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\begin{document}
\title{Classical Mechanics: Fundamentals}
\author[1]{Mario Cezar Bertin}%
\affil[1]{Instituto de FĂsica, Universidade Federal da Bahia}%
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\date{\today}
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\section*{Introduction}
{\label{357463}}
The main objective of these notes is the construction of a point of view
of classical mechanics, a point of view that allows a closer connection
with modern and contemporary physics. The two main theoretical
approaches for the understanding of nature since the beginning of the XX
century are General Relativity and Quantum Mechanics. The first is the
set of theories for high speeds and massive structures of the universe.
The later is the set of theories that allows us to approach beyond the
microscopic phenomena. Both worlds are not disconnected, since high
energy physics, as the physics done by the Large Hadron Collider,
rellies on relativistic quantum field theory, which is a quantum
mechanical theory set up by special relativity.
\par\null
Rich media available at \url{https://youtu.be/0lgs1vTzS2E}
Classical mechanics, of course, is a full discipline by itself. It can
be learned and developed without reference to any other physical theory.
Dynamical systems, orbital dynamics, chaos, and fractal dynamics are
examples of modern developments. On the other hand, we live in a world
that has molecular and atomic physics, quantum mechanics, quantum field
theory, general and special relativity, quantum information. Our
fundamental understanding of nature is coded in the standard model of
elementary particles, and the theory of the Big Bang. It is not only a
waste to study classical mechanics as a separate subject, but it is
imperative to understand how all other physical theories rely on
classical mechanics in fundamentals and techniques.
Here we explore two main ideas. The first one is that physics is the
science that maps natural phenomena into mathematical structures. The
other idea is that a physical system is only as definable as it can be
measured, and the information that defines a system is the one all
observers in a specified class of observers agree.
\section*{The particle}
{\label{274003}}\par\null
Rich media available at \url{https://youtu.be/pelK0Nn-RZc}
Some concepts that gave origin to classical mechanics are just too
classical. We must have them here even if we came to suffer
significantly to get rid of them when necessary. And the most basic of
all classical concepts is~\textbf{the particle.}
In classical mechanics, a particle is an eternal smallest element. It is
a brick that builds everything that exists. Particles are the building
blocks of any physical system. A particle has no size, no internal
structure, but has some information attached. Typically, the information
that comes with the particle is the~\textbf{mass}, but it can also
include other observables, as the~\textbf{electric charge}, the
\textbf{spin}, or additional internal charges. By eternal, we mean that
a particle cannot be created or destroyed. A particle exists.
Of course, this concept of a particle does not have any real
correspondence in nature. The real world has extended bodies, formed by
molecules, atoms of many types, which are formed by electrons, protons,
and neutrons. Protons and neutrons are then formed by other basic
structures known as quarks. Electrons are elementary by themselves.
However, quarks and electrons cannot be described as classical
particles, because eventually, this concept will not be sufficient to
specify the way they exist. To understand if we are found in the domain
of classical physics, we should learn if the above concept of a particle
is, at least, approximately accurate to experiment. If this is the case,
we may address the electron as a classical particle. Sometimes the
classical particle is a sufficient concept.
The set of information that comes with the particle depends on the
specific physical system of interest. Mass is always one of the
quantities, and it is related to the concept of inertia, which we will
explore in moments. An electron, for example, has other defining
quantities; the electric charge, and the spin. If we are treating an
electron as a classical particle, these measures must be the defining
properties of the electron. In this case, we have the
values~\(m_e\approx9,109\cdot10^{-31}\ Kg\), about~\(1836\) times lighter than
the proton,~\(q_e\approx-1.602\cdot10^{-19}\ \text{coulomb}\), which is the elementary electric charge,
and an intrinsic angular momentum, or spin, of~\(1/2\). No
other particle has the same characteristics, and all observers must
agree with them.
The origin of mass, charge, and spin can only be explained by the
relativistic quantum field theory. Therefore, in classical mechanics,
these values must be postulated. But they are the first examples of what
is called~\textbf{dynamical invariants}. The values
of~\(\left(m_e,q_e,s_e\right)\) are always the same for the electron, as for any
other particle, and they never change.
Another fundamental particle in nature is the photon, the particle of
light and electromagnetic radiation. For centuries the debate about the
nature of light opposed the particle and the wave points of view. Today,
we understand the light fundamentally as a field, which can be made a
particle when it reaches a detector. Still, it is also a wave when
interacting with slits to form interference phenomena. The photon is of
little use in classical mechanics since it has zero mass and zero
electric charge. The photon does not have a spin value, but it has, on
the other hand, a value called~\textbf{helicity}, which gives rise to
its polarization properties.
\section*{Interaction and movement}
{\label{960671}}
We know movement should be part of the classical mechanical description
because changes in the movement state of objects are part of our
everyday lives. How can we accommodate the concept in our theory?
First, we recognize that a universe with a single particle cannot
present movement states for the particle. Therefore,~\textbf{the
movement~}must be a property of a system of two or more particles. If we
have a universe with two particles, we must allow the particles to
interact, in this case, to change their respective states of movement.
In the real world, we know by experiment that two particles with values
of mass do interact by gravitation. We also know that two particles with
electric charge interact by electric and magnetic fields.
A single particle that does not interact with any other particle is
a~\textbf{free particle}. Ideally, we may have a system of two or more
free particles, i.e., particles that form a mechanical system but do not
interact with each other, but this would be a very uninteresting system.
We may have a system with several particles that interact among
themselves but do not interact with other particles or systems; in this
case, we call this a~\textbf{closed system}, or an~\textbf{isolated
system}.
We may always separate a system of particles, and form sub-systems, by
using a definite criterium. For example, a system
with~\(n\in\mathbb{N}\) particles, each with a mass~\(m_1\),
and another system with~\(k\in\mathbb{N}\) particles, each with a
mass~\(m_2\), may be seen as two sub-systems of a larger
system with~\(n+k\) particles with distinct masses. In this
case, a system with~\(n\) particles may always be seen as
\(n\) systems, each with a single particle.
\section*{The observer, the measurement, and the
observable}
{\label{534013}}
The~\textbf{observer} is a physical system by itself, which possesses
rulers, clocks, or any other measurement apparatus. The job of the
observer and its tools is to collect information about other physical
systems. We call each information possible to be collected from a system
a~\textbf{measure}. We also use the word~\textbf{measurement} for the
act of obtaining a measure.
A measure must refer to a specific characteristic of the system. For
example, some curious mind could wonder about the distribution of eye
colors in a system of~\(n\) human beings. The colors could
be brown, blue, and green, and these are the possible measures of the
measurement. At the end of all measurements,~\(m\) humans
will have brown eyes,~\(k\) humans will have blue eyes,
and~\(n-m-k\) humans will present green eyes. The eye color is
the characteristic that has been measured by the observer, and it is
called the~\textbf{observable}. This is not the kind of example we will
deal with in the classical mechanical theory but serves to illustrate
the point.
We say that a measure belongs to an observable in the sense that an
observer may perform a measurement on the observable, therefore
collecting that measure. The set of all possible measures of a single
observable is called the~\textbf{spectrum} of the observable. In this
case, a measurement is the selection of a member of the spectrum.
Here we actually find our first mathematical structure, the~\textbf{set
theory}. The spectrum of an observable is a set in the mathematical
sense, and a measure is a member of the set. A measurement, therefore,
is also the assignment of a member of the spectrum to a characteristic
of the system. The spectrum may be limited or unlimited, countable or
non-countable.
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