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VIGRE seminar, Spring 2003: Mathematical Methods in Classical Mechanics

Instructor: Peter Kuchment
Students Enrolled: Bradley Crawyer (undergrad math major) and Gaik Ambartsoumian, Jeb Belcher, Beng Ong, Thirupathi Penmethsa, Elaina Rodriguez (graduate math students).

The class will be devoted to studying the basic underlying mathematical structures of the three cornerstones of mechanics: Newtonian, Lagrangian, and (time permitting) Hamiltonian mechanics. We will discuss the basic principles of mechanics: space and time, inertial systems of coordinates, and Galilean relativity principle. Important special cases will be considered, as for instance motion in central fields, Kepler's laws of planetary motions, and others. More complex motions of rigid bodies will be studied as well.

Students will discover many amazing problems and facts of this beautiful area. For instance, they will learn the celebrated Poincare's recurrence theorem, which in particular claims that if you open a partition separating a chamber containing a gas in one half and vacuum in another, then after a while all gas molecules will again collect in the first chamber. Why do we never observe this paradoxical situation? Similarity arguments will be introduced that enable one to answer the questions like: How does the maximal distance that an animal can run without water depend o n the (linear) size L of the animal? How do the maximal running velocity on level ground and uphill depend on L? The class will use the book "Mathematical Methods of Classical Mechanics" written by V. I. Arnold, one of the leading mathematicians of the last several decades. Lectures by instructor will be intertwined with presentations by students and discussions. Prerequisites: Calculus sequence, ordinary differential equations, and vector and linear algebra. Undergraduates having working knowledge of these topics should not have problems handling the course. Graduate students would find a lot to learn as well. Grading will be based on presentations and homework problems. Hope to see you in class!