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Apr 2004

Volume 72, Issue 4, pp. 423-559

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An introduction to the theme issue

Harvey S. Leff, David P. Jackson, Kerry Browne, and Stamatis Vokos

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 423

Online Publication Date: Mar 2004

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© 2004 American Association of Physics Teachers.
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01.30.-y Physics literature and publications
01.50.-i Educational aids
45.05.+x General theory of classical mechanics of discrete systems
01.55.+b General physics
46.05.+b General theory of continuum mechanics of solids
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Sea gulls, butterflies, and grasshoppers: A brief history of the butterfly effect in nonlinear dynamics

Robert C. Hilborn

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 425 | Cited 18 times

Online Publication Date: Mar 2004

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The butterfly effect has become a popular metaphor for sensitive dependence on initial conditions—the hallmark of chaotic behavior. I describe how, where, and when this term was conceived in the 1970s. Surprisingly, the butterfly metaphor was predated by more than 70 years by the grasshopper effect. © 2004 American Association of Physics Teachers.
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05.45.-a Nonlinear dynamics and chaos
47.52.+j Chaos in fluid dynamics
01.65.+g History of science
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Symmetries and conservation laws: Consequences of Noether’s theorem

Jozef Hanc, Slavomir Tuleja, and Martina Hancova

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 428 | Cited 9 times

Online Publication Date: Mar 2004

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We derive conservation laws from symmetry operations using the principle of least action. These derivations, which are examples of Noether’s theorem, require only elementary calculus and are suitable for introductory physics. We extend these arguments to the transformation of coordinates due to uniform motion to show that a symmetry argument applies more elegantly to the Lorentz transformation than to the Galilean transformation. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
03.30.+p Special relativity
11.30.-j Symmetry and conservation laws

The use of models in problems of energy conservation

K. A. Legge and J. Petrolito

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 436 | Cited 1 time

Online Publication Date: Mar 2004

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Recent discussions regarding the misconceptions students develop about work and conservation of energy has focused attention on introductory mechanics and the use of the “center-of-mass” or particle model. This discussion has led to suggestions that the model is insufficient for introductory mechanics and that more complex models should be used. We suggest that it is not the model that requires re-working, but rather an appreciation of the use and limitations of models in general. We include some examples illustrating the misapplication of the particle model to systems with internal energy. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
45.05.+x General theory of classical mechanics of discrete systems
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Modern mechanics

Ruth W. Chabay and Bruce A. Sherwood

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 439 | Cited 8 times

Online Publication Date: Mar 2004

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We consider the goals of the introductory course in classical mechanics taken by physics majors and argue both that these goals are not well met in actual courses and that the goals themselves should be rethought. We propose alternative goals and describe an introductory “modern mechanics” course that addresses these alternative goals. Included in the description are several genres of homework problems that are nearly absent from traditional mechanics courses at both the introductory and intermediate levels. The intermediate mechanics course could be restructured to exploit a broader foundation laid by the introductory course. © 2004 American Association of Physics Teachers.
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01.40.Di Course design and evaluation
01.40.G- Curricula and evaluation
01.50.-i Educational aids

A unit on oscillations, determinism and chaos for introductory physics students

Priscilla W. Laws

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 446 | Cited 5 times

Online Publication Date: Mar 2004

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This article describes a unit on oscillations, determinism and chaos developed for calculus-based introductory physics students as part of the laboratory-centered Workshop Physics curriculum. Students begin by observing the motion of a simple pendulum with a paper clip bob with and without magnets in its vicinity. This observation provides an introduction to the contrasting concepts of Laplacian determinism and chaos. The rest of the unit involves a step-by-step study of a pendulum system that becomes increasingly complex until it is driven into chaotic motion. The time series graphs and phase plots of various configurations of the pendulum are created using a computer data acquisition system with a rotary motion sensor. These experimental results are compared to iterative spreadsheet models developed by students based on the nature of the torques the system experiences. The suitability of the unit for introductory physics students in traditional laboratory settings is discussed. © 2004 American Association of Physics Teachers.
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01.50.Pa Laboratory experiments and apparatus
05.45.Tp Time series analysis
07.05.Hd Data acquisition: hardware and software
02.30.Lt Sequences, series, and summability

