The Variational Principles of Mechanics

Cornelius Lanczos

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486134703
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Analytical mechanics is, of course, a topic of perennial interest and usefulness in physics and engineering, a discipline that boasts not only many practical applications, but much inherent mathematical beauty. Unlike many standard textbooks on advanced mechanics, however, this present text eschews a primarily technical and formalistic treatment in favor of a fundamental, historical, philosophical approach. As the author remarks, there is a tremendous treasure of philosophical meaning` behind the great theories of Euler and Lagrange, Hamilton, Jacobi, and other mathematical thinkers.
Well-written, authoritative, and scholarly, this classic treatise begins with an introduction to the variational principles of mechanics including the procedures of Euler, Lagrange, and Hamilton.
Ideal for a two-semester graduate course, the book includes a variety of problems, carefully chosen to familiarize the student with new concepts and to illuminate the general principles involved. Moreover, it offers excellent grounding for the student of mathematics, engineering, or physics who does not intend to specialize in mechanics, but wants a thorough grasp of the underlying principles.
The late Professor Lanczos (Dublin Institute of Advanced Studies) was a well-known physicist and educator who brought a superb pedagogical sense and profound grasp of the principles of mechanics to this work, now available for the first time in an inexpensive Dover paperback edition. His book will be welcomed by students, physicists, engineers, mathematicians, and anyone interested in a clear masterly exposition of this all-important discipline.

CHAPTER VIII

THE PARTIAL DIFFERENTIAL EQUATION OF HAMILTON-JACOBI

Put off thy shoes from off thy feet, for the place whereon thou standest is holy ground.

EXODUS III, 5

Introduction. We have done considerable mountain climbing. Now we are in the rarefied atmosphere of theories of excessive beauty and we are nearing a high plateau on which geometry, optics, mechanics, and wave mechanics meet on common ground. Only concentrated thinking, and a considerable amount of re-creation, will reveal the full beauty of our subject in which the last word has not yet been spoken. We start with the integration theory of Jacobi and continue with Hamilton’s own investigations in the realm of geometrical optics and mechanics. The combination of these two approaches leads to de Broglie’s and Schroedinger’s great discoveries, and we come to the end of our journey.

1. The importance of the generating function for the problem of motion. In the theory of canonical transformations no other theorem is of such importance as that a canonical transformation is completely characterized by knowing one single function S, the generating function of the transformation. This parallels the fact that the canonical equations are likewise characterized by one single function, the Hamiltonian function H. These two fundamental functions can be linked together by a definite relation. In order to solve the problem of motion it suffices to consider the Hamiltonian function and try to simplify it to a form in which the canonical equations become directly integrable. For this purpose a suitable canonical transformation can be employed. But this transformation depends on one single function. And thus the problem of solving the entire canonical set can be replaced by the problem of solving a single equation. This equation happens to be a partial differential equation.

From the practical viewpoint not much is gained. The solution of a partial differential equation—even one equation—is no easy task, and in most cases is not simpler than the original integration problem. Those problems which can be solved explicitly by means of the partial differential equation are, in the majority of cases, the same problems which can be solved also by other means. For this reason the Hamiltonian methods were long considered as of purely mathematical interest and of little practical importance. The philosophical value of these methods, the entirely new understanding they furnished for the deeper problems of mechanics, remained unnoticed except by a few scientists who were impressed by the extraordinary beauty of the Hamiltonian developments. Among these we may mention especially Jacobi; while later there were Helmholtz, Lie, Poincare, and, in our day, de Broglie and Schroedinger. In contemporary physics, the Hamiltonian methods gained recognition because of the optico-mechanical analogy which was made clear by Hamilton’s partial differential equation. Since the advent of Schroedinger’s wave mechanics, which is based essentially on Hamilton’s researches, the leading ideas of Hamiltonian mechanics have found their way into the textbooks of theoretical physics. Yet even so, the technical side of the theory is primarily stressed, at the cost of the philosophical side.

From the point of view adopted here, the purely technical side of the subject is of minor importance. The principal interest is focused on the basic significance of the theory and the interrelation of its various aspects. We shall thus discuss in succession the Jacobian and the Hamiltonian theories and exhibit the central role which the partial differential equation of Hamilton- Jacobi plays in these developments.

Summary. Canonical transformations are characterized by one single function, the generating function. The problem of finding a proper canonical transformation which simplifies the Hamiltonian function to a form in which the equations are directly integrate is thus equivalent to the problem of constructing one single function. This function is determined by a single partial differential equation. The problem of solving the entire system of canonical equations can be replaced by the problem of solving this one equation.

2. Jacobi’s transformation theory. Let us consider a conservative mechanical system with a given Hamiltonian function H which does not depend on the time t We wish to transform the mechanical variables q₁,...,q_n; p₁,...,p_n into a new set of variables Q₁,...,Q_n; P₁,...,P_n by a canonical transformation. We do not specify this canonical transformation, except for a single condition, namely, that the Hamiltonian function H shall be on...