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.
pacs:D01.30.Pp, S01.40.Fk, N01.50.H-
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,Grayson () but too often there is little or no computation included in the physics curriculum.Fuller () 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,programs () 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 textbookscomptexts () 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. Roos () 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 computation 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 without 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.
Ii Implementation Issues
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 disadvantages3M ()) 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 that are being employed. Furthermore, these software packages can be quite expensive. The other main alternative is to have students create programs from scratch using a standard programming language like Fortran, Java, or Python. Students who do their own coding must develop an understanding of the models they are studying as well as the numerical methods that are employed. This solution is generally inexpensive since free compilers are available for these languages on a variety of operating systems. Many common programming tasks can be carried out with the aid of freely-available numeric libraries, like Open Source Physics (OSP)OSP () and SciPy,SciPy () to reduce the time that must be spent creating the programs. However, in spite of these libraries there is still a great deal of programming “overhead” that must be addressed and instructors whose students have no programming background may not have sufficient time to teach the necessary programming skills. There are some intermediate solutions available, such as the Easy Java Simulations (EJS) package,EJS () which automates many programming tasks and allows users to focus on the numerical algorithms that are used. However, EJS does not offer the broad range of functionality that is available with a standard programming language or a software package like Mathematica.
We have chosen to use Mathematica and Matlab in our classical mechanics courses, but we plan to move toward using Java/OSP and EJS for some topics. We like Mathematica for its powerful symbolic mathematics capabilities, and Matlab because it is widely used in industry. Both of these packages are tools that students will likely be able to use in other classes and perhaps throughout their careers. Programming in Java with the OSP library provides an opportunity to share both programs and curricular materials with the broader physics community and contribute to (as well as benefit from) a larger project.Belloni () EJS provides simple tools for quickly creating visualizations and simulations. Each of these options, as well as the others listed above, will be more or less suited to an individual instructor’s specific situation. Instructors must be guided by the background of their students (Do they have programming experience?), their own computational experience (With what platforms is the instructor familiar?), as well as other practical considerations (Are there funds available to pay for software licenses? Will the software be supported by qualified campus staff?).
The choice of platform (or platforms) should also be guided by how the instructor intends to use the computational tools in the course. In our classical mechanics courses we use computation in two main ways: for in-class demonstrations and for student computational projects. Typically we will demonstrate computer solutions to physics problems as part of our in-class lecture. The code used to construct the solution is made available to students so that they can examine the code and “experiment” with it at their leisure. We then follow up these demonstrations by assigning computational projects that students must complete on their own or in small groups. Early in the course these projects may involve only minor modifications of the code used in the demonstrations, perhaps to study the same phenomenon in a new system. In this way the demonstration code serves as a template that helps the students complete the computational project. As the course proceeds we demand more from our students, expecting them to construct computational solutions without a template (but using computational tools they have seen before). These computational projects can easily be made the basis of formal writing assignments in which students must present their analytic and numerical work along with figures, typeset equations (possibly in /LaTeX), and a thorough discussion of the physics.
Iii Algorithms and Physics
Because the software packages we use tend to hide the details of the numerical algorithms they employ, we feel that it is important to provide students with some explicit instruction on algorithms. We focus on simple algorithms, rather than the sophisticated algorithms employed by the software packages we use, for a few important computational tasks. This instruction enhances students’ knowledge of computation because it emphasizes the fact that computations require some algorithms and that the choice of algorithm can critically affect the success of the computation. In addition, teaching students about algorithms can sometimes lead to a better understanding of important physics concepts.
As an example of how we use algorithms in our course, let us consider an object of mass oscillating in one dimension on an ideal spring with force constant . The equation of motion for this simple harmonic oscillator is easy to solve analytically, but we can also take a numerical approach to the problem. The simplest algorithm that we might consider for this purpose is the Euler algorithm:
where , , and represent the displacement from equilibrium, the velocity, and the net force at time . For this system . Although the software packages we use have their own built-in ODE solvers, we can also use these packages to implement a simple algorithm like the Euler algorithm. The solid red curve in Figure 1(a) shows the trajectory of the object in phase space (position versus velocity) generated by the Euler algorithm when kg, N/m, m, , and s. The dashed blue curve in Fig. 1(a) shows the exact solution. It is clear from this result that the Euler algorithm is unstable since the trajectory produced by the algorithm spirals continually outward away from the exact solution. This also means that the Euler algorithm does not conserve energy: it is clear that with the Euler algorithm the object’s maximum displacement and maximum speed are steadily increasing, so its total mechanical energy must also be steadily increasing.
