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E-raamat: Classical Mechanics with Calculus of Variations and Optimal Control

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Classical Mechanics with Calculus of Variations and Optimal ControlSuurem pilt
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This is an intuitively motivated presentation of many topics in classical mechanics and related areas of control theory and calculus of variations. All topics throughout the book are treated with zero tolerance for unrevealing definitions and for proofs which leave the reader in the dark.

Some areas of particular interest are: an extremely short derivation of the ellipticity of planetary orbits; a statement and an explanation of the tennis racket paradox''; a heuristic explanation (and a rigorous treatment) of the gyroscopic effect; a revealing equivalence between the dynamics of a particle and statics of a spring; a short geometrical explanation of Pontryagin's Maximum Principle, and more.

In the last couple of chapters, aimed at more advanced readers, the Hamiltonian and the momentum are compared to forces in a certain static problem. This gives a palpable physical meaning to some seemingly abstract concepts and theorems. With minimal prerequisites consisting of basic calculus and basic undergraduate physics, this book is suitable for courses from an undergraduate to a beginning graduate level, and for a mixed audience of mathematics, physics and engineering students. Much of the enjoyment of the subject lies in solving almost 200 problems in this book.

This book is published in cooperation with Mathematics Advanced Study Semesters.

Arvustused

This book can be recommended to students and also to everyone involved in preparing an introductory course in advanced classical mechanics, due to the well-selected material and, even more so, the clear presentation." - Zentralblatt Math

"One of the most valuable aspects of the book - unfortunately rare among textbooks - is that we see an author in command of his subject who shares not just the bare facts but how he thinks about them and how all the pieces fit together." - MAA Reviews

