# Physics 308a — Mechanics of Discrete and Continuous Systems

# Jerry Gollub - Fall, 2010

**
Brief Summary:** Classical mechanics of systems of particles, and also continua such as fluids. Topics from particle mechanics include Lagrangian mechanics, systems of particles, non-inertial frames of reference, and chaotic oscillations. Fluid topics include the, kinematics of fluids, the Navier-Stokes equations, vorticity, boundary layers, and instabilities.

## 1. Description

The subject of classical mechanics is part of the core of physics. This course treats some of the traditional topics of classical mechanics at an intermediate level, but it also includes some topics of more recent interest, such as chaotic dynamics, that will help students to see mechanics as a lively subject that is still rapidly evolving. The second half of the course treats the dynamics of fluids, which is widely applicable to other disciplines including astrophysics, oceanography, the atmospheric sciences, engineering, and biological systems.

*Mathematics and numerics*: The following topics are probably familiar to some degree, and will appear at various points in the course: Vector cross product; line integrals; curl of a vector function; curvilinear coordinates; eigenvalues and eigenvectors of matrices; solving linear second-order differential equations with constant coefficients; numerical integration; and writing or modifying simple programs. You will probably find Mathematica useful for homework, and you will also have an opportunity to utilize Matlab. The “Mathematical Appendices” developed by Lyle Roelofs and previous HC students may be helpful.

## 2. Course Objectives

Learn to distinguish between different levels of understanding of mechanics, from basic comprehension, to application to new situations, to critical analysis.

Contribute actively to classroom learning

Use Newtonian mechanics to predict the dynamics of one and two particle systems with known forces analytically and numerically.

Use the Lagrangian approach to solve model problems in mechanics.

Explain how chaos affects the predictability of mechanical systems, and how rotation affects apparent forces that influence motion.

Utilize the Navier-Stokes equations to explain the dynamics of fluid continua, and elucidate the underlying concepts (e.g. the stress tensor; vorticity; instability).

Apply that knowledge to some discipline that is of interest to you (e.g. astrophysics, biological physics, geophysics, etc.).

## 2. Texts and sources

**There will be two primary required texts:**

*Classical Mechanics* by John R. Taylor: __http://www.uscibooks.com/ taylor2.htm__

*Physical Hydrodynamics,* by E. Guyon , J-P Hulin, L. Petit, and C.D. Mitescu.

The first of these has been widely praised The second is in my judgment the best book available on continuum dynamics for physics students.

**Other books on traditional mechanics (on Reserve):**

Fowles and Cassiday, “Analytical Mechanics”, 6^{th} edition, 1999.

Barger and Olsson, Classical Mechanics: A Modern Perspective, 1995.

T.W.B. Kibble, *Classical Mechanics* (third edition, 1985). Written by a well known British theoretical physicist. It's rather terse, but has interesting problems. It is available in paperback for a very modest price.

S. Thornton and J. Marion, *Classical dynamics of particles and systems*, 2004.

Baierlein, *Newtonian Dynamics* (1983). Less formal than Marion and Thornton; quite readable, but not as thorough.

P. Smith and R.C. Smith, *Mechanics (1990)*. This modern book contains a treatment of chaotic dynamics, but the traditional material seems a bit dull.

**Books on Chaos**

G. Baker and J.P. Gollub, *Chaotic Dynamics: An Introduction*, second edition, 1996.

S. H. Strogatz, *Nonlinear dynamics and chaos*, 1994.

**Other books and resources on continuum mechanics**

D. Tritton, *Physical Fluid Dynamics*

M. van Dyke*, An Album of Fluid Motion*

T.E. Faber, *Fluid Dynamics for Physicists*

B. Lautrup, *Continuum Mechanics*

H. Tennekes, *The Simple Science of Flight*

S. Vogel, *Life in Moving Fluids*

F.W. Taylor, *Elementary Climate Physics*

*Multimedia Fluid Mechanics DVD*

## A blackboard site for this course will be maintained, where you can get copies of handouts, etc.

## 3. Format of the course

I like classroom interaction and believe it is important for your learning. To ensure that there is substantial interaction, and to make sure that the problem sets are integrated with the course, there is a *required* weekly tutorial (most weeks). This meeting is in addition to the regular classes. Occasionally, a lecture and tutorial may be interchanged.

## 4. Course requirements

*Regular* *attendance:* You are expected to attend all classes and tutorials. Absences for any reason *other* than illness will be considered excessive if more than two in number at class or tutorial. *If you are ill or expect to miss class due to special circumstances, a timely e-mail message to that effect is expected*. Especially in a small course, each person counts.

*Preparation*: You are asked to come to class prepared. The best learning occurs when you read or skim the upcoming material before class, and then read more carefully after class. Interaction during lectures is expected and encouraged.

*Homework*: There will be regular homework assignments. You are expected to complete the homework (or at least to make a serious effort on each problem) *on time*. Some problems will be designated as "individual" problems, for which you may not consult with classmates, but may use the library. You should *not* expect to be able to do *all* of the problems successfully.

