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Mechanical Sytems, Classical Models

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All phenomena in nature are characterized by motion; this is an essential property of matter, having infinitely many aspects. Motion can be mechanical, physical, chemical or biological, leading to various sciences of nature, mechanics being one of them. Mechanics deals with the objective laws of mechanical motion of bodies, the simplest form of motion.

In the study of a science of nature mathematics plays an important rôle. Mechanics is the first science of nature which was expressed in terms of mathematics by considering various mathematical models, associated to phenomena of the surrounding nature. Thus, its development was influenced by the use of a strong mathematical tool; on the other hand, we must observe that mechanics also influenced the introduction and the development of many mathematical notions.

In this respect, the guideline of the present book is precisely the mathematical model of mechanics. A special accent is put on the solving methodology as well as on the mathematical tools used; vectors, tensors and notions of - field theory. Continuous and discontinuous phenomena, various mechanical magnitudes are presented in a unitary form by means of the theory of distributions. Some appendices give the book an autonomy with respect to other works, special previous mathematical knowledge being not necessary.

Some applications connected to important phenomena of nature are presented, and this also gives one the possibility to solve problems of interest from the technical, engineering point of view. In this form, the book becomes we dare say a unique outline of the literature in the field; the author wishes to present the most important aspects connected with the study of mechanical systems, mechanics being regarded as a science of nature, as well as its links to other sciences of nature. Implications in technical sciences are not neglected.

Author(s): Petre P. Teodorescu
Series: Mathematical and Analytical Techniques with Applications to Engineering
Edition: 1
Publisher: Springer
Year: 2007

Language: English
Pages: 783

1402054416......Page 1
CONTENTS......Page 7
PREFACE......Page 10
1.1 Basic notions......Page 12
1.2 Mathematical model of mechanics......Page 56
2.1 Physical quantities. Units......Page 68
2.2 Homogeneity......Page 73
2.3 Similitude......Page 81
1.1 Decomposition of forces. Bases......Page 86
1.2 Products of vectors......Page 90
2.1 Moments......Page 100
2.2 Reduction of systems of forces......Page 106
1.1 Centres of mass......Page 126
1.2 Moments of inertia......Page 136
2.1 Displacements......Page 173
2.2 Constraints......Page 177
1.1 Statics of the particle......Page 212
1.2 Acceleration of the particle......Page 304
2.1 Statics of rigid solids......Page 240
2.2 Statics of threads......Page 286
1.1 Trajectory and velocity of the particle......Page 298
1.3 Particular cases of motion of a particle......Page 311
2.1 Kinematical formulae in the motion of a rigid solid......Page 316
2.2 Particular cases of motion of the rigid solid......Page 321
2.3 General motion of the rigid solid......Page 327
3.1 Relative motion of a particle......Page 341
3.2 Relative motion of the rigid solid......Page 344
3.3 Kinematics of systems of rigid solids......Page 351
1.1 Introductory notions......Page 363
1.2 General theorems......Page 375
2.1 General considerations......Page 394
2.2 Motion of the particle with one or two degrees of freedom......Page 399
1.1 Rectilinear and plane motion......Page 411
1.2 Motion of a heavy particle......Page 416
1.3 Pendulary motion......Page 432
2. Other problems of dynamics of the particle......Page 450
2.1 Tautochronous motions. Motions on a brachistochrone and on a geodesic curve......Page 451
2.2 Other applications......Page 462
2.3 Stability of equilibrium of a particle......Page 467
1.1 General results......Page 478
1.2 Other problems......Page 485
2. Motion of a particle subjected to the action of an elastic force......Page 489
2.1 Mechanical systems with two degrees of freedom......Page 490
2.2 Mechanical systems with a single degree of freedom......Page 503
1.1 Principle of universal attraction......Page 552
1.2 Theory of Newtonian potential......Page 558
2.1 Motion of celestial bodies......Page 564
2.2 Problem of artificial satellites of the Earth and of interplanetary vehicles......Page 586
2.3 Applications to the theory of motion at the atomic level......Page 596
1.1 Particle dynamics......Page 603
1.2 General theorems......Page 614
2.1 Conservative vectors. Gradient......Page 623
2.2 Differential operators of first and second order......Page 637
3.1 Mathematical model of the motion. General theorems......Page 667
3.2 Motion of a particle of variable mass in a gravitational field......Page 682
3.3 Mathematical pendulum. Motion of a particle of variable mass in a field of central forces......Page 690
3.4 Applications of Meshcherskli's generalized equation......Page 697
1.1 Vector analysis......Page 700
1.2 Exterior differential calculus......Page 712
2. Notions of field theory......Page 716
2.3 Integral formulae......Page 730
3.1 Composition of distributions......Page 738
3.2 Integral transforms in distributions......Page 740
3.3 Applications to the study of differential equations. Basic solutions......Page 743
REFERENCES......Page 752
F......Page 771
M......Page 772
P......Page 773
T......Page 774
W......Page 775
B......Page 776
E......Page 777
H......Page 778
L......Page 779
N......Page 780
R......Page 781
T......Page 782
Z......Page 783