Mathematical Methods of Classical Mechanics

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Springer Science & Business Media, 2013 M11 11 - 464 páginas
Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) and linear algebra (linear operators, quadratic forms). With the help of this apparatus, we examine all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the hamiltonian formalism. The author has tried to show the geometric, qualitative aspect of phenomena. In this respect the book is closer to courses in theoretical mechanics for theoretical physicists than to traditional courses in theoretical mechanics as taught by mathematicians.
 

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Contenido

NEWTONIAN MECHANICS
2
Chapter
14
1
50
Legendre transformations
61
15
65
Chapter 4
75
E Noethers theorem
88
22
98
Applications of the integral invariant of PoincaréCartan
240
Huygens principle
248
The HamiltonJacobi method for integrating Hamiltons canonical
258
53
260
Generating functions
266
Actionangle variables
279
Averaging
285
Averaging of perturbations
291

Small oscillations
103
28
107
Chapter 6
123
Eulers equations Poinsots description of the motion
142
30
148
Part III
160
33
170
Appendix 9
178
Integration of differential forms
181
42
191
Chapter 8
201
The Lie algebra of hamiltonian functions
214
Symplectic geometry
219
Parametric resonance in systems with many degrees of freedom
225
Chapter 9
233
Appendix 1
301
Appendix 2
317
Appendix 3
323
Symplectic structure on algebraic manifolds
346
Appendix 5
371
Appendix 6
381
Appendix 8
393
Perturbation theory of conditionally periodic motions
399
Poincarés geometric theorem its generalizations
416
Appendix 11
438
Calculus of variations
452
Appendix 13
453
Lagranges equations
459
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