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AN INTRODUCTION TO LAGRANGIAN MECHANICS

AN INTRODUCTION TOLAGRANGIAN MECHANICSA lain J. BrizardDepartment of Chemistry and PhysicsSaint Michael s College, Colchester, VT 05439 July 7, 2007iPrefaceThe original purpose of the present lecture notes on Classical MECHANICS was to sup-plement the standard undergraduate textbooks (such as Marion and Thorton sClassicalDynamics of Particles and Systems) normally used for an intermediate course in Classi-cal MECHANICS by inserting a more general and rigorous INTRODUCTION to LAGRANGIAN andHamiltonian methods suitable for undergraduate physics students at sophomore and ju-nior levels. The outcome of this effort is that the lecture notes are now meant to providea self-consistent INTRODUCTION to Classical MECHANICS without the need of any is expected that students taking this course will have hada one-year calculus-basedintroductory physics course followed by a one-semester course in Modern physics .

introductory physics course followed by a one-semester course in Modern Physics. Ideally, students should have completed their three-semester calculus sequence by the time they enroll in this course and, perhaps, have taken a course in ordinary differential equations. On the other hand, this course should be taken before a rigorous course in ...

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Transcription of AN INTRODUCTION TO LAGRANGIAN MECHANICS

1 AN INTRODUCTION TOLAGRANGIAN MECHANICSA lain J. BrizardDepartment of Chemistry and PhysicsSaint Michael s College, Colchester, VT 05439 July 7, 2007iPrefaceThe original purpose of the present lecture notes on Classical MECHANICS was to sup-plement the standard undergraduate textbooks (such as Marion and Thorton sClassicalDynamics of Particles and Systems) normally used for an intermediate course in Classi-cal MECHANICS by inserting a more general and rigorous INTRODUCTION to LAGRANGIAN andHamiltonian methods suitable for undergraduate physics students at sophomore and ju-nior levels. The outcome of this effort is that the lecture notes are now meant to providea self-consistent INTRODUCTION to Classical MECHANICS without the need of any is expected that students taking this course will have hada one-year calculus-basedintroductory physics course followed by a one-semester course in Modern physics .

2 Ideally,students should have completed their three-semester calculus sequence by the time theyenroll in this course and, perhaps, have taken a course in ordinary differential the other hand, this course should be taken before a rigorous course in QuantumMechanics in order to provide students with a sound historical perspective involving theconnection between Classical physics and Quantum , the second semesterof the sophomore year or the fall semester of the junior year provide a perfect niche for structure of the lecture notes presented here is based onachieving several a first goal, I originally wanted to model these notes afterthe wonderful monographof Landau and Lifschitz onMechanics, which is often thought to be too concise for mostundergraduate students.

3 One of the many positive characteristics of Landau and Lifschitz sMechanicsis that LAGRANGIAN MECHANICS is introduced in its first chapter and not in laterchapters as is usually done in more standard textbooks used at the sophomore/juniorundergraduate level. Consequently, LAGRANGIAN mechanicsbecomes the centerpiece of thecourse and provides a continous thread throughout the a second goal, the lecture notes introduce several numerical investigations of dynam-ical equations appearing throughout the text. These numerical investigations present aninteractive pedagogical approach, which should enable students to begin their own numer-ical investigations. As a third goal, an attempt was made to introduce relevant historicalfacts (whenever appropriate) about the pioneers of Classical MECHANICS .

4 Much of the his-torical information included in the Notes is taken from Ren e Dugas (History of MECHANICS ,1955), Wolfgang Yourgrau and Stanley Mandelstam (Variational Principles in Dynamicsand Quantum Theory, 1968), or Cornelius Lanczos (The Variational Principles of Me-chanics, 1970). In fact, from a pedagogical point of view, this historical perspective helpseducating undergraduate students in establishing the deepconnections between Classicaland Quantum MECHANICS , which are often ignored or even inverted (as can be observedwhen undergraduate students are surprised to hear that Hamiltonians have an indepen-dent classical existence). As a fourth and final goal, I wanted to keep the scope of theseiinotes limited to a one-semester course in contrast to standard textbooks, which often in-clude an extensive review of Newtonian MECHANICS as well as additional material such asHamiltonian standard topics covered in these notes are listed in order as follows: Introductionto the Calculus of Variations (Chapter 1), LAGRANGIAN MECHANICS (Chapter 2), HamiltonianMechanics (Chapter 3), Motion in a Central Field (Chapter 4), Collisions and ScatteringTheory (Chapter 5), Motion in a Non-Inertial Frame (Chapter6), Rigid Body Motion(Chapter 7), Normal-Mode Analysis (Chapter 8), and Continuous LAGRANGIAN Systems(Chapter 9).

