*PHYS 331: Theoretical Mechanics*

**Instructor and Team Leader:**Marty Johnston

Theoretical Mechanics is a junior/senior level course focusing on the study of physical systems through the application of classical mechanics. During the semester the ideas of Newton, Lagrange, and Hamilton are developed and applied to a wide spectrum of problems ranging from the motion of charged particles to coupled oscillators. Mathematical techniques are infused throughout the course; vector calculus, differential equations, matrix algebra, complex variables and variational calculus all find their way into the discussion. Because of the strong mathematical underpinnings of the field, the study often evolves into a dry, hands-off, exploration of mathematics without any solid connection to measurement. To keep problems analytically solvable, systems sometimes become so idealized they no longer resemble reality. However, by introducing computational tools into the mix, difficult mathematical problems can often be solved which greatly facilitates the study of real systems.

Over time we have bolstered the computational component of the course by adding a few computer intensive projects throughout the semester. While this change has been helpful, students clearly need additional, more systematic computational experiences. To accomplish this goal and to connect the course with real systems, we are planning new homework modules that include both computational components and observational verifications.

**Homework Module 1:**Solving Lagrange’s Equations for Falling Chains

*Conceptual goals:*Understand the application of Lagrangian dynamics to an interconnected, multi-particle system. Recognize that computational techniques allow us to model complicated systems that are intractable using analytical techniques.

*Computational goals:*Use MATLAB to numerically solve the differential equations that result from applying the Lagrangian to the physical system. Develop skills in numerical solutions to differential equations and visualization.

*Experimental goal:*Verify that the computational model predicts expected behavior.

We are designing a homework module that utilizes MATLAB to solve Lagrange’s equations for falling chains in a number of different scenarios. Students will analyze actual physical systems and solve the resulting differential equations in MATLAB. They will then compare their theoretical models with actual data they capture using high-speed video. The problems arise from a series of papers discussing the dynamics of a falling chain [1,2,3]. These papers point out conceptual errors that have been perpetuated for years in textbooks and showcase the sometimes surprising results that can only be obtained using computational techniques.

**References:**

1. W. Tomaszewski, P. Pieranski, and J. C. Geminard, “The motion of a freely falling chain tip”, Am. J. of Phys., 74, 776-783 (2006).

2. C. W. Wong and K. Yasui, “Falling Chain”, Am. J. of Phys., 74, 490-496 (2006).

3. M. G. Calkin and R. H. March, “The dynamics of falling chain: I”, Am. J. of Phys., 57, 154-159 (1989).

If you are interested in using any of the materials found in this webpage in your own course, please contact us.