2 The clockwork Universe

2.1 Mechanics and determinism

It is probably fair to say that no single individual has had a greater influence on the scientific view of the world than Isaac Newton. The main reason for Newton's prominence was his own intrinsic genius, but another important factor was the particular state of knowledge when he was, in his own phrase, 'in the prime of my age for invention'.

In 1543, a century before Newton's birth, Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favour of a heliocentric view in which the Earth moved round the Sun. By removing the Earth, and with it humankind, from the centre of creation, Copernicus had set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers. The most famous of these must surely have been Galileo, who was summoned to appear before the Inquisition in 1633, on a charge of heresy, for supporting Copernicus' ideas. As a result Galileo was 'shown the instruments of torture', and invited to renounce his declared opinion that the Earth moves around the Sun. This he did, though tradition has it that at the end of his renunciation he muttered 'Eppur si muove' ('And yet it moves').

Figure 1.3 Three views of planetary motion
The Earth-centred view of the ancient Greeks and of the Catholic church in the sixteenth century(a) The Earth-centred view of the ancient Greeks and of the Catholic church in the sixteenth century.
(b) The Copernican system, in which the planets move in collections of circles around the Sun.The Copernican system in which the planets move in collections of circles around the Sun
The Keplerian system in which a planet follows an elliptical orbit with the Sun at one focus of the ellipse(c) The Keplerian system in which a planet follows an elliptical orbit, with the Sun at one focus of the ellipse.

In the Protestant countries of Northern Europe, thought on astronomical matters was more free, and it was there in the early seventeenth century, that the German-born astronomer Johannes Kepler (1571-1630) devised a modified form of Copernicanism that was in good agreement with the best observational data available at the time. According to Kepler, the planets did move around the Sun, but their orbital paths were ellipses rather than collections of circles. This discovery, first published in 1609 in Kepler's book Astronomia Nova (The New Astronomy), was essentially an observational result. Kepler had no real reason to expect that the planets would move in ellipses, though he did speculate that they might be impelled by some kind of magnetic influence emanating from the Sun.

Kepler's ideas were underpinned by new discoveries in mathematics. Chief among these was the realization, by René Descartes, that problems in geometry can be recast as problems in algebra. Like most revolutionary ideas, the concept is disarmingly simple. Imagine a giant grid extending over the whole of space.
figure 1.4, locate the position of any point in terms of its x and y coordinatesFig 1.4 A two-dimensional coordinate system can be used to locate the position of any point in terms of its x- and y-coordinates.
Figure 1.4 shows the two-dimensional case, with a grid extending over part of the page. The grid is calibrated (in centimetres) so the position of any point can be specified by giving its x- and y- coordinates on the grid. For example, the coordinates of point A are x = 3cm and y = 4cm.

This idea becomes more powerful when we consider lines and geometrical shapes. The straight line shown in Figure 1.5 is characterized by the fact that, at each point along the line, the y-coordinate is half the -coordinate. Thus, the x- and y- coordinates of each point on the line obey the equation y = 0.5x, and this is said to be the equation of the line.
figure 1.5, a 2-D coordinate system can represent lines and other geometrical shapes by equationsrepFig 1.5 A two-dimensional coordinate system can be used to represent lines and other geometrical shapes by equations.
Similarly, the circle in Figure 1.5 is characterized by the equation


This is the beginning of a branch of mathematics, called coordinate geometry, which represents geometrical shapes by equations, and which establishes geometrical truths by combining and rearranging those equations. Sometimes, what is difficult to show using traditional geometry is easy to establish using algebra, so this 'mapping' of geometry into algebra gave scientists new ways of tackling geometrical problems, allowing them to go further than the greatest mathematicians of ancient Greece.

Newton's good fortune was to be active in physics (or 'natural philosophy' as it would then have been called) at a time when the cause of Kepler's ellipses was still unexplained and the tools of geometry were ripe for exploitation. The physics of Aristotle was clearly inadequate, and all other attempts seemed unconvincing. The new astronomy called for a new physics which Newton had the ability and the opportunity to devise. He was the right man, in the right place, at the right time.

