The motion of point masses under the influence of a potential can be computed by simple methods. However, if the trajectories are restricted by mechanical constraints such as strings, rails, crankshafts, and molecular bonds, special numerical techniques must be invoked. The need for efficient computational strategies is particularly pressing for molecular simulations, where large systems of compound molecules are tracked. The best strategy is the use of Cartesian coordinates in combination with constraint forces in the Lagrange formulation. This approach has led to the extremely successful SHAKE and RATTLE algorithms. The same ideas may be profitably applied in very different fields such as robotics, mechanics, and geometry, and the study of chaos in simple systems.
Received 26 March 2013Accepted 17 April 2013Published online 18 June 2013
This article is dedicated to the memory of Konrad Singer, exceptional human being and scientist.
Article outline: I. INTRODUCTION II. THE VERLET ALGORITHM III. LAGRANGE MULTIPLIERS A. Motion restricted to curves or surfaces B. Links and joints IV. SHAKE V. RATTLE VI. SPECIAL ALGORITHMS FOR MOLECULAR SIMULATION VII. APPLICATIONS A. Double/triple pendulum B. Robot arms C. Random walk of a two-dimensional Kramers chain D. Motion on curves or surfaces VIII. SUMMARY IX. SUGGESTED PROBLEMS A. Problem 1—Circular motion according to Langrange B. Problem 2—The pendulum: Circular motion with gravity C. Problem 3—The double pendulum D. Problem 4—Uniform motion around Sergels Torg E. Problem 5—Model gravitational well
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