Foundations of Theoretical Mechanics: The Inverse Problem in Newtonian Mechanics

by Ruggero Maria Santilli

The objective of this monograph is to present some methodological foundations of theoretical mechanics that are recommendable to graduate students prior to, or jointly with, the study of more advanced topics such as statistical mechanics, thermodynamics, and elementary particle physics. A program of this nature is inevitably centered on the methodological foundations for Newtonian systems, with particular reference to the central equations of our theories, that is, Lagrange's and Hamilton's equations. This program, realized through a study of the analytic representations in terms of Lagrange's and Hamilton's equations of generally nonconservative Newtonian systems (namely, systems with Newtonian forces not necessarily derivable from a potential function), falls within the context of the so-called Inverse Problem, and consists of three major l. The study of the necessary and sufficient conditions for the existence of a Lagrangian or Hamiltonian representation of given equations of motion with arbitrary forces; 2. The identification of the methods for the construction of a Lagrangian or Hamiltonian from given equations of motion verifying conditions 1; and 3 The analysis of the significance of the underlying methodology for other aspects of Newtonian Mechanics, e. g. , transformation theory, symmetries, and first integrals for nonconservative Newtonian systems. This first volume is devoted to the foundations of the Inverse Problem, with particular reference to aspects I and 2.

Genres: Physics

266 pages, Hardcover

First published January 1, 1978

Reviews

[Kyle · Rating 5 out of 5]

Santilli does an excellent job of explaining the problem that the book is taking on. Given a system of equations of motion (in a Newtonian sense), how can one know if there is a Lagrangian or Hamiltonian that can represent it? What types of Newtonian systems can be represented?

The author systematically takes on this problem in what I think is a very approachable way for anyone with some graduate physics education (it is advanced enough that it could stretch advanced physics undergraduates, but I think the main ideas are still approachable). Despite what you may have heard, it is possible to represent non-conservative forces with a Lagrangian or Hamiltonian. Indeed, Santilli presents the way to calculate them and the conditions necessary for a non-conservative system to be representable by a Hamiltonian/Lagrangian. Indeed the details of what it means to be representable is an interesting subject that the author explains well.

Despite being written in the late 1970's, the lessons of this book are surprisingly unknown in modern physics treatments. You often hear that Lagrangian/Hamiltonian systems are only good for conservative systems (or that variational approaches are only good for conservative forces) and this book not only gives counterexamples, but explains why such an approach is not only possible, but understandable.

Given the advanced subject and somewhat rarity of the volume, I'm not sure how many will want to read this book, but it covers the subject and problem it lays out extremely well.

(This usually goes without saying, but given the author's, shall we say unconventional, career, this review is for the book and its content alone and is not a comment or endorsement for Santilli's general ideas or works)