A1 General Theory of Motion and Force (Mechanics)
Completely free Motion of a material Point
What is motion? Location of a material point with respect to its environment.
When you see an object in motion, you note how its location changes relative to its environment, wence you define motion of an object as the process of change of location. In order to simplify the following work, imagine that there is only one body which is so small that it can be treated as a point - a material point. Before you can talk of change of location relative to an environment, you must know what altogether determines a location and how to distinguish between locations.
You will only be able to specify the location of a point with respect to an environment uniquely, if you can refer in the process to other points, lines and planes, which you know of and can consider to be immovable; for example, you specify the geographical location of a point by whether it lies to the North or South of the equator, to the East or West of the zero-meridian, above or below sea level. In this context, the equator, zero-meridian and sea level represent a system of reference in which the relative specification of a location is unique.
However, Earth itself moves and every object on its surface and in its 
interior shares its motion*,
whence the geographical location of a point on it - its distance
from the equator, from the
zero-meridian, from the sea level and above all also from other
lines and planes, which are fixed relative
to Earth, that is, partake in Earth's motion - does not change.
* We speak of Sun rise, Sun set, passage of the Sun through the meridian, although we call the motion of Earth its cause. Thus we see how little we are conscious of Earth's motion. Our resistance, which Copernicus discovered first, is explained to a large extent by the difficulty we encounter in ridding ourselves of the mental impression that Earth does not move.
During treatment of mathematical and physical problems, you relate to another system of lines and planes. For example, if you are dealing with points in a plane, say on this page, you draw (it is only one of many possibilities) two lines XX and YY which are perpendicular to each other (Fig.1, Fig.2). You call them axes, their point of intersection origin or zero of the system, the distances of a point P from the axes - with signs + or - according to their position relative to the axes - coordinates (abscissa, ordinate), the whole of it a coordinate system. You describe the location of a point according to whether it lies above or below XX and simultaneously to the left or right of YY. But how far to the right or left? In order to find this out, you must be able to measure.
Measurement of length. Unit of length.
Every measurement is a comparison of the quantity to be measured with a quantity of the same kind, defined as the respective unit. For the measurement of the straight distance between two points, it has been agreed internationally: The straight distance between two marks of the bar, made out of Platinum-Iridium and preserved at the Bureau International des Poids et Mesures in Paris is the unit of length measurement.
In other words, measurement of a straight segment determines the distance between its two end points in terms of the distance of the two marks, mentioned above. When this bar has the temperature of melting ice, this distance represents approximately the ten millionths part of the distance between one of Earth's Poles and the zero-meridian: This distance is called one metre (m). The metre (m) is thus an arbitrarily fixed (by copy of the standard as measure), reproducible, unique, normal length. The hundredth part of the metre, the centimetre (cm), is used for physical measurements as unit of length.
2. In fact, the metre of the Bureau International only approximates this definition. Its length is not 1·10-7, but 0.999,914·10-7 of Earth's quadrant. However, in practice, this deviation is irrelevant and for science only the constancy of the unit is important.
The metre relates to Earth's dimensions; if Earth
were to encounter a cosmic revolution, these would change in an
unknown he
manner. On the other
hand, the wave length of light remains unchanged and the metre
could be reconstructed in terms of it. It has bee suggested
(first by Babinet 1794-1872 1829), to use as a natural unit of length the wave length of a definite kind of
light, for example, of a definite Fraunhofer-line (Fraunhofer). According to Michelson, 1 metre of red
Cd light (wave length l = 643.85 mm) are 1,553,163.5 waves, of blue light (wave length l = 48 000
mm) 2,083,372.1 waves. The number of wave lengths (l) per cm
(in vacuum), corresponding to the equation nl = 1, is called
the wave number of l and is measured in 10-8 cm (Ångström-units,
(Å.-U; Ångström). The wave number 
= 20,000 cm-1 corresponds to the wave
length l= 5,000 Å.-U.
In order to be reliable, a ruler in daily use should depend as little as possible on the temperature and air humidity. Materials for good rulers are - ordered according to increasing utility regarding temperature conditions - brass, silver, German silver, steel, glass, Ni-steel. Timbers useful for rulers are - ordered according to increasing utility regarding air humidity conditions - poplar, oak, mahogany, beech, pine, lime, maple, fir. Walnut should not be used!