The failure of the Euler algorithm to conserve energy is related to yet another failure: the failure to preserve phase space volume. According to Liouville’s Theorem, a conservative system (such as our simple harmonic oscillator) must preserve the volume (or area) occupied by an ensemble of trajectories in phase space. Figure 1(a) shows the locations generated by the Euler algorithm for a trajectory ensemble at times , 1.8, 3.6, and 5.4 s. At the points are randomly distributed within a square region around m, . As time passes the cluster of points moves through phase space and the shape of the region becomes distorted. However, it is clear from Figure 1(a) that the size of the region also grows over time, in clear contradiction to Liouville’s Theorem. In fact, it is easy to show that the Euler algorithm doesn’t preserve phase space volume. If we treat Equation 1 as a two-dimensional map, then this map will preserve phase space volume if and only if the Jacobian of the map, defined by
has determinant equal to one. A quick calculation will show that for the Euler algorithm, in the case of the simple harmonic oscillator.
These problems with the Euler algorithm can be fixed with a simple modification, leading to what is known as the Euler-Cromer algorithm.Cromer () This algorithm simply updates the velocity first, and then uses the new velocity to update the position. The equations describing this algorithm are:
where all quantities are defined as for the Euler algorithm above. This algorithm is stable and conserves energy, on average, for oscillatory motion.Cromer () We can quickly illustrate this by applying this new algorithm to our simple harmonic oscillator. The results are shown in Figure 1(b). It is clear that the solution produced by the algorithm remains close to the exact solution, coinciding with the exact solution at every quarter period. Similarly, the total energy oscillates about the correct value with a period equal to half of the oscillator’s period. Furthermore, this algorithm appears to preserve phase space volume. This can be easily proved by showing that the Jacobian for Equation 3 has determinant equal to one.
Once these algorithms (and their differences) have been demonstrated, students should be given a chance to use them. They can explore the behavior of the algorithms as the time step is increased or decreased. They can compare the results of these algorithms with the results obtained using the built-in ODE solver supplied by the software package. They can make modifications to the model by adding drag forces or driving forces. Note that students are not restricted to drag forces linear in the velocity, or sinusoidal driving forces. In fact, it is very instructive to consider the case of a quadratic drag force (a nonlinear system) and compare the results to those obtained with a linear drag force (a linear system). If these non-conservative forces are added then neither algorithm will preserve phase-space volume, as can be easily seen by finding the determinant of the Jacobian (and noting that is no longer a function of only). Carrying out this calculation using the Euler-Cromer algorithm makes it clear that for drag forces, where , the determinant of the Jacobian is less than one indicating that phase space volume will shrink over time. This is exactly what one should expect for a dissipative system.
Other algorithms for common numerical tasks may be worth discussing in class. Algorithms for root finding (Newton-Raphson method, bisection method) and numerical integration (trapezoid approximation, Simpson’s Rule, Monte Carlo methods) are useful and simple enough to present without absorbing too much class time. Although class time spent discussing algorithms may take away from time spent discussing physics, the above example shows that it is possible to teach important physical concepts along with algorithms. As another example, the Newton-Raphson method illustrates the concept of a stable attractor which is important for understanding the dynamics of dissipative systems.
