Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Lagrangian mechanics, Gauss' principle, quadratic programming, and generalized inverses: new equations for non-holonomically constrained discrete mechanical systems

Authors: Robert E. Kalaba and Firdaus E. Udwadia
Journal: Quart. Appl. Math. 52 (1994), 229-241
MSC: Primary 70F25; Secondary 70H30, 90C90
DOI: https://doi.org/10.1090/qam/1276235
MathSciNet review: MR1276235
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Abstract: In this paper we formulate Lagrangian mechanics as a constrained quadratic minimization problem. This quadratic minimization problem is then solved using the theory of generalized inverses of matrices thereby obtaining the explicit equations of motion of constrained, discrete mechanical systems. The approach extends the boundaries of Lagrangian mechanics in that we provide a general formulation for describing the constrained motion of such systems without either the use of Lagrange multipliers or the use of quasi-coordinates. An important feature of the approach is that we do not require prior knowledge of the specific set of constraints to accomplish this formulation. This makes the equations presented here more generally useful, and perhaps more aesthetic, than the Gibbs-Appell equations which require a felicitous choice of problem-specific quasi-coordinates. The new equations of motion presented here are applicable to both the holonomic and nonholonomic constraints that Lagrangian mechanics deals with. They are obtained in terms of the usual generalized coordinates used to describe the constrained system. Furthermore, they can be integrated by any of the currently available numerical integration methods, thus yielding analytical and/or computational descriptions of the motions of constrained mechanical systems.

References [Enhancements On Off] (What's this?)

  • [1] J. L. Lagrange, Mécanique Analytique, MMe Ve Courcier, Paris, 1811
  • [2] R. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum Press, 1972
  • [3] L. A. Pars, A Treatise on Analytical Dynamics, Ox Bow Press, Connecticut, 1979
  • [4] P. Appell, Sur une forme générale des équations de la dynamique, Mem. Sci. Math., Gauthier-Villars, Paris, 1925
  • [5] Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33, Amer. Math. Soc., Providence, RI, 1972
  • [6] C. Gauss, Uber ein neues allgemeines Grundgesetz der Mechanik, J. Reine Angew. Math. 4, 232-235 (1829)
  • [7] E. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge, 1917
  • [8] C. Rao and S. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley, New York, 1972
  • [9] C. Lawson and R. Hanson, Solving Least Squares Problems, Prentice Hall, Englewoods Cliffs, NJ, 1974

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DOI: https://doi.org/10.1090/qam/1276235
Article copyright: © Copyright 1994 American Mathematical Society

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