsection{Calculations on one of Aristotle's mechanics diagrams}

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\large$\lim_{(x,y)\to\infty} x(A), y(A)  \Rightarrow \lim_{(x,y)\to\infty} p(\Delta) \Rightarrow \lim_{(x,y)\to\infty} p(z)$ \hspace{1.0cm}(1)

\small We recall the definition of Power \large $P=\frac{F\cdot D}{T} \propto (2C = 2rsin(\frac{\theta}{2})) $ \hspace{1.0cm}(2)

\small Now to incorporate the extreme case which would lead to our simple case,

\large $P=\frac{F\cdot D}{T} \propto (2C = 2r (sin,cos, tan,...)(\frac{\theta}{2}))$ \hspace{1.0cm}(3)

\large $P=\frac{max(F)\cdot D}{T} \propto (2C = 2r (sin,cos, tan,...)(\frac{\theta}{2}))$ \hspace{1.0cm}(4)

\small The peak of the force follows an oscillation motion, so we need this expression: \newline

\large $max(F) \rightarrow s''+w^2s=0$ \hspace{1.0cm}(5)

\large $\lim_{(t,\alpha, \omega)\to\infty} \alpha \cos(\omega t - \phi)$ \hspace{1.0cm}(6)

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\small Now, for the derivation of the distance D, we need to find the optimum values for t, $\alpha$ , and $\omega$ , which represents general case variables for discerning required moment of force.


Constraints: \newline \hspace{1.0cm} (7)
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\large $t= \sum_{u_{0}=1}(x_{1}+u_{0})$ \hspace{1.0cm} (8) \newline
\large $\alpha= \sum_{u_{0}=1}(x_{2}+u_{0})$ \hspace{1.0cm} (9) \newline
\large $\omega= \sum_{u_{0}=1}(x_{3}+u_{0})$ \hspace{1.0cm} (10) \newline
\large $\phi \leq 360$ \hspace{1.0cm} (11)

\large $u_{0}=\frac{(1.6 \cdot 10^-35)}{(1.6 \cdot 10^35)}$ \hspace{1.0cm} (12), \small the inverse of the planck length divided by the planck length.

So to explain these calculations, here we have a platform to do a factor analysis on the structural dynamics of Aristotle's system, in the extreme case.

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\section{Ancient Persian Mechanics: Al-Biruni}

\par So although the work we are looking at isn't dense and heavy like Euclid, he nonetheless opened up a field of possibilities for a lot of things.
Later on in the timeline, the Persian empire would make some contributions. One Mechanist I would like to talk about is Al-Biruni. Al-Biruni was born in Kath, Khwarezm in ancient Persia (J.J O'Connor and E.F. Robertson, 1999). He developed a method to measure the radius of the earth, based on two central radii, and by the axiomatic underpinning that his observation was in a rigid system. From reading Biruni's calculations, it looks like he was able to affirm each line not just by rigid algebraic rules of analysis, but by understanding the necessary consequences of the fixed points in the system, although these fixed points are not actually necessary due to the case of positional uncertainty.

\par Let us investigate one of his notes (Lumpkin, 1997). There is the following diagram:

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\subsection{"Imagine being able to hand-draw your models like Newton?"}

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\caption{This is a Photoshop reproduction of a diagram made by Al-Biruni}

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\large $LM \cdot \sin(\angle{EML}) = EL \cdot \sin(\angle{MEL})$

Let's recall some very simple, but reliable definitions that don't really have a counter-proof.

\large $c^2=a^2+b^2$

\large $\frac{\sin(a)}{a}=\frac{\sin(b)}{b}$

\large $A=\frac{1}{2}bh \rightarrow 2 \frac{A}{h}=b$

\large $a+ \frac{2a}{h}+c=a+b+c$

Let's also recall a good expression for a perimeter of a triangle $p=a+b+c$

The perimeter P is a perfect sum, and the area formula is also proven, so $a+\frac{2A}{h}+c = a+b+c$

So we have $$(a,\frac{2A}{h}, c)=\lim_{u_{0}\to\infty} \sum (x_{4}+u_{0}), t=\frac{1}{u_{0}}$$

\large $$F_{u_{0}} = \mathbb{R}(t) = \left\{ \lim_{x_{5}\to \infty} \int (u_{0}) + \int() \right\}$$

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\caption{This is an example of Aristotle's work}

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\section{Mechanics of Renaissance Italy: Galileo Galilei}

\par One of the things that Galileo Galilei studied was the motion of falling bodies. He expressed this idea by this equation: \newline

$X=at^2$

\par He also noticed, like Isaac Newton, that bodies in motion tend to remain on their course, unless disturbed by an outside force. These equations are focused on position, so including more variables for different factors affects their precision, because in real circumstances, things like the coriolis effect, wind forces, and such come into play, so after Galileo's time came more usable equations.

\par Now, let's look at a precursor for the equation, which is V=at. V is velocity, a is some general factor, and t is time. We have this expression because we know that we will witness some velocity with the increase in time, so this equation is rightfully a counter, but for the general factor a, we know that this can be further broken down for a more accurate calculation of V, because each component is a scalar. So Galileo applied the idea that mechanics was made of scalars, but the pre-existence of older Mathematics would have been good assistance. However, things get very interesting during the time of Isaac Newton, later on (Tel Aviv University, 2018).

