Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics(Theoretical and Mathematical Physics)

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: I. 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; 1. The identification of the methods for the construction of a Lagrangian or Hamiltonian from the given equations of motion; 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.

Table of Contents:
1 Elemental Mathematics.- 1.1 Existence theory for implicit functions, solutions, and derivatives in the parameters.- 1.2 Calculus of differential forms, Poincare lemma, and its converse.- 1.3 Calculus of variations, action functional, and admissible variations.- Charts:.- 1.1 A theorem on the existence, uniqueness, and continuity of the implicit functions for Newtonian systems.- 1.2 A theorem on the existence, uniqueness, and continuity of a solution of a Newtonian initial value problem.- 1.3 A theorem on the existence, uniqueness, and continuity of the derivatives with respect to parameters of solutions of Newtonian systems.- 1.4 A relationship between local and global solutions for conservative systems.- 1.5 Hilbert space approach to Newtonian Mechanics.- Examples.- Problems.- 2 Variational Approach to Self-Adjointness.- 2.1 Equations of motion, admissible paths, variational forms, adjoint systems and conditions of self-adjointness.- 2.2 Conditions of self-adjointness for fundamental and kinematical forms of Newtonian systems.- 2.3 Reformulation of the conditions of self-adjointness within the context of the calculus of differential forms.- 2.4 The problem of phase space formulations.- 2.5 General and normal forms of the equations of motion.- 2.6 Variational forms of general and normal systems.- 2.7 Conditions of self-adjointness for general and normal systems.- 2.8 Connection with self-adjointness of linear operators.- 2.9 Algebraic significance of the conditions of self-adjointness.- Charts:.- 2.1 Hausdorff, second -countable, ?-differentiable manifolds.- 2.2 Newtonian systems as vector fields on manifolds.- 2.3 Symplectic manifolds.- 2.4 Contact manifolds.- 2.5 Geometrical significance of the conditions of self-adjointness.- Examples.- Problems.- 3 The Fundamental Analytic Theorems of the Inverse Problem.- 3.1 Statement of the problem.- 3.2 The conventional Lagrange's equations.- 3.3 Self-adjointness of the conventional Lagrange's equations.- 3.4 The concept of analytic representation in configuration space.- 3.5 The fundamental analytic theorem for configuration space formulations.- 3.6 A method for the construction of a Lagrangian from the equations of motion.- 3.7 The implications of nonconservative forces for the structure of a Lagrangian.- 3.8 Direct and inverse Legendre transforms for conventional analytic representations.- 3.9 The conventional Hamilton's equations.- 3.10 Self-adjointness of the conventional Hamilton's equations.- 3.11 The concept of analytic representation in phase space.- 3.12 The fundamental analytic theorem for phase space formulations and a method for the independent construction of a Hamiltonian.- Charts.- 3.1 The controversy on the representation of nonconservative Newtonian systems with the conventional Hamilton's principle.- 3.2 The arena of applicability of Hamilton's principle.- 3.3 Generalization of Hamilton's principle to include the integrability conditions for the existence of a Lagrangian.- 3.4 Generalization of Hamilton's principle to include Lagrange's equations and their equations of variation.- 3.5 Generalization of Hamilton's principle to include Lagrange's equations, their equations of variations, and the end points contributions.- 3.6 Generalization of Hamilton's principle to include a symplectic structure.- 3.7 Generalization of Hamilton's principle for the unified treatment of the Inverse Problem in configuration and phase space.- 3.8 Self-adjointness of first-order Lagrange's equations.- 3.9 The fundamental analytic theorem for first-order equations of motion in configuration space.- 3.10 A unified treatment of the conditions of self-adjointness for first-, second-, and higher-order ordinary differential equations.- 3.11 Engels' methods for the construction of a Lagrangian.- 3.12 Mertens'approach to complex Lagrangians.- 3.13 Bateman's approach to the Inverse Problem.- 3.14 Douglas'approach to the Inverse Problem.- 3.15 Rapoport's approach to the Inverse Problem.- 3.16 Vainberg's approach to the Inverse Problem.- 3.17 Tonti's approach to the Inverse Problem.- 3.18 Analytic, algebraic and geometrical significance of the conditions of variational self-adjointness.- Examples.- Problems.- Appendix: Newtonian Systems.- A. 1 Newton's equations of motion.- A.2 Constraints.- A.3 Generalized coordinates.- A.4 Conservative systems.- A.5 Dissipative systems.- A.6 Dynamical systems.- A.7 The fundamental form of the equations of motion in configuration space.- A.l Galilean relativity.- A.2 Ignorable coordinates and conservation laws.- A.3 Impulsive motion.- A.4 Arrow of time and entropy.- A.5 Gauss principle of least constraint.- A.6 The Gibbs-Appel equations.- A.7 Virial theorem.- A.8 Liouville's theorem for conservative systems.- A.9 Generalizations of Liouville's theorem to dynamical systems.- A. 10 The method of Lagrange undetermined multipliers.- A. 11 Geometric approach to Newtonian systems.- A. 12 Tensor calculus for linear coordinate transformations.- A. 13 Tensor calculus for nonlinear coordinate transformations.- A. 14 Dynamical systems in curvilinear coordinates.- Examples.- Problems.- References.