Investigating student understanding in intermediate mechanics: Identifying the need for a tutorial approach to instruction

Bradley S. Ambrose

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 453 | Cited 5 times

Online Publication Date: Mar 2004

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The conceptual understanding and reasoning skills of advanced undergraduates as they make the transition from a traditional sequence in introductory calculus-based physics to their first course in upper-level mechanics are probed. The results thus far are consistent with findings from other investigations in upper-division courses, which indicate that persistent difficulties with fundamental concepts can hinder meaningful learning of advanced topics. To address this problem, the tutorial approach developed at the University of Washington has been adapted and incorporated into the intermediate mechanics course at Grand Valley State University. This modification has produced promising results. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
01.40.Di Course design and evaluation
01.40.G- Curricula and evaluation
45.05.+x General theory of classical mechanics of discrete systems
02.30.Vv Operational calculus

Student use of vectors in introductory mechanics

Sergio Flores, Stephen E. Kanim, and Christian H. Kautz

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 460 | Cited 3 times

Online Publication Date: Mar 2004

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Most students’ initial exposure to physics is in the context of kinematics and dynamics. An understanding of how these topics relate to each other requires the ability to reason about vectors that represent forces and kinematic quantities. We present data that suggest that after traditional instruction in mechanics many students lack this ability. Modifications to instruction can significantly improve student performance on questions about vector addition and subtraction and increase the likelihood that students employ vectors in their attempt to solve mechanics problems. However, an increased emphasis on these topics has so far been only moderately successful in promoting the level of proficiency required to understand the connection between force and acceleration as vector quantities. We describe some of the procedural and reasoning difficulties we have observed in students’ use of vectors. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
45.05.+x General theory of classical mechanics of discrete systems
02.10.Ud Linear algebra
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Square-wave excitation of a linear oscillator

Eugene I. Butikov

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 469

Online Publication Date: Mar 2004

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The forced oscillations of a torsion spring pendulum excited by an external square-wave external torque are discussed. Two different ways of determining the steady-state response of the oscillator are described and compared. The behavior of this familiar mechanical system can help a student better understand why and how a LCR-circuit transfers a square-wave voltage from input to output with a distortion of its shape.© 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
07.10.-h Mechanical instruments and equipment
45.40.Cc Rigid body and gyroscope motion
84.30.Ng Oscillators, pulse generators, and function generators

Oscillator damped by a constant-magnitude friction force

Avi Marchewka, David S. Abbott, and Robert J. Beichner

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 477 | Cited 7 times

Online Publication Date: Mar 2004

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Although a simple spring/mass system damped by a friction force of constant magnitude shares many of the characteristics of the simple and damped harmonic oscillators, its solution is not presented in most texts. Closed form solutions for the turning and stopping points can be found using an energy-based approach. A dynamical approach leads to a closed form solution for the position of the mass as a function of time. The main result is that the amplitude of the oscillator damped by a constant magnitude friction force decreases by a constant amount each swing and the motion dies out after a finite time. We present these two solutions and suggest ways that the system can be used in the classroom. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
46.55.+d Tribology and mechanical contacts
45.05.+x General theory of classical mechanics of discrete systems

The driven pendulum at arbitrary drive angles

Gordon J. VanDalen

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 484

Online Publication Date: Mar 2004

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We discuss the equation of motion of the driven pendulum and generalize it to arbitrary driving angle. The pendulum will oscillate about a stable angle other than straight down if the drive amplitude and frequency are large enough for a given drive angle. The emphasis is on the parameters associated with a simply made demonstration apparatus.© 2004 American Association of Physics Teachers.
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01.50.My Demonstration experiments and apparatus
07.10.-h Mechanical instruments and equipment

