# Space and Time

We start with the intuitive ideas of space and time we all have: We perceive that everthing there is in the physical world, has a place in some form of big container with three independent directions, which we call space. We further experience that things can change shape and position in space, the rate of which we measure using clocks recording time by periodic motion.

3D Cartesian coordinate system

We will record position in space by $x$ and time by $t$, where $x$ and $t$ represent numbers. Points in space-time can then be recorded as pairs of numbers $(x,t)$.

We live in three-dimensional or 3d space and a point $x$ can be identified by three coordinates or numbers $x_1$, $x_2$ and $x_3$, which we can collect into a triple $(x_1,x_2,x_3)$ and we can write $x=(x_1,x_2,x_3)$.

If we restrict the world to two dimensions or 2d, that is to a plane, then we need only two space coordinates $x_1$ and $x_2$, which we can collect into a pair $x=(x_1,x_2)$.

If we restrict the world further to one space dimension or 1d, that is to a line, then just one coordinate is enough and we have $x=x_1$.

We start using rational numbers expressed as decimal numbers like

• $3.142=\frac{3142}{1000}=3+10^{-1}+4\times10^{-2}+2\times 10^{-3}$

using the base $10$ and denoting here multiplication by $\times$. As usual $10^{-2}=\frac{1}{100}$, and more generally e.g. $10^3=10\times 10\times 10$, $10^{-3}=\frac{1}{10^3}$.

We recall that a rational number $r$ is the quotient $r=\frac{p}{q}$ of two integers $p$ and $q$ (of the form $0,\pm 1,\pm 2,\pm 3,...$) with $q\neq 0$, and the non-negative integers $0,1,2,3,...$ are called natural numbers.

We denote the set of natural numbers by $N$, and the set of rational numbers by $Q$.

# GPS

The Global Positioning System GPS gives your coordinates (latitude and longitude and altitude) on Earth at the press of a button on your GPS receiver at your current position. How does it work?

# SI Standards of Length and Time

The SI Standard of unit of time is second, which is the duration of a certain number of oscillationsof a certain caesium atom. More precisely:

• one second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom at a temperature of 0         Mechanical clock with wheels, gears, escapement

The SI Standard of unit of length is a lightsecond, which is the distance traveled by light in one second, and meter as

• the distance traveled by light during a time interval of 1/299 792 458 of a second.

Entering into the Inner World

# Coordinate Systems in 1d, 2d and 3d

With a laser-beam we can set up coordinate systems along a 1d line, in a 2d plane and in 3d space.

In 1d, for example a horisontal line, we mark the coordinates in meter using a laser beam and a clock measuring the time it takes for light to pass from a given point, which we called the origin, to different points to the right and left marking the points to the left with a minus sign.

In 2d, for example a horisontal plane, we choose two perpendicular 1d directions which we mark separately as in 1d:

Cartesian 2D coordinate system in a plane.

In 3d we choose three perpendicular directions and mark each direction as in 1d. We can think of these directions as South-North, East-West, down-up.

# The Time Step

We will denote by $dt$smallest unit of time, which can be different in different situations. The smallest $dt$ we can measure is the time of one oscillation of a caesium atom, about $0.0000000001 =10^{-10}$ seconds. Depending on the setting, $dt$ may be one second, minute, hour, day, month, year,…

We will describe as Newton’s World of Mechanics the physical world with a smallest $dt$ as indicated, and by Leibniz’ World a mathematical fictional world where $dt$ is assumed to be smaller than any given finite value, or vanishingly small.

We shall find that in Leibniz fictional world with a vanishingly small time unit, many mathematical expressions and formulas become easier to manipulate and easier to understand on a conceptual level, than in Newton’s real world with a finite smallest time unit.

In computational simulations we have to use a finite time step, since the computer can only perform a finite number of operations per time unit.

We shall use tools from Leibniz world when we construct computational digital simulations of Newton’s real analog world, because these are efficient tools, but in the computational simulations effectively use a finite time step or time unit.

Leibniz world is like the ideal world of Plato, which is useful for thinking but untouchable in reality or simulation. IT is like the world we can imagine by using language, which is different from the real world. But we also know that the real world can be more remarkable than any world we may imagine.

Caesium reference clock at NIST Laboratory, Colorado.

The Caesium atom is large and vibrates slowly. It also reacts with water.

# Point, Vector and Distance = Vector Norm

Let $x=(x_1,x_2,x_3)\in Q^3$, where $Q^3$ is the set of triples $(x_1,x_2,x_3)$ with $x_i\in Q$, be a point in a 3d coordinate system. We can to the point $x$ associate a vector also denoted by $x$ as the directed straight line segment from the origin $O=(0,0,0)$ to the point $x$, or arrow from $O$ to $x$. We write $x\in Q^3$ also for a vector $x$.

By Pythagoras Theorem, the distance from $O$ to the point $x$, which is also the length or norm $\vert x\vert$ of the vector $x$, is given by

• $\vert x\vert =\sqrt{x_1^2 +x_2^2+x_3^2}\quad \mbox{or }\vert x\vert ^2=x_1^2 +x_2^2+x_3^2$,

which we can think of as the length of the straight line from the origin to the point $x$, also referred to as the vector $x$.

# Scalar Product

If $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$ are two vectors in 3d space, that is $x,y\in Q^3$, then we define their scalar product $x\cdot y$ as follows

• $x\cdot y=x_1y_1+x_2y_2+x_3y_3$

We will say that if $x\cdot y = 0$, then the vectors $x$ and $y$ are orthogonal or perpendicular. We note that the length of the vector $x$, or distance from the origin $(0,0)$ to the point $(x_1,x_2)$ is defined in terms of the scalarproduct as follows:

• $x\cdot x=\vert x\vert^2$.

The distance between two points $x$ and $y$ is then given by $\vert x-y\vert$, where $x-y=(x_1-y_1,x_2-y_2,x_3-y_3)$.

Further, the angle $\theta$ between two (non-zero) vectors $x$ and $y$ (with $x$ an arrow with tail at $(0,0,0)$ and head at $(x_1,x_2,x_3)$) is connected to the scalar product $x\cdot y$ by the following formula (which you will derive yourself below):

• $\cos(\theta )=\frac{x\cdot y}{\vert x\vert \vert y\vert}$.

The central quantities of geometry of distance and angle, are thus computable in terms of the scalar product. Neat!

# Change of Position/Time Unit = Velocity

Velocity $v$ is defined as change $dx$ of position $x$ per unit time step $dt$, that is,

• $v = \frac{dx}{dt}$.

Nude time stepping down a staircase by Marcel Duchamp.

# Change of Velocity/Time Unit: Acceleration

Accelleration $a$ is defined as change $dv$ of velocity $v$ per unit time step $dt$, that is,

• $a = \frac{dv}{dt}$.

# Particles and Forces

Everything which happens in physical space can be thought of as an interaction between materialparticles each one occupying a specific point in space at a given time, with the interaction mediated by certain forces.

# Newton’s 2nd Law: F=Ma

The most basic law of physics is Newton’s 2nd Law stating that

• $F = M a$

where $F$ is force, $M$ is mass and $a$ is accelleration. Since $a =\frac{dv}{dt}$ Newton’s 2nd law can be written

• $\frac{dv}{dt}= \frac{F}{M}$,

or normalizing to $M=1$,

• $\frac{dv}{dt}= F$.

This law connects the world of particles, the world of velocities of particles, with the world of forces.

If we think of the world as consisting of particles interacting by forces, we understand that somehow the effect of forces acting on particles must be specified and Newton’s 2nd is the basic law making this specification.