For years before Newton, people had been trying to understand the world from a scientific perspective, discovering laws that would help explain why things happen in the way that they do. Bits of knowledge were assembled, but there was no clear idea how these bits related to one another; understanding was fragmentary. Newton's great achievement was to provide a synthesis of scientific knowledge. He did not claim to have all the answers, but he discovered a convincing quantitative framework that seemed to underlie everything else. For the first time, scientists felt they understood the fundamentals, and it seemed that future advances would merely fill in the details of Newtonís grand vision. Before Newton, few could have imagined that such a world-view would be possible. Later generations looked back with envy at Newton's good fortune. As the great Italian-French scientist Joseph Lagrange remarked:

'There is only one Universe... It can happen to only one man in the world's history to be the interpreter of its laws.'

At the core of Newton's world-view is the belief that all the motion we see around us can be explained in terms of a single set of laws. We cannot give the details of these laws now, but it is appropriate to mention three key points:

1. Newton concentrated not so much on motion, as on deviation from steady motion - deviation that occurs, for example, when an object speeds up, or slows down, or veers off in a new direction.

2. Wherever deviation from steady motion occurred, Newton looked for a cause. Slowing down, for example, might be caused by braking. He described such a cause as a force. We are all familiar with the idea of applying a force, whenever we use our muscles to push or pull anything.

3. Finally Newton produced a quantitative link between force and deviation from steady motion and, at least in the case of gravity, quantified the force by proposing his famous law of universal gravitation.

In keeping with his grand vision, Newton proposed just one law for gravity - a law that worked for every scrap of matter in the Universe, for moons and planets as well as for apples and the Earth. By combining this law with his general laws of motion, Newton was able to demonstrate mathematically that a single planet would move around the Sun in an elliptical orbit, just as Kepler claimed each of the planets did. Moreover, thanks to the understanding that gravity was the cause of planetary motion, Newtonian physics was able to predict that gravitational attractions between the planets would cause small departures from the purely elliptical motion that Kepler had described. In this way, Newton was able to explain Kepler's results and to go beyond them.

In the hands of Newton's successors, notably the French scientist Pierre Simon Laplace (1749-1827), Newtonís discoveries became the basis for a detailed and comprehensive study of mechanics (the study of force and motion). The upshot of all this was a mechanical world-view that regarded the Universe as something that unfolded according to mathematical laws with all the precision and inevitability of a well-made clock. The detailed character of the Newtonian laws was such that once this majestic clockwork had been set in motion, its future development was, in principle, entirely predictable. This property of Newtonian mechanics is called determinism. It had an enormously important implication. Given an accurate description of the character, position and velocity of every particle in the Universe at some particular moment (i.e. the initial condition of the Universe), and an understanding of the forces that operated between those particles, the subsequent development of the Universe could be predicted with as much accuracy as desired.
 An orrery (a mechanical model of the Solar System) Figure 1.10 An orrery (a mechanical model of the Solar System) can be taken as a metaphor for the clockwork Universe of Newtonian mechanics.
Needless to say, obtaining a completely detailed description of the entire Universe at any one time was not a realistic undertaking, nor was solving all the equations required to predict its future course. But that wasn't the point. It was enough that the future was ordained. If you accepted the proposition that humans were entirely physical systems, composed of particles of matter obeying physical laws of motion,

Two other books in the Physical World series, Describing Motion and Predicting Motion, develop these ideas in greater detail

then in principle, every future human action would be already determined by the past. For some this was the ultimate indication of God: where there was a design there must be a Designer, where there was a clock there must have been a Clockmaker. For others it was just the opposite, a denial of the doctrine of free will which asserts that human beings are free to determine their own actions. Even for those without religious convictions, the notion that our every thought and action was pre-determined in principle, even if unpredictable in practice, made the Newtonian Universe seem strangely discordant with our everyday experience of the vagaries of human life.

