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Lecture Notes on Classical Mechanics (A Work in Progress)

Lecture Notes on Classical Mechanics (A Work in Progress) Daniel ArovasDepartment of PhysicsUniversity of California, San DiegoMay 8, Preface .. xiii0 Reference Lagrangian Mechanics (mostly) .. Hamiltonian Mechanics (mostly) .. Mathematics ..21 Introduction to Introduction and Review .. Newton s laws of motion .. Aside : mass .. Examples of Motion in One Dimension .. Uniform force .. Uniform force with linear frictional damping .. Uniform force with quadratic frictional damping .. Crossed electric and magnetic fields .. Pause for Reflection ..102 Systems of Work-Energy Theorem .. Conservative and Nonconservative Forces .. Example : integratingF= U.

Contents 0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 0 Reference Materials 1 0.1 Lagrangian Mechanics (mostly ...

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Transcription of Lecture Notes on Classical Mechanics (A Work in Progress)

1 Lecture Notes on Classical Mechanics (A Work in Progress) Daniel ArovasDepartment of PhysicsUniversity of California, San DiegoMay 8, Preface .. xiii0 Reference Lagrangian Mechanics (mostly) .. Hamiltonian Mechanics (mostly) .. Mathematics ..21 Introduction to Introduction and Review .. Newton s laws of motion .. Aside : mass .. Examples of Motion in One Dimension .. Uniform force .. Uniform force with linear frictional damping .. Uniform force with quadratic frictional damping .. Crossed electric and magnetic fields .. Pause for Reflection ..102 Systems of Work-Energy Theorem .. Conservative and Nonconservative Forces .. Example : integratingF= U.

2 Conservative Forces in Many Particle Systems .. Linear and Angular Momentum .. Scaling of Solutions for Homogeneous Potentials .. Euler s theorem for homogeneous functions .. Scaled equations of motion .. Appendix I : Curvilinear Orthogonal Coordinates .. Example : spherical coordinates .. Vector calculus : grad, div, curl .. Common curvilinear orthogonal systems .. Rectangular coordinates .. Cylindrical coordinates .. Spherical coordinates .. Kinetic energy .. 253 One-Dimensional Conservative Description as a Dynamical System .. Example : harmonic oscillator .. One-Dimensional Mechanics as a Dynamical System.

3 Sketching phase curves .. Fixed Points and their Vicinity .. Linearized dynamics in the vicinity of a fixed point .. Examples of Conservative One-Dimensional Systems .. Harmonic oscillator .. Pendulum .. Other potentials .. 364 Linear Damped Harmonic Oscillator .. Classes of damped harmonic motion .. Remarks on the case of critical damping .. Phase portraits for the damped harmonic oscillator .. Damped Harmonic Oscillator with Forcing .. Resonant forcing .. Examples .. General solution by Green s function method .. General Linear Autonomous Inhomogeneous ODEs .. Kramers-Kr onig Relations (advanced material ).

4 595 Calculus of Snell s Law .. Functions and Functionals .. Functional Taylor series .. Examples from the Calculus of Variations .. Example 1 : minimal surface of revolution .. Example 2 : geodesic on a surface of revolution .. Example 3 : brachistochrone .. Ocean waves .. Appendix : More on Functionals .. 726 Lagrangian Generalized Coordinates .. Hamilton s Principle .. Invariance of the equations of motion .. Remarks on the order of the equations of motion .. Lagrangian for a free particle .. Conserved Quantities .. Momentum conservation .. Energy conservation .. Choosing Generalized Coordinates.

5 How to Solve Mechanics Problems .. Examples .. One-dimensional motion .. Central force in two dimensions .. A sliding point mass on a sliding wedge .. A pendulum attached to a mass on a spring .. The double pendulum .. The thingy .. Appendix : Virial Theorem .. 947 Noether s Continuous Symmetry Implies Conserved Charges .. Examples of one-parameter families of transformations .. Conservation of Linear and Angular Momentum .. Advanced Discussion : Invariance ofS.. The Hamiltonian .. IsH=T+U? .. Example: A bead on a rotating hoop .. Charged Particle in a Magnetic Field .. Fast Perturbations : Rapidly Oscillating Fields.

6 Example : pendulum with oscillating support .. Field Theory: Systems with Several Independent Variables .. Gross-Pitaevskii model ..1148 Constraints and Variational Calculus .. Constrained Extremization of Functions .. Extremization of Functionals : Integral Constraints .. Extremization of Functionals : Holonomic Constraints .. Examples of extremization with constraints .. Application to Mechanics .. Constraints and conservation laws .. Worked Examples .. One cylinder rolling off another .. Frictionless motion along a curve .. Disk rolling down an inclined plane .. Pendulum with nonrigid support .. Falling ladder.

7 Point mass inside rolling hoop .. 1379 Central Forces and Orbital Reduction to a one-body problem .. Center-of-mass (CM) and relative coordinates .. Solution to the CM problem .. Solution to the relative coordinate problem .. Almost Circular Orbits .. Precession in a Soluble Model .. The Kepler Problem:U(r) = kr 1.. Geometric shape of orbits .. Laplace-Runge-Lenz vector .. Kepler orbits are conic sections .. Period of bound Kepler orbits .. Escape velocity .. Satellites and spacecraft .. Two examples of orbital Mechanics .. Appendix I : Mission to Neptune .. I. Earth to Jupiter .. II. Encounter with Jupiter.

8 III. Jupiter to Neptune .. Appendix II : Restricted Three-Body Problem .. 16610 Small Coupled Coordinates .. Expansion about Static Equilibrium .. Method of Small Oscillations .. Can you really just choose an A so that both these wonderful thingshappen in and .. to elaborate? .. Finding the modal matrix .. Example: Masses and Springs .. Example: Double Pendulum .. Zero Modes .. Example of zero mode oscillations .. Chain of Mass Points .. Continuum limit .. Appendix I : General Formulation .. Appendix II : Additional Examples .. Right Triatomic Molecule .. Triple Pendulum .. Equilateral Linear Triatomic Molecule.

9 Aside : Christoffel Symbols .. 19911 Elastic Center of Mass Frame .. Central Force Scattering .. Hard sphere scattering .. Rutherford scattering .. Transformation to laboratory coordinates .. 20812 Noninertial Reference Accelerated Coordinate Systems .. Translations .. Motion on the surface of the earth .. Spherical Polar Coordinates .. Centrifugal Force .. Rotating tube of fluid .. The Coriolis Force .. Foucault s pendulum .. 22013 Rigid Body Motion and Rotational Rigid Bodies .. Examples of rigid bodies .. The Inertia Tensor .. Coordinate transformations .. The case of no fixed point .. Parallel Axis Theorem .. Example.

10 General planar mass distribution .. Principal Axes of Inertia .. Euler s Equations .. Example .. Euler s Angles .. Torque-free symmetric top .. Symmetric top with one point fixed .. Rolling and Skidding Motion of Real Tops .. Rolling tops .. Skidding tops .. Tippie-top .. 24414 Continuum Strings .. d Alembert s Solution to the Wave Equation .. Energy density and energy current .. Reflection at an interface .. Mass point on a string .. Interface between strings of different mass density .. Finite Strings : Bernoulli s Solution .. Sturm-Liouville Theory .. Variational method .. Continua in Higher Dimensions.


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