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Unformatted text preview: Physics 507: Theoretical Mechanics Contents I. Lagrangian Mechanics 2 A. Equations of Motion 2 B. Symmetries and Conservation Laws 2 C. Back to Newton 3 II. Classical Integrable Problems 4 A. Motion in 1D 4 B. Motion in Central Field 4 1. Two-body problem 4 2. Mapping onto one-dimensional problem 5 3. Keplers Problem 5 C. Scaling & Virial Theorem 7 D. Scattering in Central Field 7 1. Rutherfords Formula 8 2. Switching to the Laboratory reference frame. 8 E. Perturbative methods 8 III. Rotational Motion & Mechanics of Rigid Body 10 A. Fixed-axis rotation 10 B. Fixed-point rotation 10 C. Euler equations 11 D. Heavy Spinning Top 12 E. Non-inertial Reference Frames 13 IV. Oscillations 14 A. Free oscillations 14 B. Dissipation and driving force. 15 1. 1D problem 15 2. Oscillations under periodic force. 16 3. D > 1 system with dumping 16 C. Parametric resonance 17 V. Hamiltonian Mechanics 18 A. Deterministic dynamical system 18 B. Hamiltonian Formalism 18 C. Semiclassical (WKB) approximation in Quantum Mechanics 20 D. Poisson Brackets 20 E. Canonical Transformations 22 VI. Introduction to Chaos Theory 24 A. Major Concepts and Tools 24 B. Chaos in Hamiltonian Systems 24 C. Non-Hamiltonian Chaos: Bifurcations and Attractors 25 2 I. LAGRANGIAN MECHANICS A. Equations of Motion Principle of least action ( S = R L ( q, q, t ) d t ) results in Lagrange Equation of motion(LE): q i ( t ) t 2 Z t 1 L ( q, q, t ) d t = 0 d d t L q i = L q i (1) Here q ( q 1 , q 2 ..., q N ) are generalized coordinates, p i = L / q i is the generalized momentum conjugated to q i , L /q i is the generalized force, and q i is the generalized velocity. LE generalizes the second Newtons Law. Important: L does not depend on q i , ... q i etc. As a result, LE is the second order di f erential equation for q , i.e. it determines function q i ( q, q, t ) . B. Symmetries and Conservation Laws Time translation invariance ( t t + t ) means that L /t = 0 : L = L t t = t L q i L q i q i L q i = t d d t L q i L q i = 0 . (2) we obtain the law of conservation of mechanicanl energy : E q i p i L = const (3) Noether Theorem: If L is invariant under certain coordinate transformation ( group ): q i q i +(d q i / d s ) s , than: L ( q, q, t ) = s L q i d q i d s + L q i d q i d s = s d d t d q i d s L q i = 0 (4) Therefore, d q i d s L q i = const (5) Example 1: translational invariance: r a r a + r ( r a are Cartesian coordinates of the a-th particle). r is the same for all particles, and it de f nes three independent translations ( x , y , z ). From Noether theorem, L = r d d t X a L r a = 0 , one gets conservation of total linear momentum ( p a L / r a ): P = X a p a = const (6)...
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This note was uploaded on 09/28/2008 for the course PHYS 507 taught by Professor Anon during the Spring '04 term at Cornell University (Engineering School).
- Spring '04