2.2 Energy and conservation

Newtonian mechanics is concerned with explaining motion, yet it contains within it the much simpler idea that some things never change. Take the concept of mass, for example, which appears throughout Newtonian mechanics, including the law of gravitation. In Newtonian mechanics, mass is conserved. This means that the mass of the Universe is constant and the mass of any specified collection of particles is constant, no matter how much rearrangement occurs within the system. A chemist might take one kilogram of hydrogen and let it react with eight kilograms of oxygen to produce water. According to the law of
conservation of mass , nine kilograms of water will be produced, the same as the total mass of the ingredients (1 kg + 8 kg = 9 kg). You may think this is trivial, but it is not. Conservation laws are rare and wonderful things. There is no general law of conservation of volume for example. The initial volume of the hydrogen and oxygen is far greater than the final volume of the water. The fact that mass is conserved really is a deep discovery about the checks and balances that exist in our Universe.

The conservation of energy is dealt with in detail in Predicting Motion

Newtonian mechanics introduced several other important conservation laws, including the celebrated law of conservation of energy. Not too surprisingly, this law states that the total energy of the Universe is constant and the total energy of an isolated system of particles is constant. But the full meaning of these words will only become apparent once the concept of energy has been properly defined.

For the moment, it is sufficient to note that we all have some familiarity with the concept of energy. We pay money for gas, electricity and petrol precisely because they are sources of energy, and we use that energy to heat and light our homes and to drive cars. From this, it is apparent that energy has many different forms - chemical energy in gas or electrical energy can be converted into light energy, thermal energy, or the energy of a whirring vacuum cleaner. It is possible to change energy from one form into another but, crucially, when all these forms are properly quantified, the total amount of energy remains constant. Energy is neither created or destroyed because it is a conserved quantity.

Perhaps the simplest form of energy is kinetic energy : the energy associated with motion. If a particle has mass m and speed v, its kinetic energy is given by the formula

equation 1.3

Suppose the particle hits a wall and is brought to a sudden halt. It then has no speed and no kinetic energy, but the initial energy has not been lost. Rather, it has been converted into other forms of energy, such as those associated with sound and heat.

The conservation of energy can be illustrated by considering a stone that is thrown vertically upwards. The stone starts out with a certain amount of kinetic energy, but as it climbs it slows down and its kinetic energy decreases. What happens to this energy? The answer is that there is another form of energy called potential energy, which in this case is associated with the downward pull of gravity and increases as the stone climbs. On the upward part of its journey, the stone's kinetic energy is gradually converted into potential energy until, at the top of its flight, the stone is momentarily at rest. At this point, the stone has no kinetic energy and its potential energy is at its highest. On the way down, potential energy is converted back into kinetic energy, as the stone loses height and gains speed. Assuming that no other forms of energy are involved, by the time the stone returns to its initial height, all of its initial kinetic energy is recovered and the stone is once again travelling at its initial speed. Figure 1.11 shows how the kinetic and potential energies of the stone vary during its up-and-down flight. The total energy, formed by adding the kinetic and potential energies together, is also shown. You can see quite clearly that energy is converted from one form to another while the total energy remains fixed.
figure 1.11
Figure 1.11 A stone is thrown vertically upwards and falls down again. The graph shows how the kinetic energy, potential energy and total energy vary as a stone travels up and down again. (For convenience, the potential energy is taken to be zero when the stone is launched, and when it is caught again.)
One of the consequences of the conservation of energy is that it makes sense to think of storing energy in order to have a ready supply whenever required. Figure 1.12 shows several examples of energy storage in action.

Figure 1.12 Some examples of energy storage
figure 1.12aa) A hydroelectric scheme in which the gravitational potential energy of water descending from a high lake is used to drive generators that produce electricity.
(b) Petrol, a liquid from which it is easy to extract chemical energy.figure 1.12b
figure 1.12c(c) An electrical dry cell which stores electrical energy.