Iv Computational Projects
Computational assignments should be chosen with care to ensure that they take full advantage of what computing offers. One way to do this is to use computation to break out of the restrictions imposed by the need for an analytic solution (or approximate solution). We have already mentioned the case of the harmonic oscillator with a quadratic drag force. Two other examples of this type are motion in a non-inertial reference frame and motion of a charged particle in electric and magnetic fields. Traditional courses tend to focus on cases where analytic solutions are possible, such as the Eastward deflection of an object that drops from rest to Earth or the cyclotron motion of an electron in a magnetic field. Computation allows students to solve more realistic problems such as long-range projectile motion on Earth and the motion of an electron in combined electric and magnetic fields. Solving such problems computationally not only allows students to tackle realistic problems, it also allows students to visualize the motion in these systems. Figure 2 shows two calculated trajectories, one with non-inertial forces and one without, for a long-range projectile fired due East from Rome, Georgia. Visualizing this motion helps students get a sense of the magnitude of the effect of non-inertial forces. Figure 3 shows an OSP application that simulates a charged particle moving in the presence of constant (but general) three dimensional electric and magnetic fields. Students can experiment with the parameters of the calculation without performing a tedious analytic analysis.
Some topics cannot be taught at all without computation. The most obvious example of this is chaos. Computational solutions of systems like the driven, damped pendulum can be used to illustrate important concepts of dissipative chaos such as sensitive dependence on initial conditions, period doubling bifurcations, Poincaré sections, and strange attractors. Simple iterated function systems like the logistic map can be used to delve deeper into dissipative chaos with bifurcation diagrams and Lyapunov exponents. While these topics are now included in many textbooks,texts (); Hasbun () they cannot really be taught effectively without letting students carry out some computations. Once students have mastered the necessary computational skills, even the more advanced topic of Hamiltonian chaos becomes accessible through the study of two-dimensional area-preserving maps.Timberlake () In fact, students who have already seen the Euler-Cromer algorithm will be familiar with some of the properties of area-preserving maps.
Computational projects can also be chosen to introduce important numerical methods. Numerical integration can be introduced in the context of finding the period of a pendulum undergoing large amplitude oscillations. Numerical root-finding techniques can be used to determine the time of flight for a projectile with air resistance. Numerical determination of eigenvalues and eigenvectors can be used in the context of finding principle axes for rigid-body motion or normal modes of coupled oscillator systems.
Many textbooks now have computational exercises among their end-of-the-chapter homework problems, and these can be a valuable source of ideas for computational projects.texts () One of us (JH) has written a textbook for classical mechanics that explicitly incorporates computation throughout the book and includes a wide variety of computational projects.Hasbun () In addition, anyone can obtain most of our computational materials (except solutions to student projects) written in Matlabmscript () or Mathematicamathematica (). We plan to update these websites with new materials as they are created. We also plan to port these materials to Java/OSP or EJS. These materials will be made available through the BQ-OSP database.bq () Instructors wishing to include computation in their physics courses are also encouraged to consult computational physics textscomptexts () and the Open Source Physics website.OSP ()
A course in intermediate classical mechanics provides an excellent opportunity for including computation without requiring the resources and commitment needed for developing a new computational course or program. Many topics that are typically covered in an intermediate classical mechanics course can benefit from a computational approach. The examples discussed in this article provide just a small sample. The intermediate classical mechanics course is also a good place to introduce computation because it typically comes early in the curriculum. Ideally this course would be preceded by a computational physics course and followed by other upper-level physics courses that include computational work. We have found that even if computation is required only in classical mechanics students will use the computational skills they have gained to solve problems in a wide variety of other physics and mathematics courses. An early exposure to computational work can also lead to opportunities for undergraduate research in computational physics.
Each instructor must decide how much computation to include, which computational topics to cover, and how computation will be used in the course. These decisions must be based on a wide variety of factors, including the computational skills of the students, the computational background of the instructor, the computational resources (both hardware and software) available, and time and content coverage constraints. Nonetheless, we feel that it is both possible and important to include computation somewhere in the physics curriculum. We hope that this article provides some guidance for those who wish to include computation in the curriculum, but cannot create a dedicated course or program in computational physics.
Acknowledgements.One of us (JH) wishes to thank David M. Cook for his inspiring workshop dealing with computational physics and which he attended. He also wishes to thank Wolfgang Christian and Francisco Esquembre for their invaluable help in developing OSP Java applications following OSP workshops held by Wolfgang Christian, Mario Belloni, and Anne Cox.
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- (15) The matlab code used is available upon request by e-mailing the author. OSP applications not included in Ref. Hasbun, will be made freely available here: www.westga.edu/~jhasbun/osp/osp.htm in tandem with the textbook release.
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