*Tutorials*: These sessions will be devoted to discussing homework, both to provide assistance, and to give you practice in oral presentation. The class will be divided into working groups. Each week, one problem will be assigned to each problem-solving group for presentation. The groups meet in advance of the tutorial to confer, and one or more members of the group will present the problem. Presentation is expected to rotate so that over time, everyone presents equally often. Tutorial groups are expected to meet before tutorial. *Note that there isn't time to present solutions in detail. Rather a sketch or list of key issues should be given, with advice, partial answers, and pitfalls to avoid. You can then respond to questions.*

Everyone can earn high esteem from the instructor and classmates by contributing generously to the discussion, volunteering ideas, and also by posing questions when you're mystified. We will strive for a collegial atmosphere in which corrections are proffered gently and accepted graciously without embarrassment. Attendance is mandatory, and it is expected that you will have worked on *each* problem (not necessarily successfully) in advance. I recognize that you will work harder before tutorial on the problems for which your group is responsible, but in preparing your final solutions, you will make up for this.

*Exams:* There will be *two* take-home exams, of which one is at the end of the semester. Each will be 2-3 hours in duration, and will be “closed book” except for a page of notes.

*Fluids Paper:* A paper in which you present an application of fluid dynamics from a neighboring discipline (geophysics, atmospheric science, astrophysics, etc.).

## 5. Grading

*Grading philosophy:* When evaluating students' work, I look for effort, engagement, and mastery on an individual basis. The performance of one student does not affect the grades received by others. Historically, the average grade of students in upper level courses I have taught is around 3.3 (B+ on a letter grade scale), but each class is different.

*Weights:* The following weights will be applied to the various components.

Exams (2) 25%, 25%

Homework 25%

Fluids Paper 25%

*Homework*: I (and perhaps an assistant) will read the "individual" problems and check a subset of the others. The “individual” problems will be weighted more. __Please note that clarity and the use of explanatory prose is expected.__

*Late assignments:* Homework is generally due at the class following tutorial. Homework turned in up to one week late will receive 75% of the earned grade. (Solutions are posted on that day.) Beyond that, you may obtain up to 50% credit, and it is permissible in this case to refer to the posted solutions for ideas, but not to copy the solutions. A*dditional* *exception*: you may have a "free extension" for up to one week without penalty once in the term. Please save it for a high pressure time. Simply turn in a sheet of paper with your name, the assignment number, and the remark that you are taking the "free extension."

Finally, in case of illness (e.g.. fever, etc., *not* just minor colds or tiredness due to inadequate sleep), you may request extra time, without using your free extension. I will generally grant credible requests, but please ask only when you have a serious need.

*Absences* (for reasons other than illness) will be considered in the final evaluation if more than two were noted.

## 6. Honor code

You are allowed and encouraged to discuss homework with each other (except on "individual" problems), but it would be wise to attempt them yourself first. Your written work must be your own, though it may be influenced by prior discussion*. This means, for example, that you may not look at another student's written work while preparing your own.* Two assignments that have *substantially identical wording or equations in one paragraph or more* would be presumed in violation of the honor code.

*Sources* (e.g. books) or other assistance should be acknowledged. However, you don't need to acknowledge the help of students in your problem-solving group since this interaction is understood.

The purpose of these provisions is to enhance your learning experience, and to enable the instructor to tell how well each student is doing.

## 7. Interacting with the instructor

*Electronic mail* (sent to *jgollub*) is always welcome and will generally be answered between 2 and 6 p.m. on weekdays and often on Sunday afternoon or evening.

*Phone*: You may phone my office at 896-1196. Please leave a message if I'm not there, along with a suggested time in the *evening* when I can return your call. It's OK to call me at home (649-4159) *from 9 to 11 p.m.* occasionally, Sunday through Thursday.

*Office:* I will be *very* happy to speak with you immediately after any class period (at 10:30 a.m.) , or to arrange an appointment at a mutually agreeable time. Please also do not feel shy about dropping in or about phoning; if busy, I'll suggest another time.

I look forward to working with you. *I appreciate your feedback, and urge you to communicate any concerns without delay.*

*Disabilities:* Students who think they may need accommodations because of a disability should meet with me privately *early* in the semester, and should also contact Rick Webb, Coordinator, Office of Disabilities Services to verify their eligibility for (and the nature of) an accommodation. Exams in this course will be generously timed.

## 8. Sequence of Topics

This is an approximate list. I may adjust the content and rate depending on progress in the course.

**Material from Taylor (by week)**

Date | Topic | Reading |
---|---|---|

8/30 | Basics | Ch. 1-2 |

9/6 | Momentum; angular momentum; energy | Ch. 3-4 |

9/13 | Lagrangian mechanics | Ch. 6-7 |

9/20 | Central forces | Ch. 8 |

9/27 | Non-inertial frames of reference | Ch. 9 |

10/4 | Chaos | Ch. 12 |

10/11 | BREAK | Ch. 12 |

10/18 | Scattering theory | Ch. 14 |

**Material from Guyon et al.**

Date | Topic | Reading |
---|---|---|

10/25 | Kinematics of fluids | Ch. 3 |

11/1 | Navier-Stokes equations | Ch. 4 |

11/8 | Conservation laws in fluids | Ch. 5 |

11/15 | Vorticity | Ch. 7 |

11/29 | Low Reynolds number flow | Ch. 8 |

12/6 | Instabilities | Ch. 10 |