The smallest unit, shown on rulers in daily use, is the millimetre (Fig. 3). In order to read on precision scales to tenths of millimetres, that is, on AB the length of EC, you use the Nonius auxiliary ruler CD. Its smallest subdivision is by a certain amount smaller than the main ruler's units; it can be moved along the latter. Tenth of millimetres are determined by means of the Nonius in which 10 subdivision equal 9 mm, that is, its smallest unit is 1/10 mm shorter that that of the main scale. During measurement, A on the main scale AB is made to coincide with the end E of the length EC to be measured; if C falls between two millimetres of AB, you move the Nonius so that its 0 coincides with C, that is, its lies in the same location between two millimetre marks as C. You then read off the entire millimetres on the main scale. How many tenths of millimetres is the zero of the Nonius now from the read millimetre mark (310)? Answer: You look for that mark on which it coincides with a mark of the main scale. In Fig. 3, it is the sixth, that is, the zero less 6/10 mm from the millimetre mark 310, whence EC is 310.6 mm long.
Cathetometer, Comparator.
These are are most important tools for length measurements (Greek:: kaJetos = perpendicular; metron = measure). With reference to Fig. 2, you measure the difference in the lengths of the z-coordinates of two points; in other words, you measure the vertical distance between the two horizontal planes in which the two points lie. In principle, it is a vertical scale with millimetre sub-divisions and a Nonius, which can be rotated about a vertical axis and along which you can move a horizontally directed telescope.
The main
scale must be perfectly vertical, the
axis of the telescope perfectly horizontal. You point the telescope first at one point
so that it coincides with two intersecting lines (across the
lense of the telescope) and read off the the height of the
telescope above the zero of the scale. You then rotate the
Cathetometer about the vertical axis until the second point is at
the intersecting lines and read off the second height off the
telescope. The difference of the two readings is the required
vertical distance.
Dulong and Alexis Thérèse Petit 1816 invented the Cathetometer during their work on the thermal expansion of mercury (Hg) in oder to measure exactly the difference in the heights of two mercury columns. Since then, it has been frequently changed and improved. - In order to check the subdivisions of a given ruler, say a 1 m ruler, you compare it with a standard ruler, which you consider to be exact. The Compactor is used for such activities. It is essentially a horizontal precision ruler, along which you place the ruler to be compared.
Similar to the Cathetometer, the precision measurement is done by means of a microscope which can be moved horizontally and which is focused at the ends of the two segments to be compared. The Cathetometer and Compactor are necessary tools of the techniques of measurement and precision engineering. They demand professional handling, great care and experience.
Astronomical Coordinates
Fig. 5 explains the description of the location of a point in space in terms of three orthogonal coordinates. The coordinates used to specify the locations of stars differ. The observer envisages them to lie on the surface of a sphere, at the centre of which he observes. He relates their positions either to his horizon or - more often - to the celestial equator and a definite meridian. The vertical through his place of observation P (Fig. 5) meets the sphere at the Zenith above and the Nadir below. The plane through the centre of Earth perpendicular to the Zenith-Nadir line, extended to the celestial sphere, is the true horizon of P ( parallel to it tangentially through x is the apparent horizon). A plane, which is vertical to the Horizon and contains the Zenith-Nadir Line and the Axis of Earth, intersects the vault of the heaven in a large circle, the meridian of P.
The line in which this plane intersects the horizontal plane is called the midday line of P. Its intersections with the vault are the North Point ( below the North Pole) and the South Point on the opposite side. The South Point is the starting point of the system. The largest circles which are perpendicular to the horizon, that is. pass through the Zenith, are called Circles of height, circles parallel to the Horizon Azimuthal circles. In this system, the two coordinates of the star S are its Height and its Azimuth.
The Height is reckoned from the Horizon to the Zenith at 90º, the Azimuth from the South Point of the Meridian to the West, North, East and South at 0º to 360º. ( Instead of the Height, one can also employ the Zenith distance; both complement each other to 90º).
The Height and Azimuth of a star change
continuously due to Earth's rotation. Therefore the astronomer
employs a different reference system, his coordinates
corresponding to the geographical longitude and latitude of a
location on Earth; he uses the Celestial Equator -
Earth's Equator transferred to the heaven - and an initial
meridian, which passes through the poles of the sky and the Night
Points3, the starting point being the Spring Point (
).
Largest circles at right angle to the Equator
(through the Poles) are called Circles of Declination, circles
parallel to the Equator (which become smaller towards the Poles) Parallel Circles. The vertical distance of a star from the Equator
(reckoned towards the North Pole at 90º) is called Declination (corresponding to the Height in the Horizontal System),
the distance from the first Meridian, from the Spring Night Point
reckoned from West
to South, to East to 360º is called Right Aszension. (Instead
of the Declination, one also employs the Distance from the Pole).
