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Additional Physical Format: | Online version: Santilli, Ruggero Maria, 1935- Inverse problem in Newtonian mechanics. New York : Springer-Verlag, ©1978 (OCoLC)561917143 Online version: Santilli, Ruggero Maria, 1935- Inverse problem in Newtonian mechanics. New York : Springer-Verlag, ©1978 (OCoLC)606150479 |
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Material Type: | Internet resource |

Document Type: | Book, Internet Resource |

All Authors / Contributors: |
Ruggero Maria Santilli |

ISBN: | 0387088741 9780387088747 3540088741 9783540088745 |

OCLC Number: | 9020170 |

Description: | xix, 266 pages ; 24 cm. |

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' 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. |

Series Title: | Texts and monographs in physics. |

Responsibility: | Ruggero Maria Santilli. |

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