5 Each chapter contains a problem set with variable level of difficulty; sectionsidentified with an asterisk may be omitted for a one-semestercourse. Lastly, in orderto ensure a self-contained presentation, a summary of mathematical methods associatedwith linear algebra, and trigonometic and elliptic functions is presented in Appendix B presents a brief summary of the derivation of the Schr odinger equation basedon the LAGRANGIAN formalism developed by R. P. innovative topics not normally discussed in standard undergraduate textbooksare included throughout the notes. In Chapter 1, a complete discussion of Fermat s Princi-ple of Least Time is presented, from which a generalization of Snell s Law for light refrac-tion through a nonuniform medium is derived and the equations of geometric optics areobtained.

6 We note that Fermat s Principle proves to be an ideal INTRODUCTION to variationalmethods in the undergraduate physics curriculum since students are already familiar withSnell s Law of light Chapter 2, we establish the connection between Fermat s Principle and Maupertuis-Jacobi s Principle of Least Action. In particular, Jacobi s Principle introduces a geometricrepresentation of single-particle dynamics that establishes a clear pre-relativistic connectionbetween Geometry and physics . Next, the nature of mechanical forces is discussed withinthe context of d Alembert s Principle, which is based on a dynamical generalization of thePrinciple of Virtual Work. Lastly, the fundamental link between the energy-momentumconservation laws and the symmetries of the LAGRANGIAN function is first discussed throughNoether s Theorem and then Routh s procedure to eliminate ignorable coordinates is ap-plied to a LAGRANGIAN with Chapter 3, the problem of charged-particle motion in an electromagnetic field isinvestigated by the LAGRANGIAN method in the three-dimensional configuration space andthe Hamiltonian method in the six-dimensional phase important physicalexample presents a clear link between the two INTRODUCTION to the Calculus of Foundations of the Calculus of Variations.

7 A Simple Minimization Problem .. Methods of the Calculus of Variations .. Path of Shortest Distance and Geodesic Equation .. Classical Variational Problems .. Isoperimetric Problem .. Brachistochrone Problem .. Fermat s Principle of Least Time .. Light Propagation in a Nonuniform Medium .. Snell s Law .. Application of Fermat s Principle .. Geometric Formulation of Ray Optics .. Frenet-Serret Curvature of Light Path .. Light Propagation in Spherical Geometry .. Geodesic Representation of Light Propagation .. Eikonal Representation .. Problems .. 282 LAGRANGIAN Maupertuis-Jacobi s Principle of Least Action .. Maupertuis principle .. Jacobi s principle .. D Alembert s Principle.

8 Principle of Virtual Work .. Lagrange s Equations from D Alembert s Principle .. Hamilton s Principle and Euler-Lagrange Equations .. Constraint Forces .. Generalized Coordinates in Configuration Space .. Constrained Motion on a Surface .. Euler-Lagrange Equations .. LAGRANGIAN MECHANICS in Curvilinear Coordinates .. LAGRANGIAN MECHANICS in Configuration Space .. Example I: Pendulum .. Example II: Bead on a Rotating Hoop .. Example III: Rotating Pendulum .. Example IV: Compound Atwood Machine .. Example V: Pendulum with Oscillating Fulcrum .. Symmetries and Conservation Laws .. Energy Conservation Law .. Momentum Conservation Laws .. Invariance Properties of a LAGRANGIAN .

9 LAGRANGIAN MECHANICS with Symmetries .. Routh s Procedure for Eliminating Ignorable Coordinates .. LAGRANGIAN MECHANICS in the Center-of-Mass Frame .. Problems .. 643 Hamiltonian Canonical Hamilton s Equations .. Legendre Transformation* .. Hamiltonian Optics and Wave-Particle Duality* .. Particle Motion in an Electromagnetic Field* .. Euler-Lagrange Equations .. Energy Conservation Law .. Gauge Invariance .. Canonical Hamilton s Equations .. One-degree-of-freedom Hamiltonian Dynamics .. Simple Harmonic Oscillator .. Pendulum .. Constrained Motion on the Surface of a Cone .. Charged Spherical Pendulum in a Magnetic Field .. LAGRANGIAN and Routhian .. Routh-Euler-Lagrange equations.

10 Hamiltonian .. Problems .. 894 Motion in a Central-Force Motion in a Central-Force Field .. LAGRANGIAN Formalism .. Hamiltonian Formalism .. Turning Points .. Homogeneous Central Potentials .. The Virial Theorem .. General Properties of Homogeneous Potentials .. Kepler Problem .. Bounded Keplerian Orbits .. Unbounded Keplerian Orbits .. Laplace-Runge-Lenz Vector .. Isotropic Simple Harmonic Oscillator .. Internal Reflection inside a Well .. Problems .. 1115 Collisions and Scattering Two-Particle Collisions in the LAB Frame .. Two-Particle Collisions in the CM Frame .. Connection between the CM and LAB Frames .. Scattering Cross Sections .. Definitions .. Scattering Cross Sections in CM and LAB Frames.


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