Theory and examples of intrinsically nonlinear oscillators

Pirooz Mohazzabi

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 492 | Cited 5 times

Online Publication Date: Mar 2004

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Contrary to the general claim that small oscillations in any system can be approximately treated in terms of simple harmonic motion, it is shown that in principle there are infinitely many oscillating systems for which this approximation is not valid. The theory of intrinsically nonlinear oscillators is discussed and examples are given in several areas including elasticity, electrodynamics, and gravitation. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
03.65.Ge Solutions of wave equations: bound states
46.25.-y Static elasticity
03.50.De Classical electromagnetism, Maxwell equations
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Real-time finite difference bifurcation diagrams from analog electronic circuits

Edward H. Hellen

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 499 | Cited 2 times

Online Publication Date: Mar 2004

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Bifurcation diagrams are a convenient way of displaying the variety of behaviors exhibited by nonlinear systems. One of the simplest nonlinear systems is a finite difference equation with a quadratic return map. This system exhibits a range of behaviors: stability, periodic oscillations, and chaos. We present simple inexpensive electronic circuits that perform analog computations of bifurcation diagrams for finite difference equations with quadratic return maps. These bifurcation diagrams, including one for the logistic equation, are easily displayed on an oscilloscope and agree well with analytical and computational predictions. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
05.45.-a Nonlinear dynamics and chaos

Precision measurements of a simple chaotic circuit

Ken Kiers, Dory Schmidt, and J. C. Sprott

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 503 | Cited 7 times

Online Publication Date: Mar 2004

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We describe a simple nonlinear electrical circuit that can be used to study chaotic phenomena. The circuit employs simple electronic elements such as diodes, resistors, and operational amplifiers, and is easy to construct. A novel feature of the circuit is its use of an almost ideal nonlinear element, which is straightforward to model theoretically and leads to excellent agreement between experiment and theory. For example, comparisons of bifurcation points and power spectra give agreement to within 1%. The circuit yields a broad range of behavior and is well suited for qualitative demonstrations and as a serious research tool.© 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
05.45.-a Nonlinear dynamics and chaos
84.32.Ff Conductors, resistors (including thermistors, varistors, and photoresistors)
84.30.Le Amplifiers
84.47.+w Vacuum tubes
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Deriving Lagrange’s equations using elementary calculus

Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 510 | Cited 4 times

Online Publication Date: Mar 2004

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We derive Lagrange’s equations of motion from the principle of least action using elementary calculus rather than the calculus of variations. We also demonstrate the conditions under which energy and momentum are constants of the motion. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
45.10.Hj Perturbation and fractional calculus methods

From conservation of energy to the principle of least action: A story line

Jozef Hanc and Edwin F. Taylor

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 514 | Cited 3 times

Online Publication Date: Mar 2004

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We outline a story line that introduces Newtonian mechanics by employing conservation of energy to predict the motion of a particle in a one-dimensional potential. We show that incorporating constraints and constants of the motion into the energy expression allows us to analyze more complicated systems. A heuristic transition embeds kinetic and potential energy into the still more powerful principle of least action. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
01.40.G- Curricula and evaluation
45.20.D- Newtonian mechanics
45.05.+x General theory of classical mechanics of discrete systems

Getting the most action out of least action: A proposal

Thomas A. Moore

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 522 | Cited 3 times

Online Publication Date: Mar 2004

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Lagrangian methods lie at the foundation of contemporary theoretical physics. Several recent articles have explored the possibility of making the principle of least action and Lagrangian methods a part of the first-year physics curriculum. I examine some of this proposal’s implications for subsequent courses in the undergraduate physics major, and focus on the influence that this proposal might have on the selection of topics and the opportunities this proposal presents for teaching these courses in a more contemporary way. Many of these ideas are relevant even if students first learn Lagrangian methods in a sophomore mechanics course. © 2004 American Association of Physics Teachers.
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01.40.G- Curricula and evaluation
01.40.Di Course design and evaluation
01.50.-i Educational aids
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A simple model for stochastic coherence and stochastic resonance

