Analytical Mechanics (Undergraduate Lecture Notes in Physics)

Book Preface

This textbook presents what Joseph-Louis Lagrange called Analytical Mechanics. Historically this was a great advance beyond the methods of Euclidean geometry employed by Isaac Newton in the Philosophiae Naturalis Principia Mathematica. With the methods of Lagrange and Leonhard Euler, we could actually perform calculations. Lagrange and Euler used the calculus and did not require the formidable expertise in the use of geometry that Newton possessed.

The step introduced by William Rowan Hamilton simplified the formulation. Hamilton’s ideas also represent a great step forward in our understanding of the meaning of Analytical Mechanics. This, coupled with the simplification added by Carl Gustav Jacobi, provided us with a pathway to the more modern uses of Analytical Mechanics including applications to astrophysics, complex systems, and chaos.

Our approach will introduce a modern version of what was done in the 18th and 19th centuries. We will follow essentially the historical development because the ideas unfold most logically if we do so. We will, however, pay more attention to the development of Analytical Mechanics as a valuable tool than to a historical study.

Our final step will be the relativistic formulation of Analytical Mechanics. That is an absolute necessity in any complete study of Analytical Mechanics.

Logically we begin this text with a chapter on the history of mechanics. Many texts include brief historical comments or even added pages outlining individual contributions. That is certainly an improvement on the anecdotes that our professors often passed on to us without citation. Those anecdotes piqued our interest and added flavor. But they lacked a continuity of thought and that all-important accuracy that we prize. Analytical Mechanics is the oldest of the sciences. And the history stretches from the beginnings of philosophy in Miletus in 600 BCE to the advances in scientific thought introduced in the Prussian Academy and in Great Britain. I have sincerely endeavored to shorten this, as any serious student will easily recognize. But I still worry about the length.

Because my own understanding of science has been greatly enriched by studies in history, I cannot recommend that a professor ignore the first chapter completely. The student should understand something of the interesting and tortured individual Newton was. And we cannot really comprehend the origins of the ideas that gave birth to Analytical Mechanics without encountering the work of Pierre Maupertuis, Johann Bernoulli,1 Euler, and Lagrange. The sections of Hamilton and Jacobi may be held until after the students have gained an appreciation for the methods of Analytical Mechanics. But those sections will be of interest to students as they encounter the chapter on the Hamilton-Jacobi approach. They should see the simplicity of what Jacobi brought and his great respect for the ideas of Hamilton. Then to emphasize the importance these ideas, I include an outline of Erwin Schrödinger’s original published derivation of his wave equation from the Hamilton-Jacobi equation. With the caveat surrounding a second variation, the quantum theory is buried in the theory of Hamilton and Jacobi.