The Declination of a fixed star yields its constant distance from
the Equator, whence it is constant; the Right Aszension of a star
is also constant, because the Spring Point has a fixed position
on the Equator and takes part in the daily rotation of the sky.
3. The two Night Points are located where the Ecliptic (path of Earth around the Sun) intersects the Celestial Equator. Around the 21. March, the Sun is located in the one Spring Point, around the 23. September in the other.
The coordinates in such a reference system are circular arcs, whence their measurement involves that of angles. In Astronomy, it is done by means of telescopes, which are provided with correspondingly located circular scales4. For terrestrial work, for example land survey, when the location of points in space (triangulation marks) is the objective, one relates to a horizontal system and employs a theodolite, a small, azimuthally orientated telescope, which has vertical and horizontal angle scales. You use it to determine the height and azimuth of points.
4 Telescope orientation in the celestial coordinate system. There are two ways of erecting a telescope for sighting of a star. Both have in common the erection of a telescope, which must rotate about two, perpendicular axes (cf. Fig. 5. a and b). The orientations of the intersection of the axes differ - corresponding to the differences between the azimuthal and Equator systems. Smaller telescopes are orientated in such a manner that one of the axes points at the Zenith, the other lies parallel to the horizontal plane. (azimuthal orientation), large telescopes so that one axis points at the pole (Pol-rectaszension, Hour Axis), the other (Dclination Axis) lies parallel to the equatorial plane. In the latter set up (Fraunhofer), the stars apparently rotate as a consequence of Earth's rotation about the axis through the poles. If you point the telescope (by rotation about the Declination Axis) at a star and then rotate it continuously at the rate of Earth's rotation about the Polar Axis, then the star remains in sight all the time (because of this continuous rotation of the telescope, this orientation is called parallactic) . For this purpose, the azimuthally orientated telescope would have to be rotated all the time about two axes and about each at different rates; therefore this set up is only applied for special astronomical observations for terrestrial purposes. Instead of following the stars with the telescope, Hale uses in his vertical tower telescope a flat, equatorially oriented mirror the plane of which is parallel to the axis of the earth. This mirror reflects the light from the star onto a second, fixed mirror, which is located at a definite angle over the objective and reflects the light along the vertical axis of the telescope. This system of mirrors is called a Zölostat (Lippmann).and has a horizontal angle scale. With it, you can determine the height and azimuth of the point of interest.
We have encountered different coordinate systems: One with the Equator, Zero Meridian and Sea Level, linked to Earth, the azimuthally and the equatorial systems of astronomical coordinates, linked to outer space (Fig. 5) and the space, rectangular system of Fig. 2. Depending on our needs, we can construct any kind of coordinate system which, after all, is only a means for the determination of relative positions of points or bodies. For example, the same astronomical observations relate the Copernican system to coordinates fixed in the Sun, the Ptolemaic system to coordinates linked to the earth. Both are right, but the first serves its purpose better, because it is clearer; the positions in space, occupied consecutively by a planet, yield in the Sun connected system another, clearer orbit system linked to Earth.
You can see here the significance of relating the same process to different coordinate systems. Computational and descriptive physics does this all the time. Replacement of one coordinate system by another (transition from one to another) means changes of the coordinates of individual points. The point P (Fig. 6) has in the xy-coordinate system other coordinates (distances from the axes) than in the x'y'-system.
The conversion of the coordinates in one system into those in another system, their transformation, is simple, but you must know, in order to construct the relevant equations, the relative position of the two systems, that is, how far their origins are from each other, by what angles their axes have been rotated relative each other. You see immediately from Fig. 6b that in this case x' = x-a, y' = y-b,. These are the equations of that transformation.
For a rectangular system with three axes, the origin of which has been moved by a, b, c along the x-, y- and z-axes in their positive directions, the formulae are: x' = x - a, y' = y - b, z' = z - c. Transformation formulae for more complicated systems with rotation are valid in all cases.
The coordinates of a point do not have any individual significance; they are only distances from the axes of the respective coordinate systems and differ from one system to another. However, quantities, which have individually geometric significance, differ in this respect; they are the same in all systems, they are invariant under coordinate transformations. An example is the distance s of the point P from the origin O in the two systems (Fig. 6c), rotated with respect to each other; it is an invariant. You have that s² = x²+ y²= x' ²+ y'²; the same formula applies to all coordinate systems with a common origin. Similarly, the relative distance between the two points P and Q, remains the same, even after transposition and rotation of one system with respect to the other. This fact is computationally and geometrically readily confirmed. Thus, invariants represent independent, geometrical facts and are without a relationship to arbitrarily selected coordinate systems.
