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.
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ISBN-13
978-3-540-09482-1 (9783540094821)
DOI
10.1007/978-3-642-86760-6
Schweitzer Klassifikation
4 Birkhoff's Equations.- 4.1 Statement of the problem.- 4.2 Birkhoff's equations.- 4.3 Birkhofflan representations of Newtonian systems.- 4.4 Isotopic and genotopic transformations of first-order systems.- 4.5 Direct universality of Birkhoff's equations.- Charts:.- 4.1 Lack of algebraic character of nonautonomous Birkhoff's equations.- 4.2 Algebraic significance of isotopic and genotopic transformations.- 4.3 Havas's theorem of universality of the inverse problem for systems of arbitrary order and dimensionality.- 4.4 Rudiments of differential geometry.- 4.5 Global treatment of Hamilton's equations.- 4.6 Global treatment of Birkhoff's equations.- 4.7 Lie-admissible/symplectic-admissible generalization of Birkhoff's equations for nonlocal nonpotential systems.- Examples 98 Problems.- 5 Transformation Theory of Birkhoff's Equations.- 5.1 Statement of the problem.- 5.2 Transformation theory of Hamilton's equations.- 5.3 Transformation theory of Birkhoff's equations.- Charts:.- 5.1 Need to generalize the contemporary formulation of Lie's theory.- 5.2 Isotopic generalization of the universal enveloping associative algebra.- 5.3 Isotopic generalization of Lie's first, second, and third theorems.- 5.4 Isotopic generalizations of enveloping algebras, Lie algebras, and Lie groups in classical and quantum mechanics.- 5.5 Darboux's theorem of the symplectic and contact geometries.- 5.6 Some definition of canonical transformations.- 5.7 Isotopic and genotopic transformations of variational principles 188 Examples.- Problems.- 6 Generalization of Galilei's Relativity.- 6.1 Generalization of Hamilton-Jacobi theory.- 6.2 Indirect universality of Hamilton's equations.- 6.3 Generalization of Galilei's relativity.- Charts:.- 6.1 Applications to hadron physics.- 6.2 Applications to statistical mechanics.- 6.3 Applications to space mechanics.- 6.4 Applications to engineering.- 6.5 Applications to biophysics.- Examples.- Problems.- Appendix A: Indirect Lagrangian Representations.- A.1 Indirect Lagrangian representations within fixed local variables.- A.2 Isotopic transformations of a Lagrangian.- A.3 Indirect Lagrangian representations via the use of the transformation theory.- Charts:.- A.1 Analytic Newtonian systems.- A.2 Analytic extensions of Lagrangian and Hamiltonian functions to complex variables.- A.3 The Cauchy-Kovalevski theorem.- A.4 Kobussen's treatment of Darboux's theorem of universality for one-dimensional systems.- A.5 Vanderbauwhede's functional approach to the inverse problem.- A.6 Symmetries.- A.7 Lie's construction of symmetries of given equations of motion.- A.8 First integrals and conservation laws.- A.9 Noether's construction of first integrals from given symmetries.- A.10 Isotopic transformations, symmetries, and first integrals.- A.11 Lack of a unique relationship between space-time symmetries and physical laws340 A.12 Classification of the breakings of space-time symmetries in Newtonian mechanics344 Examples.- Problems.- References.