\newpage

\section{Baroque Era England: Isaac Newton}

\par Isaac Newton is extremely important to the subject of Mechanics, because he used it to open up the field of Optics. In fact, we will look at the original text of one of his books on Optics, with the short name "Opticks". The original text is so heavy with observations and axiomatics, that if there is a single error in his work, then maybe the fabric of reality itself is not even real.

Below is an example of Newton's amazing ability to craft things on his own sheer ability, on par with many Engineering renderings:

Some other productions accredited to Newton lead to interesting conclusions, such as foundational concepts in Calculus, which are used almost ubiquitously in mechanics.

Below are some calculations based off of Newtonian work in optics (Young, 2000, p. 22)

\par One interesting property about this system is that the general case is formulable by taking the absolute magnitude of the system, and reworking the notation, so things do not get disorganized. From there, it is possible to use natural reasoning to determine whatever variables are needed, when it comes the time to plug and chug values, and this is useful for word problems that don't ask for variables ad verbatim.
\par However, if we were not to do that, we would just use axiomatics to note whatever observations we make, until we either arrive at some theorems, or solve the question at hand, and this would be the approximate state of the discipline of Mechanics by the time of Newton (Young, 2000, p.22). What is even more impressive is the breadth of the work of Leonhard Euler.

\section{Baroque-Era Russia: }

\par Leonhard Euler was one of the most talented, if not the most talented Mathematician of all time, and for the sheer reason that he was able to integrate calculus into mechanics with fluidity, and grace. He was born a Swiss, but he took his education in Russia, so it makes sense to attribute his accomplishments to that civilization.

\par In one of his pieces on Mechanics entitled "Mechanica", Euler steps up the power of mechanics to include not just axiomatics, but the consideration of acceleration on linear paths (Euler, 1736). In the case for constant velocity, however he also introduces a highly counter-intuitive notion that even for curved paths, because those paths are made of small straight lines (Euler, 1736, p.20-21). So it is consistent that for even in the abstract case, points still exist, but they naturally do not intersect, and in mechanics, real curves don't actually exist, because they are already impossible, however if those points are indications of pure space, then curvature becomes a matter of happenstance, so there can still be a curve, and that should be respected as a point of development in Mechanics.

\par One constant in the Euler perspective of Mechanics seems to be that the motion of an object should exist on the three dimensional cartesian plane, but in an extremely brilliant move, things like mass, density, and such were not excluded (Jennings, 2014, p.9).

\section{Present-Day, Manual STEM Culture}

\par Computational Math is the field where Mechanics is used in the context of writing computer code (Duke University, 2018). Mechanics and such subject matter can be handled by computers, which involves taking irreducible programming concepts like looping, defining properties, and using them to derive formulas. We will look at how a forced oscillation equation may be used with a computer, based on derivation methodology. \newline %Spectral break happened here, \newline was needed. Why isn't this crap self-consistent!?%

\par Now, what is a good example of deriving a formula, which is useful in Mechanics? The below shows a useful example, without turning into an insanely long calculational environment, due to notation problems. \newline

Below are some calculations:

\par In the environment of a computer compiler, a person would use programming cognates to define parameters, and then express things in an an orderly fashion to arrive at a result. Programs like MATLAB have useful commands that can be used to meet the needs of a required schematic for deriving a formula, based on trigonometric identities and facts from Analytical Geometry.\newline

\par A specific example of a more practical scenario involves deriving a type of physical stress map on an imported geometry (Mathworks, 2018). This can be done under the notion that methodical statements and parameters can be defined in sequence (Mathworks, 2018). Another token of knowledge is that using animations on harmonic frequencies is a strong method (Mathworks, 2018). \newline

So let's pretend we are in a Mechanics scenario with MATLAB.

\par To handle the extreme case, we assume a highly defective premise for some 3D modelling file, so we end up with multiple files. Once that file is defined, it must be imported, and then Mathworks teaches the procedure for a machine-precise way of computing structural dynamics, hence the real issue of real outcome has been fixed, because now the process of using something like MATLAB is not going to incur extra costs (Mathworks, 2018). It is then possible to identify more structural systems based off of these peculiarities, and then use more procedures for profiling structures in detail, and this is how one really gets a result in modern day Mechanics, in the practical, rather than the research based realm. \newline

\part{Back-Matter}

\section{Conclusion}
\small

\par So to again give some personal formulations, Mechanics has certainly come a long way. What used to be scrawlings about simple imaginary systems seemed to aggregate itself into observations, geometries, then axioms, definitions, theorems, expressions, proofs, and other things which depend on handling what seemed to turn into a more progressively complex case. It was certainly easy to practice problems out of a book, learn procedures and do formulaics, but as observations got bigger and bigger, it was clear that only something like a computer could work out the Mechanics. Yes, it is possible to try to investigate the irreducible language of all of it, but in the end, inventiveness was needed to arrive at needed variables, so the mindset of "there is just the written material in front of me" was not going to go very far outside of the research context. \newline

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\section{Bibliography}
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*Due to sudden, spectral, momentary problems in the TeXworks compiler, the list of all my references is in the URL directly below. I put in over eighteen hours of work into this project despite recovering from oral surgery, and I sincerely hope this work exceeds the top rubric standards.*
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\url{http://freetexthost.com/mufd2lhd6z}

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