Series Foreword: MASS and REU at Penn State University v
Preface xiii
Chapter 1 One Degree of Freedom
1(74)
§ 1 The setup
1(1)
§ 2 Equations of motion
2(4)
§ 3 Potential energy
6(4)
§ 4 Kinetic energy
10(2)
§ 5 Conservation of total energy
12(2)
§ 6 The phase plane
14(4)
§ 7 Lagrangian equations of motion
18(1)
§ 8 The variational meaning of the Euler--Lagrange equation
19(2)
§ 9 Euler--Lagrange equations -- general theory
21(2)
§ 10 Noether's theorem/Energy conservation
23(1)
§ 11 Hamiltonian equations of motion
24(2)
§ 12 The phase flow
26(2)
§ 13 The divergence
28(3)
§ 14 A lemma on moving domains
31(3)
§ 15 Divergence as a measure of expansion
34(1)
§ 16 Liouville's theorem
35(1)
§ 17 The "uncertainty principle" of classical mechanics
36(3)
§ 18 Can one hear the shape of the potential?
39(3)
§ 19 A dynamics-statics equivalence
42(6)
§ 20
Chapter summary
48(2)
§ 21 Problems
50(25)
Chapter 2 More Degrees of Freedom
75(66)
§ 1 Newton's laws
75(2)
§ 2 Center of mass
77(2)
§ 3 Newton's second law for multi-particle systems
79(1)
§ 4 Angular momentum, torque
80(1)
§ 5 Rotational version of Newton's second law; conservation of the angular momentum
81(3)
§ 6 Circular motion: angular position, velocity, acceleration
84(1)
§ 7 Energy and angular momentum of rotation
85(2)
§ 8 The rotational -- translational analogy
87(1)
§ 9 Potential force fields
87(3)
§ 10 Some physical remarks
90(1)
§ 11 Conservation of energy
91(1)
§ 12 Central force fields; conservation of angular momentum
92(2)
§ 13 Kepler's problem
94(2)
§ 14 Kepler's trajectories are conies: a short proof
96(4)
§ 15 Motion in linear central fields
100(4)
§ 16 Linear vibrations: derivation of the equations
104(1)
§ 17 A nonholonomic system
105(3)
§ 18 The modal decomposition of vibrations
108(3)
§ 19 Lissajous' figures and Chebyshev's polynomials
111(2)
§ 20 Invariant 2-tori in R4
113(4)
§ 21 Rayleigh's quotient and a physical interpretation
117(2)
§ 22 The Coriolis and the centrifugal forces
119(3)
§ 23 Miscellaneous examples
122(4)
§ 24 Problems
126(15)
Chapter 3 Rigid Body Motion
141(26)
§ 1 Reference frames, angular velocity
142(1)
§ 2 The tensor of inertia
143(4)
§ 3 The kinetic energy
147(1)
§ 4 Dynamics in the body frame
148(2)
§ 5 Euler's equations of motion
150(1)
§ 6 The tennis racket paradox
151(1)
§ 7 Poinsot's description of free rigid body motion
152(2)
§ 8 The gyroscopic effect -- an intuitive explanation
154(2)
§ 9 The gyroscopic torque
156(1)
§ 10 Speed of precession
157(2)
§ 11 The gyrocompass
159(2)
§ 12 Problems
161(6)
Chapter 4 Variational Principles of Mechanics
167(16)
§ 1 The setting
167(1)
§ 2 Lagrange's equations
168(1)
§ 3 Examples
169(2)
§ 4 Hamilton's principle
171(1)
§ 5 Hamilton's principle ⇔ Euler--Lagrange equations
172(1)
§ 6 Advantages of Hamilton's principle
173(1)
§ 7 Maupertuis' principle -- some history
174(1)
§ 8 Maupertuis' principle on an example
175(1)
§ 9 Maupertuis' principle -- a more general statement
176(1)
§ 10 Discussion of the Maupertuis principle
177(2)
§ 11 Problems
179(4)
Chapter 5 Classical Problems of Calculus of Variations
183(28)
§ 1 Introduction and an overview
183(1)
§ 2 Dido's problem -- a historical note
184(1)
§ 3 A special class of Lagrangians
185(2)
§ 4 The shortest way to the smallest integral
187(2)
§ 5 The brachistochrone problem
189(3)
§ 6 Johann Bernoulli's solution of the brachistochrone problem
192(2)
§ 7 Geodesies in Poincare's metric
194(2)
§ 8 The soap film, or the minimal surface of revolution
196(4)
§ 9 The catenary: formulating the problem
200(1)
§ 10 Minimizing with constraints -- Lagrange multipliers
201(2)
§ 11 Catenary -- the solution
203(1)
§ 12 An elementary solution for the catenary
204(1)
§ 13 Problems
205(6)
Chapter 6 The Conditions of Legendre and Jacobi for a Minimum
211(22)
§ 1 Conjugate points
212(3)
§ 2 The Legendre and the Jacobi conditions
215(2)
§ 3 Quadratic functional: the fundamental theorem
217(2)
§ 4 Sufficient conditions for a minimum for a general functional
219(3)
§ 5 Necessity of the Legendre condition for a minimum
222(1)
§ 6 Necessity of the Jacobi condition for a minimum
223(3)
§ 7 Some intuition on positivity of functionals
226(3)
§ 8 Problems
229(4)
Chapter 7 Optimal Control
233(24)
§ 1 Formulation of the problem
233(2)
§ 2 The Maximum Principle
235(2)
§ 3 A geometrical explanation of the Maximum Principle
237(7)
§ 4 Example 1: a smooth landing
244(2)
§ 5 Example 2: stopping a harmonic oscillator
246(4)
§ 6 Huygens's principle vs. Maximum Principle
250(2)
§ 7 Background on linearized and adjoint equations
252(2)
§ 8 Problems
254(3)
Chapter 8 Heuristic Foundations of Hamiltonian Mechanics
257(38)
§ 1 Some fundamental questions
257(1)
§ 2 The main idea
258(5)
§ 3 The Legendre transform, the Hamiltonian, the momentum
263(1)
§4 Properties of the Legendre transform
264(2)
§ 5 The Hamilton--Jacobi equation
266(1)
§ 6 Noether's theorem
267(3)
§ 7 Conservation of energy
270(1)
§ 8 Poincare's integral invariants
271(2)
§ 9 The generating function
273(1)
§ 10 Hamilton's equations
273(1)
§ 11 Hamiltonian mechanics as the "spring theory"
274(6)
§ 12 The optical-mechanical analogy
280(4)
§ 13 Hamilton--Jacobi equation leading to the Schrodinger equation
284(3)
§ 14 Examples and Problems
287(8)
Bibliography 295(2)
Index 297
Mark Levi, Pennsylvania State University, University Park, PA, USA
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