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- From the Postscript to ‘The Restless Universe’(1951)
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.
In principle, according to Newtonian mechanics, it is possible to predict the entire future behaviour of the Universe provided the initial positions and velocities of all the particles in it are known, and the laws describing their interactions are known. List at least two reasons why this goal is, in practice, beyond our reach. This question is unusually open-ended, with no single correct answer. In preparing our answer we have taken the opportunity to expand the discussion slightly, but would not expect you to include all the points listed below.
There are several reasons why it would be impossible, in practice, to follow through the Newtonian programme of predicting the entire future of the Universe. Nowadays we know that the Universe contains an immense number of fundamental particles. The visible Universe contains about 10 80 protons and the same number of electrons. Measuring the positions and velocities of such a large number of particles is an unimaginable task.
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Indeed, there may be some regions of the Universe that are so remote that we cannot yet know of their existence because the Universe is too young for light from them to have reached us! Small errors in measurement may, at first, produce only small errors in prediction, but the errors are cumulative and will eventually become serious. Nowadays we know that some simple systems are extremely sensitive to the initial conditions. In some cases, it is impossible to measure the initial conditions accurately enough to make anything more than a very short-term prediction.
Such systems are said to be chaotic.
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Newton only produced an explicit formula for gravitational forces. His work was later extended to cover electromagnetic forces, but our understanding of the interactions between particles remains partial and a complete understanding elusive. No computer could ever attempt an exact solution. And even if the calculations could be done, there would be nowhere to store all the results. For all these reasons, we cannot hope to predict the exact future of the Universe. That would be too much to expect.
The Restless Universe (Palomar Transient Factory) | Palomar College Planetarium
The great successes of physics emerge when we ask specific questions about systems that are simple, or can be thought of as being simple, so that the mathematical analysis remains feasible for humans and their computers. 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. 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. 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.
From the Postscript to ‘The Restless Universe’(1951)
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 nor 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. 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 13 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. 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 14 shows several examples of energy storage in action. From the time of Newton until the end of the nineteenth century the development of physics consisted essentially of the refinement and extension of the mechanical view of the Universe.
There were many stages in this process but one of the most interesting came towards its end with the realisation that the cosmic clockwork was inevitably unwinding and running down. The source of this realisation was the development of thermodynamics. The first half of the nineteenth century was a period of great economic and industrial growth. The steam engine, invented in the previous century, was becoming increasingly common in locomotives, mines and factories; power was becoming available on demand.
A major priority for engineers was to produce more efficient engines, in order to deliver more useful power for less expenditure on fuel. Thermodynamics emerged as a study of the basic principles determining energy flows and the efficiency of machines. This may seem like a big idea in engineering rather than a big idea in physics. Certainly, thermodynamics is important to engineers, and continues to guide the design of engines of all sorts, but thermodynamics is just as important to physicists.
It explains a wealth of natural phenomena, from the freezing of water to the evaporation of a black hole, and casts light on concepts like temperature, heat and spontaneous processes, which do not fit naturally into the Newtonian world-view. It is still instructive to return to the origins of the subject. Speaking very roughly, a steam engine is a device that uses fuel to convert water into steam and uses the resulting expansion in volume to drive a piston.
The kinetic energy of the piston is exploited using a variety of mechanical devices - gears, drive belts, camshafts and so on, but thermodynamics concentrates on the early stages of the process, where heat is used to create kinetic energy. To begin with there was much dispute about the nature of heat. Many people thought of it as a sort of fluid which could flow from one body to another. Eventually, it became clear that no such fluid exists and that heat is best defined as energy transferred because of a temperature difference.
This scientific definition of the word 'heat' is slightly different from everyday usage, so it may help to consider a specific example. Think of a hot steak veggie-burger, if you prefer resting on a cold plate. The steak cools down and the plate warms up as energy flows from the steak to the plate.
The energy transferred in this way is called heat. By contrast, work is energy transferred by non-thermal means. For example, if you rub the plate vigorously with a cloth, the energy of the plate will increase and it will get slightly warmer. But, this energy transfer is not caused by a temperature difference between the plate and the cloth, so energy transferred by rubbing is classified as work rather than heat.
In general, the total energy gained by a system such as the plate is the sum of the heat and the work transferred to it. It is worth emphasising that heat and work are not themselves properties of a system.
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We cannot examine a plate and deduce that it has received so much energy from heat and so much energy from work. All that counts is that the plate has a total amount of energy, and that any increase in this energy is the sum of the heat and work transferred to the plate.
This understanding of heat, work and energy is incorporated in the first law of thermodynamics. When all types of energy transfer, including work and heat, are taken into account, the energy of an isolated system remains constant. From a modern perspective, we can see that this is just another way of stating the law of conservation of energy with the explicit recognition of heat as a quantity of energy to be included, alongside work, in any energy audit. Inventors should take note: an engine may convert energy from one form to another, but it cannot produce energy from nothing.
The kinetic energy of the piston of a steam engine, for instance, has been paid for in advance by the heat transferred to the steam. Given this modern understanding of heat as energy transferred in a particular way, you might wonder why we bother to distinguish between heat and work at all. The reason is that heat can be used to define another important quantity: entropy. We cannot define entropy properly in this introductory survey. In very broad terms you can think of entropy as a measure of 'disorder' - the random motion of molecules in steam corresponds to more disorder, and hence more entropy, than the more orderly motion of molecules in ice.
Interestingly enough, there is a connection between entropy and heat: whenever heat is transferred to a body, the entropy of that body increases. The term entropy was deliberately chosen to be reminiscent of energy, though the differences between the two quantities are just as important as their similarities.