The motion of material bodies formed the subject of some of the earliest researchpursued by the pioneers of physics. From their efforts there has evolved a vastfield known as analytical mechanics or dynamics, or simply, mechanics. In thepresent century the term "classical mechanics" has come into wide use to denotethis branch of physics in contradistinction to the newer physical theories, espe-especially quantum mechanics. We shall follow this usage, interpreting the name toinclude the type of mechanics arising out of the special theory of relativity. It isthe purpose of this book to develop the structure of classical mechanics and tooutline some of its applications of present-day interest in pure physics. Basic toany presentation of mechanics are a number of fundamental physical concepts,such as space, time, simultaneity, mass, and force. For the most part, however,these concepts will not be analyzed critically here; rather, they will be assumed asundefined terms whose meanings are fa

The coerstone of mode cosmology is the belief that the place that we occupy in theUniverse is in no way special. This is known as the cosmological principle, and it is anidea which is both powerful and simple. It is intriguing, then, that for the bulk of the historyof civilization it was believed that we occupy a very special location, usually the centre, inthe scheme of things.The ancient Greeks, in a model further developed by the Alexandrian Ptolemy, believedthat the Earth must lie at the centre of the cosmos. It would be circled by the Moon, the Sunand the planets, and then the ‘fixed’ stars would be yet further away. A complex combina-tion of circular motions, Ptolemy’s Epicycles, was devised in order to explain the motionsof the planets, especially the phenomenon of retrograde motion where planets appear totemporarily reverse their direction of motion. It was not until the early 1500s that Coper-nicus stated forcefully the view, initiated nearly two thousand years befor

For most of history, astronomers have had to rely on light in the visible part of the spec-trum in order to study the Universe. One of the great astronomical achievements of the20th century was the exploitation of the full electromagnetic spectrum for astronomicalmeasurements. We now have instruments capable of making observations of radio waves,microwaves, infrared light, visible light, ultraviolet light, X-rays and gamma rays, whichall correspond to light waves of different (in this case increasing) frequency. We are evenentering an epoch where we can go beyond the electromagnetic spectrum and receive in-formation of other types. A remarkable feature of observations of a nearby supeova in1987 was that it was also seen through detection of neutrinos, an extraordinarily weaklyinteracting type of particle normally associated with radioactive decay. Very high energycosmic rays, consisting of highly-relativistic elementary particles, are now routinely de-tected, though as yet there is n

It is perfectly possible to discuss cosmology without having already leaed general rel-ativity. In fact, the most crucial equation, the Friedma equation which describes theexpansion of the Universe, tus out to be the same when derived from Newton’s theoryof gravity as it is when derived from the equations of general relativity. The Newtonianderivation is, however, some way from being completely rigorous, and general relativity isrequired to fully patch it up, a detail that need not conce us at this stage. Readers withsome familiarity with general relativity may wish to study the relativistic derivation givenin Advanced Topic 1.In Newtonian gravity all matter attracts, with the force exerted by an object of mass Mon one of mass m given by the famous relationshipF =GM m,r 2(3.1)where r is the distance between the objects and G is Newton’s gravitational constant. Thatis, gravity obeys an inverse square law. Because a force on an object induces an accelerationwhich is also prop

We now consider the real meaning of the constant k which appears in the Friedmaequationa ̇a2=8πGkρ − 2 .3a(4.1)While the Newtonian derivation in the last chapter introduced this as a measure of theenergy per particle, the true interpretation, apparent in the context of general relativity, isthat it measures the curvature of space. General relativity tells us that gravity is due to thecurvature of four-dimensional space–time, and a full analysis can be found in any generalrelativity textbook. Here I will be purely descriptive, and focus on the interpretation ofk as measuring the curvature of the three spatial dimensions. Further details of generalrelativistic cosmology can be found in Advanced Topic 1.We have demanded that our model universes be both homogeneous and isotropic. Thesimplest type of geometry which can have this property is what is called a flat geometry,in which the normal rules of Euclidean geometry apply. However, it tus out that theassumption of isotropy is n

In Chapter 3 we derived the equations satisfied by an expanding isotropic gas. They are theFriedman equation 1a ̇a2=8πGkρ − 2 ,3a(5.1)which goves the time evolution of the scale factor a(t), and the fluid equationρ ̇ + 3a ̇pρ + 2ac= 0 ,(5.2)which gives us the evolution of the mass density ρ(t). I stress that, despite their Newtonianderivation, these are the real equations used by cosmologists, more traditionally derivedvia the equations of general relativity as described in Advanced Topic 1.This chapter finds and discusses some simple solutions to these equations, which willbe used extensively during the book. However, while they have wide applicability duringthe early stages of the Universe, it will tu out that these are not sufficient to describe thepresent state of the Universe, for which an extra ingredient, the cosmological constant, willbe needed. It is introduced in Chapter 7.Before finding solutions to these equations, we can study two of their implications.5.1Hub

The Big Bang model does not give a unique description of our present Universe, but ratherleaves quantities such as the present expansion rate, or the present composition of the Uni-verse, to be fixed by observation.It is a standard practice to specify cosmological models via a few parameters, which onethen tries to determine observationally to decide which version of the model best describesour Universe. In this chapter and the next I’ll discuss the most commonly-considered pa-rameters, including ones we have already seen and new ones.6.1The expansion rate H 0The Hubble constant H 0 tells us the present expansion rate of the Universe, making it avital cosmological parameter as measurements of other properties of the Universe oftendepend on it. It also ought to be the easiest to measure, since all galaxies are supposed toobey v = H 0 r. So all we have to do is measure the velocities and distances of as manygalaxies as we can and get an answer. However, each measurement has its problem