The essential part of the second volume of this important monograph [for part I (1978) see Zbl 0401.70015] consists of the introduction, three chapters and one very comprehensive appendix. The numeration of the chapters is simply the continuation of that of the previous volume, which contains also three chapters. In the introduction the author concisely recalls the classification of forces and formulates the theorems on self- adjoint Newtonian forces (potential forces). In the present volume non- potential, or in the terminology of the author ”non-self adjoint”, systems are studied from various points of view. Mainly Newtonian forces are taken into account, that is, forces which do not depend on accelerations \(\ddot q^ i\), \(i=1,...,n\). Here \(q^ i\) stand for generalized coordinates of a holonomic system.
In chapter 4, after recalling the basic results established in the former volume and concerning Hamilton’s equations, the so called Birkhoff equations are introduced. These equations are given by \[ (1)\quad(\partial R_{\nu}/\partial a^{\mu}-\partial R_{\mu}/\partial a^{\nu})\dot a^{\nu}-(\partial B/\partial a^{\mu}+\partial R_{\mu}/\partial t)=0, \] where \(\mu,\nu =1,2,...,2n\), \(R_{\nu}=R_{\nu}(t,a)\), \(B=B(t,a)\), \(a=(a^ 1,...,a^{2n})\), \(a=(\underline r,\underline y)=(r^ 1,...,r^ n;\quad y_ 1,...,y_ n),\) while t denotes time. If we set \(\underline y=\underline p\), \(\underline R=(\underline p,\underline 0)\), \(B=H=H(t,\underline r,\underline p)\), where H is the Hamiltonian, then we recover Hamilton’s equations. Birkhoff’s equations can be derived from the most general first-order variational principle \(\delta A=0\), where A is linear in \(\dot a\). The functional A is called Pfaffian action. The equations of motion of a holonomic, not necessarily conservative system can be represented in the form of a first-order system \(C_{\mu \nu}(t,a)\dot a^{\nu}+D_{\mu}(t,a)=0.\)
Under suitable assumptions, it can be proved that this system is self-adjoint if and only if it can be written in form of equ. (1). The essential results presented in chapter 4 concern the possibility to do so, which is not possible in general. Therefore a more general representation is introduced: \[ (2)\quad [(\partial R_{\nu})/(\partial a^{\mu})-(\partial R_{\mu})/(\partial a^{\nu})\dot a^{\nu}- (\partial B/(\partial a^{\mu})+(\partial R_{\mu})/(\partial t)]_{SA}\equiv \]
\[ \equiv [h^{\alpha}_{\mu}(C_{\alpha \nu}\dot a^{\nu}+D_{\alpha})_{NSA}]_{SA}, \] where SA means self-adjoint, NSA means non-self-adjoint, and \((h^{\alpha}_{\mu})=(h^{\alpha}_{\mu}(t,a))\) is a regular matrix. The representation (2) is called direct when the matrix \((h^{\alpha}_{\mu})\) is the unit matrix, otherwise it is called indirect. Non-self-adjointness is here due to the presence of non-potential forces. The functions B, \(R_{\nu}\) and \(h^{\alpha}_{\mu}\) can, in general, be determined only approximately. Algebraic and geometric study of Birkhoffian systems is a further objective of chapter 4. Seven ”charts”, or appendices, render the monograph self-contained, and on the other hand they contain more advanced or sometimes more detailed discussions of questions raised in the main text. The role of charts in the remaining chapters and in the appendix is similar.
In the second volume mainly local, generally non-potential, forces are taken into account. Generalization of Birkhoff’s equations to non-local non-potential system is presented in chart 4.7. This very general case requires new algebraic and geometric methods. For instance, the so called Lie-admissible algebras must be introduced.
Chapter 5 is entitled ”Transformation theory of Birkhoff’s equations”. Before proceeding to this particular topic, the contemporary approach to the theory of canonical transformations of Hamilton’s equations is reviewed in an elegant manner. The insufficiency of canonical transformations is pointed out. Also the canonoid, or not quite canonical transformations are briefly discussed. The main section of this chapter deals with various aspects of equations resulting from transformations of Birkhoff’s equations. Particularly it is shown that the general, non- canonical transformations \(t\to t'(t,a),\quad a^{\mu}\to a^{\prime\mu}(t,a)\) transform Hamilton’s equations into Birkhoff’s equations. In seven charts appended to chapter 5 such topics as a generalization of Lie’s theory including the universal enveloping algebras, Darboux’s theorem of the symplectic and contact geometries, isotopic and genotopic transformations of variational principles are discussed. One chart presents eight definitions of canonical transformations existing in the physical and mathematical literature; they are not all equivalent. A drawback of chapter 5 is the lack of physical examples. In the present form this chapter will rather appeal to mathematically inclined readers.
In chapter 6 essentially three topics are studied. Firstly, having in mind a generalization of the well-known canonical quantization rules, a formally admissible Birkhoffian generalization of the Hamilton-Jacobi equations is investigated. Hence, it is then possible to introduce Birkhoffian quantization rules, which yield the hadronic generalization of Schrödinger equations. The second topic is related to the theorem on indirect universality of Hamilton’s equations. Roughly speaking, this deep theorem states that ordinary differential equations admit an indirect Hamiltonian representation. The third topic concerns Galileian relativity. It turns out that Galilei’s relativity imposes generally excessive restrictions on the acting forces. Therefore a possible generalization of Galileian relativity is proposed and discussed. Galilei’s relativity is an expression of some techniques of Hamiltonian mechanics. Hence we infer that a possible generalization can be constructed via Birkhoffian mechanics. The charts completing chapter 6 briefly indicate some applications of the Birkhoffian mechanics to hadron physics, statistical mechanics, space mechanics, biophysics and engineering.
The comprehensive appendix includes 5 charts dealing mainly with various aspects of Lagrangian representations of a non-self-adjoint system of the ordinary differential equations \(A_{ki}(t,q,\dot q)\ddot q^ i+B_ k(t,q,\dot q)=0.\) In the remaining six charts conservation laws and symmetries for self-adjoint and non-self-adjoint systems are concisely dealt with. When studying the appendix, the reader must pay attention to the numeration of the formulas when referring to them in the text. This referring is often erroneous.
It should be added that each chapter and the appendix are completed with several instructive examples and problems for solving by one’s single- handed efforts.
The number of footnotes is, in my opinion, too large. At least part of them could be included in the main text. The assertion: ”Engineering systems are non-self-adjoint as a rule...”, seems to me exaggerated.
The opera magna by the author is most probably one of the first monographs in the world literature which deals so thoroughly and extensively with non-potential systems. This inspiring monograph, which is, let me add, very carefully edited by Springer-Verlag, will be indispensable for researchers having to do with variational problems of theoretical mechanics, continuum and structural mechanics, various fields of physics, biomechanics biophysics and biomathematics. I think that for mathematicians interested in rather mathematical aspects of the calculus of variations this monograph will also be very useful. The monograph can also be used as the textbook for various graduate courses on variational methods in mechanics and theoretical physics and as a supplement to courses of the calculus of variations. [Slightly abridged version, the complete review is available on demand.]