Benno Moiseiwitsch's Variational Principles
1. Preface to Variational Principles (1966).
The first chapter of this book is devoted to the variational formulation of classical as well as relativistic mechanics. It introduces variational principles through Hamilton's principle and the principle of least action, showing their equivalence to the dynamical equations of Lagrange and Hamilton, and also includes an examination of mechanics from the view point of differential geometry. This leads to a variational treatment of geodesic lines in Riemannian space and of the motion of a particle in a gravitational field.
In the second chapter we turn to the subject of optics and consider Fermat's principle of least time. The analogy between dynamics and geometrical optics is then discussed and the wave equations of Schrödinger, Klein-Gordon and Dirac are evolved. This is followed by an examination of the role of Hamilton's principle in quantum mechanics.
Through the use of the variational principle, the next chapter develops the Lagrangian and Hamiltonian formulations of the general field equations of physics and then considers particular applications to the equations of wave motion in classical dynamics, to the electromagnetic field equations, to the diffusion equation and to the various equations of wave mechanics. A brief discussion of the Schwinger dynamical principle in the theory of quantized fields concludes the part of the book dealing with the field equations of mathematical physics.
The remaining part of the book is concerned with discrete and continuous eigenvalue problems. After summarizing the theory of the small oscillations of a dynamical system at the beginning of the fourth chapter, Rayleigh's principle is proved and the Ritz variational method developed for the Sturm-Liouville equation. The more general problem of the eigenenergies of a quantum mechanical system is now discussed, upper bounds to the eigenenergies obtained and lower bounds to the ground state eigenenergy derived. The problem of determining the eigenenergies of atomic systems is then investigated and the special case of the two-electron system treated in considerable detail, the remarkable accuracy with which the energies of such systems have been calculated by using the Ritz variational method being emphasized. We then turn our attention to the energy curves of molecules and consider the cases of the hydrogen molecular ion and the hydrogen molecule as examples. The chapter ends with a discussion of the relationship between perturbation theory and the variational method.
The last chapter deals with the use of variational principles in the theory of scattering, a subject which has received much attention recently. Variational principles for the scattering amplitude and for the phase shift due to Hulthen, Kahn, Schwinger and others are established. The special case of the scattering of particles having vanishing energy is treated in some detail, upper bounds to the scattering length being derived and application being made to the elastic scattering of electrons and positrons by hydrogen atoms and to the elastic scattering of neutrons by deuterons. Finally we examine time-dependent scattering theory and look at variational principles for the collision operator and the transition matrix which arise therein.
Inevitably, in order to remain within the confines of a small volume, it has proved necessary to omit much interesting material from the present work owing to the large number of different applications of variational principles that have been carried out recently. An effort has been made to keep the treatment of variational principles at a reasonably elementary level, the general aim being to provide a fairly broad view of the way in which variational principles have been applied to various problems in theoretical physics, although a certain amount of emphasis has been given to their role in the quantum theory of scattering.
The search for unifying concepts and general principles has been one of the most important aims of mathematicians from early times and, as before, I would like to emphasize that some of the most aesthetically satisfying and illuminating general principles are the variational principles that are the subject of this book. Variational principles are concerned with the maximum and minimum properties, or more generally the stationary properties, of an extensive range of quantities of mathematical and physical interest spanning a wide field of applications.
Not only are these variational principles characterized by an elegant mathematical structure, but also they often possess the greatest practical utility in the solution of important problems in physics and chemistry and provide the most reliable method of accurately determining the values of many physical quantities of fundamental importance such as energy eigenvalues.
Variational principles have an ancient history and arose in antiquity in connection with the solutions of the isoperimetric and the geodesic problems. The isoperimetric problem, proposed by Greek mathematicians of the second century B.C., was originally concerned with the determination of the shape of a plane closed curve possessing a given perimeter encompassing the greatest possible area, namely the circle, while the geodesic problem was concerned with determining the curve that produced the shortest path between a pair of given points on the surface of the earth, namely an arc of a great circle on a sphere.
These two problems have many generalizations. For example, the triangle with a given perimeter enclosing the greatest area is equilateral and, in general, the polygon having a given perimeter that encloses the greatest area is a regular polygon. Also, the closed surface possessing a given area that surrounds the greatest volume is a sphere. Further, the shortest path connecting two points on a circular cylinder is a helix, a result that is related to the shape of the DNA molecule.
Another ancient variational problem, first solved by the Greek mathematician and scientist Heron of Alexandria in the first century A.D., is the determination of the path of a light ray reflected by a plane mirror. This is an example of the principle of least time that was later named after the seventeenth century mathematician Fermat who examined the refraction of a light ray at an interface separating two different media and showed that it takes the route of the shortest possible time.
However, this variational principle is not necessarily a minimum principle since a light ray that passes from one focal point to the other focal point of an ellipse by reflection at a mirror within the ellipse, which is tangential to the ellipse at its minor axis, takes the trajectory with the maximum possible total distance between the foci connected by two straight-line segments with a vertex at the mirror. Thus we have here an example of a principle of maximum time.
Many elementary examples of variational principles also occur in mechanics. There is the hanging chain problem that is concerned with finding the shape of a uniform chain hanging between two points under gravity and was found to be the catenary by Jakob Bernoulli in the seventeenth century and can be solved by minimizing the potential energy of the chain. Also, there is the brachistochrone problem proposed and solved by Jakob Bernoulli for determining the shape of a smooth wire connecting two points such that a bead takes the shortest possible time to travel from the initial to the final point of the wire. The solution to this problem is a cycloid and was solved by a number of other mathematicians including Leibnitz, de l'Hôpital, and Isaac Newton as well as Jakob's brother Johann.
An attempt to generalize Fermat's principle in optics to mechanics was first made by Maupertuis who introduced in 1744 the concept of action that he defined as mvs, the product of mass, velocity and distance. This was put on a firm mathematical basis by Euler, also in 1744, and led to the principle of least action where the action was defined to be + mv ds, and later to the work of Lagrange and Hamilton and to Hamilton's principle, giving rise to the development of the subject of the calculus of variations.
The determination of frequency eigenvalues was discussed by Lord Rayleigh in his book The Theory of Sound in 1877 and led to Rayleigh's principle, which states that the fundamental frequency of a vibrating system is necessarily increased if constraints are imposed upon it and this gave rise to variational principles for determining the frequencies of vibration of strings and membranes as well as other mechanical systems, and subsequently to the Ritz variational method that has often been employed for evaluating the energy eigenvalues of quantum mechanical systems such as atoms and molecules. More recently, variational principles have been used with much success in the solution of various atomic scattering problems.
Thus variational principles are of fundamental importance for our understanding of the physical world and are of the greatest value for the accurate determination of its properties and I hope that my book goes some way towards demonstrating this.
B L Moiseiwitsch
Belfast
January 2004
Mathematics of Computation 21 (98) (1967), 284-286.
JOC/EFR October 2016
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