Robert C. Hilborn

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 528 | Cited 8 times

Online Publication Date: Mar 2004

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I describe a simple iterated map that displays two important noise-induced effects for nonlinear systems: stochastic coherence and stochastic resonance. The model requires only modest computational capabilities and some knowledge of nonlinear dynamics and illustrates the constructive role of noise in nonlinear systems. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
05.45.-a Nonlinear dynamics and chaos
05.40.Ca Noise
02.50.Ey Stochastic processes

Dimensional analysis, falling bodies, and the fine art of not solving differential equations

Craig F. Bohren

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 534 | Cited 9 times

Online Publication Date: Mar 2004

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Dimensional analysis is a simple, physically transparent and intuitive method for obtaining approximate solutions to physics problems, especially in mechanics. It may—indeed sometimes should—precede or even supplant mathematical analysis. And yet dimensional analysis usually is given short shrift in physics textbooks, presented mostly as a diagnostic tool for finding errors in solutions rather than in finding solutions in the first place. Dimensional analysis is especially well suited to estimating the magnitude of errors associated with the inevitable simplifying assumptions in physics problems. For example, dimensional arguments quickly yield estimates for the errors in the simple expression math for the descent time of a body dropped from a height h on a spherical, rotating planet with an atmosphere as a consequence of ignoring the variation of the acceleration due to gravity g with height, rotation, relativity, and atmospheric drag. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
45.40.-f Dynamics and kinematics of rigid bodies

Weakly nonlinear oscillations: A perturbative approach

Peter B. Kahn and Yair Zarmi

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 538 | Cited 4 times

Online Publication Date: Mar 2004

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The perturbative analysis of a one-dimensional harmonic oscillator subject to a small nonlinear perturbation is developed within the framework of two popular methods: normal forms and multiple time scales. The systems analyzed are the Duffing oscillator, an energy conserving oscillatory system, the cubically damped oscillator, a system that exhibits damped oscillations, and the Van der Pol oscillator, which represents limit-cycle systems. Special emphasis is given to the exploitation of the freedom inherent in the calculation of the higher-order terms in the expansion and to the comparison of the application of the two methods to the three systems. © 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
05.45.Xt Synchronization; coupled oscillators
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.30.Hq Ordinary differential equations
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Synchronous analog I/O for acquisition of chaotic data in periodically driven systems

Robert DeSerio

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 553 | Cited 1 time

Online Publication Date: Mar 2004

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A new data acquisition technique has been designed for measuring chaotic attractors in periodically driven systems. Analog output hardware generates a continuous periodic waveform used in the drive while analog input hardware digitizes one or more chaotic waveforms characterizing the system phase space variables. With both processes synchronized to a common clock, the construction of multiple Poincaré sections is reduced to the simple process of redimensioning the chaotic waveforms and taking slices in one of the array indices. The implementation described here is applied to an electronic Duffing oscillator running at 1.6 kHz and provides quality Poincaré sections in near real time.© 2004 American Association of Physics Teachers.
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01.50.-i Educational aids
07.05.Hd Data acquisition: hardware and software
05.45.Gg Control of chaos, applications of chaos
05.45.Xt Synchronization; coupled oscillators
84.30.Bv Circuit theory
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Post-Use Review: Classical Mechanics

John R. Taylor, Author and Gayle Cook, Reviewer

American Journal of Physics -- April 2004 -- Volume 72, Issue 4, pp. 559

Online Publication Date: Mar 2004

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Abstract Unavailable
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01.30.Vv Book reviews
47.00.00 Fluid dynamics
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