Mathematical Methods of Classical Mechanics

Portada
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.
 

Contenido

Part
2
Legendre transformations
14
Chapter
15
Investigation of motion in a central field
33
Motions of a system of n points
44
The method of similarity
50
Liouvilles theorem
68
Holonomic constraints
85
Generating functions
266
Actionangle variables
279
Averaging
285
Averaging of perturbations
291
Appendix 1
301
Appendix 2
310
Geodesics of leftinvariant metrics on Lie groups and
318
Symplectic structure on algebraic manifolds
346

Linearization
98
91
111
Inertial forces and the Coriolis force
129
Eulers equations Poinsots description of the motion
142
Part III
160
Appendix 9
178
Integration of differential forms
181
Symplectic geometry
219
Parametric resonance in systems with many degrees of freedom
225
Chapter 9
233
Applications of the integral invariant of PoincaréCartan
240
Huygens principle
248
The HamiltonJacobi method for integrating Hamiltons canonical
258
Appendix 5
371
Appendix 6
381
Appendix 8
393
Perturbation theory of conditionally periodic motions
399
Chapter 8
401
Poincarés geometric theorem its generalizations
416
Appendix 11
438
Calculus of variations
452
Appendix 13
453
163
458
Lagranges equations
459
Derechos de autor

Otras ediciones - Ver todas

Términos y frases comunes

Información bibliográfica