David M. Cook

Computation in undergraduate physics: The Lawrence approach

The physics program at Lawrence University has introduced sophisticated computational techniques throughout its curriculum. Distinguishing features of the Lawrence approach include a focus on flexible, general purpose computational packages; application to theory and experiment; extensive use for preparing reports; and distribution throughout the curriculum. Most importantly, computation is introduced early enough so that students subsequently use computers independently on their own initiative. A required sophomore course in computational mechanics provides a uniform orientation to symbolic and numerical tools, and an elective junior/senior course in computational physics is offered. Students’ use of computational resources in independent studies and summer research experiences and positive comments from recent graduates provide evidence of the success and value of these curricular inclusions.

Trajectory of Charged Particles in Weakly Inhomogeneous Magnetic Fields

The detailed time dependence of the trajectory, and particularly of the acceleration, of a charged particle moving nonrelativistically in a weakly inhomogeneous, slowly varying electromagnetic field is calculated by a method that involves superimposing a conventional perturbation expansion in the weak field on the expansion of Berkowitz and Gardner and of Kruskal, the latter expansion being asymptotically valid in the limit of small Larmor radius and slowly varying fields. Fourier transforms are introduced to facilitate isolation of rapidly varying terms from the most general solutions for the coefficients in the expansion. The calculations are carried to first order in the weak inhomogeneity and (at least formally) to all orders in Larmor radius. The resulting velocity reproduces the familiar transverse drift velocities of first‐order orbit theory, emphasizes that in the presence of fields varying slowly in time the polarization and curvature drifts are to be calculated from the time derivative of the fi...

Computation in Undergraduate Physics: The Lawrence Approach

Most efforts using computers in physics curricula focus on introductory courses or individual upper-level courses. In contrast, for a dozen years the Lawrence Department of Physics has been striving to embed the use of general purpose graphical, symbolic, and numeric computational tools throughout our curriculum. Developed with support from the (US) National Science Foundation, the Keck Foundation, and Lawrence University, our approach involves introducing freshman to tools for data acquisition and analysis, offering sophomores a course that introduces them to symbolic, numerical, and visualization tools, incorporating computational approaches alongside traditional approaches to problems in many intermediate and upper level courses, and making computational resources available so that students come to see them as tools to be used routinely on their own initiative whenever their use seems appropriate. A text reflecting the developments at Lawrence is in preparation, will undergo beta testing in 2001-02, and will be published in January, 2003. Details about the Lawrence curricular approach and the emerging text can be found from links at www.lawrence.edu/dept/physics.

Least Squares Fitting of Data to Pseudolinear Relationships

A general method is presented for applying the principle of least squares to determine the constants C, D, E, and k that optimize the fit of experimental data to a relationship between a dependent variable y and an independent variable x expressible either in the form h(y) =  Cf(x) + Dg(x) + E or in the form h(y) = Cf(k; x) + Dg(k; x) + E, where h, f, and g are arbitrary functions of the indicated arguments. Analysis of the first form is essentially identical with analysis of a bilinear form, and an analytic solution can be obtained; analysis of the second form involves an iterative process in which the value of k is assumed, tried, and successively revised until the principle of least squares is satisfied to a specified accuracy. The resulting formulation makes the broad utility of the principle of least squares more apparent than is often the case, stresses particularly the importance of weighting to the correct application of the principle and can be easily specialized to cover a wide variety of common...