Math 490, Fall 2018

MATH 490:	Special Topics in Mathematics
	Analytical Mechanics

Fall 2018 Course information

Class Time/Place:	MoWe 4:00pm–5:15pm, MP 008
Office:	MP 406
Phone:	410–455–2405
Email:	rostamian@umbc.edu
Office hours:	MoWe 3:00–4:00, or by appointment

Course Description

In this course we learn about the fascinating ideas and mathematical techniques developed by Euler, Lagrange, Gibbs, and Appell in formulating and extending Newton's ideas of the analysis of motion, with an emphasis on rigid body mechanics. We will trace the path followed by the pioneers of the subject, mainly:

Isaac Newton (1642–1727): who was mainly concerned with the motion of point masses;
Leonhard Euler (1707–1783): who extended Newton's ideas to the motion of rigid bodies (also to fluids—but that's another story);
Joseph-Louis Lagrange (1736–1813): whose analytic approach to mechanics shifted the study of motion from geometry to calculus;
Paul Émile Appell (1855–1930): whose reformulation of the equations of motion—anticipated by Josiah Willard Gibbs (1839–1903)—is particularly suited to the study of the motion of rigid bodies subject to nonholonomic constraints.

We will introduce the algebra quaternions (due to William Rowan Hamilton (1805–1865)) to keep track of the orientation of a rigid body in its motion. This has the advantage over the more commonly used Euler angles since it avoids the pesky gimbal lock problem.

Computational component

Except for some special cases, the differential equations of motion of a rigid body are generally too complex to obtain by hand, much less to solve in any form. In this course we rely on modern CAS (Computer Algebra System) software to (a) derive the equations; (b) to solve then numerically; and (c) to animate the motion on the computer screen.

Nowadays the two most popular CAS software are Maple and Mathematica. They are quite comparable in their capabilities as far as the needs of this course go. I use Maple in my own work, and that's what I will use in class. You don't need to have a prior knowledge of Maple—I will devote some time to tutorials—but you should be prepared to learn and use it to do your homework.

If you have a knowledge of Mathematica, you should be able to translate into it pretty much everything I do in Maple. Unfortunately I don't know enough about Mathematica to do any handholding for you.

Requirements

You will need to:

have a reasonably good understanding of Newton's dynamics (force equals mass times acceleration) and be able to solve simple problems such as determining the trajectory of a ball thrown with a given velocity at a given angle relative to the horizontal;
be able to solve first and second order linear differential equations, such as y'' + c y' + y = 0;
be comfortable with the concepts learned in multivariable calculus such partial derivatives, the chain rule, vectors in 3D, parametric equations of curves in 3D, calculating a tangent vector to a curve in space;

The course's official prerequisites are Phys 121 (Introductory Physics), (Math 225 (Differential Equations), Math 251 (Multivariable Calculus), all passed with grades of B or better. CMSC 201 is recommended.

Lecture notes

The material of this course does not come from any one book, therefore it is impossible to prescribe a textbook. Instead, I will attempt to write lecture notes shortly before or after each class and make it available for download. You should take notes in class to supplement what I provide.

Here is the 2018–12–02 version of the lecture notes: lecture-notes.pdf

Maple

UMBC has a campuswide license for Maple, so you don't need to buy it; it's available on all the machines in the GL labs (but not in the library).

However…

with the money that you save on not buying a textbook, you may go ahead a buy your own copy of Maple to run on your laptop or whatever. It should cost about $75. Then you won't have to go to the GL lab to do your homework.

Grading

I will periodically assign homework, and then collect and grade them. Your course grade will be based on cumulative performance on these assignments. There are no exams.

Sample demos

Stabilizing an inverted pendulum

An upside-down pendulum, or even an upside-down double-pendulum(!) may be stabilized by shaking its pivot up and down in just the right way.

Destabilizing a pendulum

In these two experiments we shake the pendulum's pivot horizontally. With the right choice of parameters, the natural “hanging down” position of the pendulum becomes unstable, and thus is begins to osccilate off-center.

A tossed baton

This baton has unequal masses (red and blue) attached to its endpoints. The animation illustrates the elaborate paths taken the two masses. Nevertheless, the center of mass of the system traces out an ordinary parabola.

[tossed baton with paths]

Coin rolling on the floor

This coin rolls without slipping on a flat horizontal floor. You may note that the coin's tilt relative to the ground varies (oscillates) as it rolls.

[rolling coin]

An unbalanced ball rolling on the floor

This homogeneous ball of mass M has an off-center slug of mass m embedded in it. The ball rolls without slipping on a flat horizontal floor.

The equations of motion are obtained through the Gibbs-Appell formulation of dynamics. The ball's orientation is tracked through quaternions.

[unbalanced ball]

The flipping T-handle

The vidoe Dancing T-hanle made on the International Space Station demonstrates the instability of rotation of a rigid body about the axis corresponding to the intermediae principal moment of inertia. We simulate that experiment by deriving the equations of motion of the object through the Gipps-Appell formulation, and then solve the equations in Maple and produce an animation. Here is the result:

[T-handle flip]

A spinning top

We derive the equations of motion of a spinning top through the Gibbs-Appell formulation of dynamics. The top's orientation is tracked through quaternions.

[spinning top]

Youtube videos

Miscellaneous notes

Registrar's Office Dates and Deadlines

The Official UMBC Honors Code

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal.

For detailed policies on academic integrity consult: Academic Integrity

Student Disability Services (SDS)

Services for students with disabilities are provided for all students qualified under the Americans with Disabilities Act of 1990, the ADAA of 2009, and Section 504 of the Rehabilitation Act who request and are eligible for accommodations. The Office of Student Disability Services is the UMBC department designated to coordinate accommodations that would allow for students to have equal access and inclusion in their courses.