Foundations of Theoretical Mechanics II: Birkhoffian Generalization of Hamiltonian Mechanics

Description

Given a local, generally nonconservative, Newtonian system, it has been established in the preceding volume of this work [1978; MR0514210] 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