classical-langrange approach - Lectures on Classical...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Lectures on Classical Mechanics by John C. Baez notes by Derek K. Wise Department of Mathematics University of California, Riverside LaTeXed by Blair Smith Department of Physics and Astronomy Louisiana State University 2005
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
i c 2005 John C. Baez & Derek K. Wise
Image of page 2
ii
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
iii Preface These are notes for a mathematics graduate course on classical mechanics. I’ve taught this course twice recently. The first time I focused on the Hamiltonian approach. This time I started with the Lagrangian approach, with a heavy emphasis on action principles, and derived the Hamiltonian approach from that. Derek Wise took notes. The chapters in this L A T E X version are in the same order as the weekly lectures, but I’ve merged weeks together, and sometimes split them over chapter, to obtain a more textbook feel to these notes. For reference, the weekly lectures are outlined here. Week 1: (Mar. 28, 30, Apr. 1)—The Lagrangian approach to classical mechanics: deriving F = ma from the requirement that the particle’s path be a critical point of the action. The prehistory of the Lagrangian approach: D’Alembert’s “principle of least energy” in statics, Fermat’s “principle of least time” in optics, and how D’Alembert generalized his principle from statics to dynamics using the concept of “inertia force”. Week 2: (Apr. 4, 6, 8)—Deriving the Euler-Lagrange equations for a particle on an arbitrary manifold. Generalized momentum and force. Noether’s theorem on conserved quantities coming from symmetries. Examples of conserved quantities: energy, momen- tum and angular momentum. Week 3 (Apr. 11, 13, 15)—Example problems: (1) The Atwood machine. (2) A frictionless mass on a table attached to a string threaded through a hole in the table, with a mass hanging on the string. (3) A special-relativistic free particle: two Lagrangians, one with reparametrization invariance as a gauge symmetry. (4) A special-relativistic charged particle in an electromagnetic field. Week 4 (Apr. 18, 20, 22)—More example problems: (4) A special-relativistic charged particle in an electromagnetic field in special relativity, continued. (5) A general-relativistic free particle. Week 5 (Apr. 25, 27, 29)—How Jacobi unified Fermat’s principle of least time and Lagrange’s principle of least action by seeing the classical mechanics of a particle in a potential as a special case of optics with a position-dependent index of refraction. The ubiquity of geodesic motion. Kaluza-Klein theory. From Lagrangians to Hamiltonians. Week 6 (May 2, 4, 6)—From Lagrangians to Hamiltonians, continued. Regular and strongly regular Lagrangians. The cotangent bundle as phase space. Hamilton’s equa- tions. Getting Hamilton’s equations directly from a least action principle. Week 7 (May 9, 11, 13)—Waves versus particles: the Hamilton-Jacobi equation.
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern