Foundations Of Theoretical Mechanics I: The Inverse Problem In Newtonian Mechanics (theoretical And Mathematical Physics)

The conditions under which a function admits a Lagrangian or Hamiltonian are known as "[conditions of variational self-adjointness](https://physics.stackexchange.com/a/287993/10024)." * * * The author states that this monograph grew out of his uneasiness in teaching classical Newtonian mechanics. So many techniques (analytical, variational, algebraic) are applicable only to systems with forces derivable from a potential. The usual Lagrangian representation of a physical system is at best a crude approximation of reality. The physical reality is full of nonconservative systems with nonholonomic constraints. The author attempts to reformulate various problems of physics from the so-called inverse calculus of variations point of view. Given a complete set of solutions of ordinary differential equations of order r:Fk(x,y,y′,y′′,⋯,y(r))=0r\colon F_k(x,y,y',y'',\cdots,y^{(r)})=0, determine whether a functional A(y)=∫x1x2L(x,y,y′⋯yr−1)dxA(y)=\int_{x_1}^{x_2}L(x,y,y'\cdots y^{r-1})\,dx exists which admits these solutions as extremals. The author offers a careful discussion of the selfadjointness conditions, their algebraic and variational significance. In many respects this monograph is unique. The relations between the properties of the Lagrangians and the solutions of Jacobi's equations have not been investigated before to the best of the reviewer's knowledge. The Morse-Feshbach Lagrangian is introduced and some rather puzzling symmetry breaking phenomena under the gauge transformation φ→φeiωe\varphi\rightarrow\varphi e^{i\omega e} are pointed out in a footnote referring to the author's research. The central theme concerns the theorem on necessary and sufficient conditions for the existence of a Lagrangian, and of a Hamiltonian. The author makes no claim to originality in the introduction. In this respect he is much too modest. The monograph surveys many known facts, but does present them from a new point of view, and a large section of the material presented here originated with the author's research. Reviewed by [Vadim Komkov](http://ams.rice.edu/mathscinet/search/author.html?mrauthid=104470) 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 aspects: 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