There is a growing consensus that physics majors need to learn computational skills, but many departments are still devoid of computation in their physics curriculum. Some departments may lack the resources or commitment to create a dedicated course or program in computational physics. One way around this difficulty is to include computation in a standard upper-level physics course. An intermediate classical mechanics course is particularly well suited for including computation. We discuss the ways we have used computation in our classical mechanics courses, focusing on how computational work can improve students' understanding of physics as well as their computational skills. We present examples of computational problems that serve these two purposes. In addition, we provide information about resources for instructors who would like to include computation in their courses.
Deep Dive into Computation in Classical Mechanics.
There is a growing consensus that physics majors need to learn computational skills, but many departments are still devoid of computation in their physics curriculum. Some departments may lack the resources or commitment to create a dedicated course or program in computational physics. One way around this difficulty is to include computation in a standard upper-level physics course. An intermediate classical mechanics course is particularly well suited for including computation. We discuss the ways we have used computation in our classical mechanics courses, focusing on how computational work can improve students’ understanding of physics as well as their computational skills. We present examples of computational problems that serve these two purposes. In addition, we provide information about resources for instructors who would like to include computation in their courses.
The primary purpose of this article is to suggest a method for incorporating computation into upper-level classical mechanics courses. There is an emerging consensus in the physics community that computational skills are important for physicists, 1 but too often there is little or no computation included in the physics curriculum. 2 Probably the best way to strengthen the computational component of the physics curriculum is to incorporate computation into all components of the curriculum. Several institutions have developed large-scale computational physics programs, 3 which typically include at least one dedicated computational physics course as well as computational components in other upper-level courses. Other institutions offer a single computational physics course along with an otherwise traditional curriculum. With a variety of good computational physics textbooks 4 available it may appear that there is no reason not to offer at least one course in computational physics. Unfortunately, the reality is that there are constraints that may prevent some departments from offering such a course. These constraints may involve the number of physics faculty, the computational background of the physics faculty, low student enrollment in physics courses, or even political resistance to curricular changes. For faculty who face these constraints but desire to include some computation in the physics curriculum, the best approach may be to incorporate computation into existing courses. 5 Even departments that have a computational physics course may be looking for ways to increase the role of computation in the physics curriculum. We believe that the intermediate classical mechanics course, typically taken by students in their sophomore or junior year, is ideally suited for incorporating computation.
Computation can contribute to a variety of learning goals, and different approaches to incorporating compu-tation into physics courses will likely emphasize different goals. One goal of including computation in physics courses is simply to improve students’ learning of physics concepts. Computer visualizations and interactive simulations are particularly useful in this regard. Computation can also open the door to new and important topics such as nonlinear dynamics and chaos. Including computation in physics courses can also increase physics students’ familiarity with widely-used computational tools and help students see how these tools can be applied to solving physics problems. Finally, computation can be used to introduce students to important numerical algorithms. In many cases these algorithms can provide insight into important physics concepts, in addition to serving as tools for carrying out computations. Although all of these learning goals can be achieved through a dedicated course in computational physics, as mentioned above this may not be a practical approach in all departments. Incorporating computation into a standard course may be an effective way to introduce computation into the physics curriculum, perhaps as a temporary solution until a dedicated computational course can be added. Computation can be included in a standard course even if the students have no computational background (although, of course, more can be done if the students have had a course in programming or computational physics). We feel that the standard (sophomore/junior level) intermediate classical mechanics course is well suited for including computation. Students in classical mechanics can benefit tremendously from computer visualizations, which help them to build intuition about classical dynamics. Classical mechanics provides an excellent forum for introducing a variety of important numerical tools such as ODE solvers, root finding, numerical integration, numerical linear algebra, etc. Some important topics in modern classical mechanics, like chaos, cannot be effectively taught with-arXiv:0708.2498v1 [physics.ed-ph] 18 Aug 2007 out computation. Another advantage of the classical mechanics course is that it is typically taught early in the upper-level physics curriculum, often immediately after the introductory sequence. Introducing students to computation at this early stage gives them the opportunity to use their computational skills in later physics and mathematics courses.
To begin using computation in a classical mechanics course one must first make choices about which computational tools to use and how to use them. Choosing the right computational platform can be difficult, as each platform has advantages and disadvantages. Symbolic and numeric mathematics software packages (such as Mathematica, Maple, and Matlab, each of which has its own advantages and disadvantages 6 ) offer quick and relatively easy ways to perform computational work, provide a wide range of computational tools, and include high-quality visualization tools. The downside is that these tools can be used with little understanding of the numerical methods t
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