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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
J. L. Lagrange, Mécanique Analytique, MMe Ve Courcier, Paris, 1811
R. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum Press, 1972
L. A. Pars, A Treatise on Analytical Dynamics, Ox Bow Press, Connecticut, 1979
P. Appell, Sur une forme générale des équations de la dynamique, Mem. Sci. Math., Gauthier-Villars, Paris, 1925
Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33, Amer. Math. Soc., Providence, RI, 1972
C. Gauss, Uber ein neues allgemeines Grundgesetz der Mechanik, J. Reine Angew. Math. 4, 232–235 (1829)
E. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge, 1917
C. Rao and S. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley, New York, 1972
C. Lawson and R. Hanson, Solving Least Squares Problems, Prentice Hall, Englewoods Cliffs, NJ, 1974
J. L. Lagrange, Mécanique Analytique, MMe Ve Courcier, Paris, 1811
R. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum Press, 1972
L. A. Pars, A Treatise on Analytical Dynamics, Ox Bow Press, Connecticut, 1979
P. Appell, Sur une forme générale des équations de la dynamique, Mem. Sci. Math., Gauthier-Villars, Paris, 1925
Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33, Amer. Math. Soc., Providence, RI, 1972
C. Gauss, Uber ein neues allgemeines Grundgesetz der Mechanik, J. Reine Angew. Math. 4, 232–235 (1829)
E. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge, 1917
C. Rao and S. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley, New York, 1972
C. Lawson and R. Hanson, Solving Least Squares Problems, Prentice Hall, Englewoods Cliffs, NJ, 1974
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
70F25,
70H30,
90C90
Retrieve articles in all journals
with MSC:
70F25,
70H30,
90C90
Additional Information
Article copyright:
© Copyright 1994
American Mathematical Society