Introduction.

The most celebrated of all dynamical problems is known as the Problem of Three Bodies, and may be enunciated as follows:

The practical importance of this problem arises from its applications to Celestial Mechanics: the bodies which constitute the solar system attract each other according to the Newtonian law, and (as they have approximately the form of spheres, whose dimensions are very small compared with the distances which separate them) it is usual to consider the problem of determining their motion in an ideal form, in which the bodies are replaced by particles of masses equal to the masses of the respective bodies and occupying the positions of their centres of gravity.

The problem of three bodies cannot be solved in finite terms by means of any of the functions at present known to analysis. This difficulty has stimulated research to such an extent, that since the year 1750 over 800 memoirs, many of them bearing the names of the greatest mathematicians, have been published on the subject.

The displacements of rigid bodies.

The name Analytical Dynamics is given to that branch of knowledge in which the motions of material bodies, considered as due to the mutual interactions of the bodies, are discussed by the aid of mathematical analysis.

It is natural to begin this discussion by considering the various possible types of motion in themselves, leaving out of account for a time the causes to which the initiation of motion may be ascribed; this preliminary enquiry constitutes the science of Kinematics. The object of the present chapter is to establish a number of kinematical theorems which will be required in the rest of the work.

We shall say that a material body is rigid when the mutual distance of every pair of specified points in it is invariable, so that the body does not expand or contract or change its shape in any way, although it may change its position with reference to surrounding objects.

If a rigid body is moved from one position to another, the change of position is called a displacement of the body.

Edmund Taylor Whittaker (1873–1956) was one of the most remarkable mathematical polymaths of modern times. He was a professional mathematician of outstanding scholarship and originality in mathematical analysis and in the mathematics of general relativity and of several other parts of mathematical physics, as well as of classical mechanics. For several years he was a professional astronomer directing an observatory. He then became a celebrated head of a mathematical department. He was a pioneer of the teaching of numerical mathematics and of mathematical statistics. He was a leading historian of mathematical physics.

As a person, Whittaker was without doubt the most influential figure of his time in the mathematical community of the British Isles. This was partly because of the singular sequence of his appointments. As a young lecturer in Cambridge from 1896 to 1906 he had as colleagues or pupils nearly all the leading British mathematicians of two generations. As Royal Astronomer of Ireland from 1906 to 1912, helped by his temperamental affinity, he gained intimate knowledge of the Irish academic world at what proved to be for it a notable vintage period. Then in Edinburgh from 1912 to 1946 he directed what became the leading individual British school of mathematics – those in London, Oxford and Cambridge being more fragmented; there a high proportion of all British mathematicians came under his influence in one capacity or another. Also he became personally acquainted with leading mathematicians throughout the world.

Introduction.

We shall now pass to the study of the general form and disposition of the orbits of dynamical systems. For simplicity we shall in the present chapter chiefly consider the motion of a particle which is free to move in a plane under the action of conservative forces, but many of the results obtained can be readily extended to more general dynamical systems.

It has already been observed (§ 104) that the determination of the motion of a particle with two degrees of freedom under the action of conservative forces is reducible to the problem of finding the geodesies on a surface with a given line-element; an account of the properties of geodesies might therefore be regarded as falling within the scope of the discussion. Many of these properties are however of no importance for our present purpose: and as the theory of geodesies is fully treated in many works on Differential Geometry, we shall only consider those theorems which are of general dynamical interest.

Periodic solutions.

Great interest has attached in recent years to the, investigation of those particular modes of motion of dynamical systems in which the same configuration of the system is repeated at regular intervals of time, so that the motion is purely periodic. Such modes of motion are called periodic solutions.

References to work published since 1927 have been inserted, and some errors corrected. For the detection of these I must thank many correspondents.

The ideas of rest and motion.

In the previous chapter we have frequently used the terms “fixed” and “moving” as applied to systems. So long as we are occupied with purely kinematical considerations, it is unnecessary to enter into the ultimate significance of these words; all that is meant is, that we consider the displacement of the “moving” systems, so far as it affects their configuration with respect to the systems which are called “fixed,” leaving on one side the question of what is meant by absolute “fixity.”

When however we come to consider the motion of bodies as due to specific causes, this question can no longer be disregarded.

In popular language the word “fixed” is generally used of terrestrial objects to denote invariable position relative to the surface of the earth at the place considered. But the earth is rotating on its axis, and at the same time revolving round the Sun, while the Sun in turn, accompanied by all the planets, is moving with a large velocity along some not very accurately known direction in space. It seems hopeless therefore to attempt to find anything which can be really considered to be “at rest.”

In the nineteenth century it was supposed that the aether of space (the vehicle of light and of electric and magnetic actions) was (apart from small vibratory motions) stagnant, and so was capable of providing a basis for absolute fixity.

The motion of systems with one degree of freedom: motion round a fixed axis, etc.

We now proceed to apply the principles which have been developed in the foregoing chapters in order to determine the motion of holonomic systems of rigid bodies in those cases which admit of solution by quadratures.

It is natural to consider first those systems which have only one degree of freedom. We have seen (§ 42) that such a system is immediately soluble by quadratures when it possesses an integral of energy: and this principle is sufficient for the integration in most cases. Sometimes, however (e.g. when we are dealing with systems in which one of the surfaces or curves of constraint is forced to move in a given manner), the problem as originally formulated does not possess an integral of energy, but can be reduced (e.g. by the theorem of § 29) to another problem for which the integral of energy holds; when this reduction has been performed, the problem can be integrated as before.

The following examples will illustrate the application of these principles.

(i) Motion of a rigid body round a fixed axis.

The trajectories of a dynamical system.

The chief object of investigation in Dynamics is the gradual change in time of the coordinates (q1, q2, …, qn) which specify the configuration of a dynamical system. When the system has three (or less than three) degrees of freedom, there is often a gain in clearness when we avail ourselves of a geometrical representation of the problem: if a point be taken whose rectangular coordinates referred to fixed axes are the coordinates (q1, q2, q3) of the given dynamical system, the path of this point in space can be regarded as illustrating the successive states of the system. In the same way when n > 3 we can still regard the motion of the system as represented by the path of a point whose coordinates are (q1, q2, …, qn) in space of n dimensions; this path is called the trajectory of the system, and its introduction makes it natural to use geometrical terms such as “intersection,” “adjacent,” etc., when speaking of the relations of different states or types of motion in the system.

Hamilton's principle, for conservative holonomic systems.

Consider any conservative holonomic dynamical system whose configuration at any instant is specified by n independent coordinates (q1, q2, …, qn), and let L be the kinetic potential which characterises its motion. Let a given are AB in space of n dimensions represent part of a trajectory of the system, and let CD be part of an adjacent are which is not necessarily a trajectory: it would however of course be possible to make CD a trajectory by subjecting the system to additional constraints.