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.

Chapter 9 presents an example of how the results of the first eight chapters can be applied to a physical theory – optics. It is all in the nature of applications, and can be omitted without any effect on the understanding of what follows.

Theories of optics

In the history of physics it is often the case that, when an older theory is superseded by a newer one, the older theory retains its validity, either as an approximation to the newer theory, an approximation that is valid for an interesting range of circumstances, or as a special case of the newer theory. Thus Newtonian mechanics can be regarded as an approximation to relativistic mechanics, valid when the velocities that arise are very small in comparison to the velocity of light. Similarly, Newtonian mechanics can be regarded as an approximation to quantum mechanics, valid when the bodies in question are sufficiently large. Kepler's laws of planetary motion are a special case of Newton's laws, valid for the inverse square law of force between two bodies. Kepler's laws can also be regarded as an approximation to the laws of motion derived from Newtonian mechanics when we ignore the effects of the planets on each other's motion.

This book, with apologies for the pretentious title, represents the text of a course we have been teaching at Harvard for the past eight years. The course is aimed at students with an interest in physics who have a good grounding in one-variable calculus. Some prior acquaintance with linear algebra is helpful but not necessary. Most of the students simultaneously take an intensive course in physics and so are able to integrate the material learned here with their physics education. This also is helpful but not necessary. The main topics of the course are the theory and physical application of linear algebra, and of the calculus of several variables, particularly the exterior calculus. Our pedagogical approach follows the ‘spiral method’ wherein we cover the same topic several times at increasing levels of sophistication and range of application, rather than the ‘rectilinear approach’ of strict logical order. There are, we hope, no vicious circles of logical error, but we will frequently develop a special case of a subject, and then return to it for a more general definition and setting only after a broader perspective can be achieved through the introduction of related topics. This makes some demands of patience and faith on the part of the student. But we hope that, at the end, the student is rewarded by a deeper intuitive understanding of the subject as a whole.