Foundations Of Theoretical Mechanics Ii: Birkhoffian Generalizations Of Hamiltonian Mechanics (theoretical And Mathematical Physics)

Given a local, generally nonconservative, Newtonian system, it has been established in the preceding volume of this work [1978; [MR0514210](http://ams.rice.edu/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=514210&loc=fromrevtext)] that, if and only if certain integrability conditions are satisfied, a Hamiltonian representation of the system can be obtained without the prior computation of a Lagrangian. Going on in this way, requiring now analyticity but dropping the previous integrability conditions, which were too restrictive from a physical point of view, the author proves the universal existence of a generalization of the Hamiltonian representation, a generalization preserving the experimenter's coordinates. He calls the equations obtained Birkhoff's equations in honor of G. D. Birkhoff, who introduced them in 1927, and shows that they generate a Lie-structure and a symplectic structure, in the same way as Hamilton's equations do; but the symplectic tensor is no longer the usual one, so that the algebra is said to be Lie-isotopic, and the geometry symplectic-isotopic. A general transformation theory is then developed, within the framework of which the classical canonical transformations appear so restrictive that their exclusive utilization would even prevent the user from finding certain purely Hamiltonian representations. Finally, a step-by-step generalization of the Hamilton-Jacobi theory and of Galilean relativity is proposed and discussed. A pleasant pedagogical organization has been chosen for the book. In each chapter, a basic presentation may be found in the main part of the text; then advanced topics are given in a number of separate complementary parts, called charts by the author; and finally follows a detailed treatment of examples. Most proofs and results are presented in local coordinates, for pedagogical reasons surely, but also and principally because the distinction between the Birkhoffian and the Hamiltonian descriptions, which is the very theme of the book, is completely lost in the global coordinate-free formulation of geometry. The richness of the work is remarkable. Though it is devoted to the Birkhoffian, Lie-isotopic and symplectic-isotopic representation of systems, it also emphasizes the limitations of this type of representation and introduces the reader, in some of the charts, to a further generalization, which this time is no longer of the Lie-type, but of a new type termed Birkhoffian-admissible, Lie-admissible and symplectic-admissible. It also shows that, if the constraint of preserving the experimenter's coordinates is removed, a Hamiltonian representation is equally possible in all cases. Thus the reader is offered an infinite variety of representations for the same generally nonconservative, physical system. When he chooses, the reader will obviously be sensible to the degree of computational feasibility of the different solutions offered but—and the author insists on this point—he will probably be most interested in the representations which allow a direct physical interpretation of all variables and functions at hand, which does not seem to be always the case for current representations. It must finally be stressed that many of the results presented are due to the author himself and that the very spirit of the book is also original. Reviewed by [Jean Fronteau](http://ams.rice.edu/mathscinet/search/author.html?mrauthid=69750) In the preceding volume, l I identified necessary and sufficient conditions for the existence of a representation of given Newtonian systems via a variational principle, the so-called conditions of variational self-adjointness. A primary objective of this volume is to establish that all Newtonian systems satisfying certain locality, regularity, and smoothness conditions, whether conservative or nonconservative, can be treated via conventional variational principles, Lie algebra techniques, and symplectic geometrical formulations. This volume therefore resolves a controversy on the repre sentational capabilities of conventional variational principles that has been 2 lingering in the literature for over a century, as reported in Chart 1. 3. 1. The primary results of this volume are the following. In Chapter 4,3 I prove a Theorem of Direct Universality of the Inverse Problem. It establishes the existence, via a variational principle, of a representation for all Newtonian systems of the class admitted (universality) in the coordinates and time variables of the experimenter (direct universality). The underlying analytic equations turn out to be a generalization of conventional Hamilton equations (those without external terms) which: (a) admit the most general possible action functional for first-order systems; (b) possess a Lie algebra structure in the most general possible, regular realization of the product; and (c) 1 Santilli (1978a). As was the case for Volume I, the references are listed at the end of this volume, first in chronological order and then in alphabetic order