### Online Materials

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### Syllabus IV

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### Study Guide

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### Text Book

### Tai L. Chow

### Classical Mechanics Second Edition CRC Press Francis & Taylor NY NY

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### goto

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### to download course materials or watch the lectures

## PHY337/637

## analyticalmEchanics

Instructor: Dr. David Carroll

Lecture: 103 Olin Hall

Time: 12:30 - 1:45 T/TH

Tutorial: Tuesdays

Ave. Out of Class Prep Time: 4 hours/class

Office Hours by Appointment: email

214 Olin Physical Laboratory, Reynolda Campus

Welcome to PHYS 337/637

This course introduces advanced methods in classical mechanics: specifically Lagrangian and Hamiltonian formulations of kinematics as well as non-inertial problems, non-integrable/chaotic problems and coupled oscillators. The approach is mathematically detailed and formal, with some focus on the underlying symmetries and geometries that will be of particular importance in gaining some deeper understanding of quantum mechanics and relativity. There will be some reference made to current problems in astrophysics and cosmology. The graduate 639 section is an excellent refresher for those preparing for the graduate qualifier; it requires a few additional assignments, but is essentially the same material. This course runs 1/2 a semester and is evaluated midterm (October).

Thematic Outline

(syllabus contains lecture times and topics along with accompanying tutorial assignments)

I. Setting up the problem

Extended bodies, Euler angles, and non inertial reference frames

Generalized coordinates, constraints to motion, and configuration space

Kinetic Energy in Generalized Coordinates and Generalized Momentum

II. Getting to a principle of motion

The principle of Virtual Work: D'Alembert and Maurpertuis

Variational Principles in mechanics and the Action

The full Lagrangian Formulation of Mechanics and Lagrangian Equations of Motion

III. Implications of this principle

Nonuniqueness of the Lagrangian

Integrals of Motion and Conservation Laws

Scale Invariance

Nonconservative Systems and Generalized Potential

Charged Particle in Electromagnetic Field

Forces of Constraint and Lagrange’s Multipliers

Lagrangian versus Newtonian Approach to Classical Mechanics

IV. Expressing configuration space in phase space

Descriptions of Motion in Phase Spaces and the Legendre Transform

Hamiltonian Formulation of Mechanics: The Hamiltonian of a Dynamic System

Hamilton’s Equations of Motion

V. Important details in phase space

Integrals of Motion and Conservation Theorems

Canonical Transformations

Poisson Brackets and Quantum Mechanics

Phase Space and Liouville’s Theorem

Time Reversal in Mechanics

VI. Advanced Applications according to time

Grades

The structure of the class is:

1. T/TH 12:30 to 1:15 Lectures

2. T/TH 1:15 - 1:45 class discussion and quizzes

3. Weekly Tutorials: Engagement and Homework (see below)

4. In class quizzes, Final Exam

Weighting:

30% final exam,

30% in-class quizzes,

40% tutorials/HW.

Prerequisites

This class has several prerequisites and given the very fast pace nature of the teaching as well as the advanced level of the materials, it is important to feel comfortable with the previous two mechanics classes you have taken. We will assume a working knowledge of Newton's Laws and their application (see first two chapters of your text for a refresher). The topics of most importance to you will be:

i. The freebody diagram

ii. Newton’s 3 main laws

iii. Kinetic and potential energy

iv. Collisions

v. *Extended bodies and Euler Angles

vi. Gravitational problems and Orbits

vii. *Harmonic and nonharmonic oscillations

* These topics are often not discussed in depth in the first part of advanced mechanics, but they will be really important for our discussions. So we will review these.

WHY is this interesting?

Newtonian mechanics isolates the forces applied to a body, decomposes those forces into their vector components, and then analyzes the motion of the body based upon Newton's second Law. To be predictive, we have to be precise about the nature of the forces (contact forces, gravitational forces, centripetal forces, and more) and their initial values. The equation of motion from Newton’s laws then provides us with a unique and unambiguous, space-time path of each object.

This works really well for some systems, and has formed the backbone of our "everyday" experience and observations. It does have two major flaws though. (1) First is that the approach does not allow for a simple scaling of complexity in its solutions. Fields such as astrophysics, fluid dynamics, and atmospheric science, can have many particles, a variety of interactions, and even using computers, the Newton II approach can be a very difficult way to solve problems. (2) Second, Newton II is only an approximation. Objects that are very small, moving very fast or are very massive (or near something very massive) do not move like F = ma. So while in some circumstances it works, in many, many others it doesn't.

Analytical Mechanics uses a different set of principles, symmetries and constraints to understand the possible motions of objects. The physical quantity of the system of most interest is the flow of energy between kinetic and potential forms and the action of objects as they move along trajectories. These concepts are a little easier to work with in configuration, and momentum/phase spaces, they allow for problem scaling with numbers of particles and they can be written in terms of relativistic and quantum mechanical invariants. Many also find an underlying elegance to the approach: the path of motion in space and time is dictated by the minimization of the flow of energy from one form to another...

The Tutorial System

This course uses a tutorial system of education. These are weekly meetings between one or two students and a tutor (the lecturer in this case) to discuss the weekly lectures and address assignments that are given by the tutor (generally in the form of HW). You must prepare for the tutorial weekly and be ready to explain answers to problems in detail. Note that this makes it pretty hard to "use the internet" to solve the HW. So it is strongly recommended that you work each problem yourself without too much outside aid. Of course working together in small groups can be helpful, just don't copy an answer without having actually worked on the problem first.