Computing plays an increasingly significant role in our undergraduate physics curriculum. Several required physics courses now have substantial computing components.
Our objectives are two-fold: First, we aim to raise the level of computer literacy of our graduates, both in the use of programming languages and in the use of symbolic computing systems such as Maple. Second, use of computer-based examples, demonstrations, and problems helps our students develop deeper understanding and intuition about physical and mathematical principles.
The Maple system, an easily learned but versatile symbolic, numerical and graphical computing system, is introduced in the course Physical Analysis, taken by all sophomores majoring in physics. In this course, taught by Prof. Michael Procario, students develop analytical and modeling skills through detailed analysis of systems such as the damped driven harmonic oscillator.
For example, the student can use Maple to differentiate the position- vs.-time function r(t) quickly and easily to obtain the velocity v(t) and acceleration a(t) functions and to graph them. Similarly, the time-varying total energy of a damped driven oscillator, a somewhat complicated function, can easily be plotted. Normal-mode frequencies and eigenvectors of a system of coupled oscillators can also be obtained using Maple.
In the junior-level courses Physical Mechanics I and II, taught by Prof. Thomas Ferguson, students use Maple's versatile analytic and graphical capabilities as a calculational tool and to enhance their intuition about the behavior of physical systems.
An example is a particle in a Lennard-Jones potential, often used to describe interatomic interactions. Students plot the function by hand and find the equilibrium point. They check their result by using Maple to plot the function and to find the minimum by differentiation.
For a given total energy, Maple is used to find the turning points of the motion, both graphically and numerically. Then students set up the appropriate integral for the time of travel between the turning points and evaluate it numerically using Maple. Finally, they expand the potential in a Taylor series, both by hand and using Maple; this expansion is then used to approximate the turning points and the travel time.
Physical Mechanics also introduces the programming language cT, developed by our Prof. Bruce Sherwood and his colleagues in the Center for Innovation in Learning. The cT language is similar to C and FORTRAN; but it also features versatile graphics capabilities and easy portability of programs across various types of computers. Students write cT programs to compute particle trajectories numerically and display the results through ordinary or animated graphs.
The junior-level course Mathematical Methods of
Physics makes extensive use of Maple for in-class demonstrations and homework problems.
Prof. Hugh Young cites Fourier series as an example of the usefulness of Maple. Its
versatile graphing capability helps students develop intuition about convergence of
series, and its ability to carry out symbolic evaluation of complicated integrals enables
students to look at a wide variety of examples and applications.
Maple's animated-graphic capabilities also provide an ideal medium for visualizing such physical phenomena as normal modes of vibrating membranes (shown in the figures) and nonequilibrium heat and diffusion problems.
All three professors stress the need to balance computing versus physics, to avoid having the computing content of the courses overshadow the physics and to provide valuable enrichment to our undergraduate programs.
