The explicit Gibbs-Appell equation and generalized inverse forms
Authors:
F. E. Udwadia and R. E. Kalaba
Journal:
Quart. Appl. Math. 56 (1998), 277-288
MSC:
Primary 70F25; Secondary 70H35
DOI:
https://doi.org/10.1090/qam/1622570
MathSciNet review:
MR1622570
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Abstract: This paper develops an extended form of the Gibbs-Appell equation and shows that it is equivalent to the generalized inverse equation of motion. Both equations are shown to follow from Gauss’s principle. An example to highlight the two equivalent, though different, equations of motion is provided. Conceptual differences between the equations, and differences in their practical application to physical situations are discussed.
P. Appell, Traité de Mécanique Rationelle, Third edition, Paris, 1911
P. Appell, Exemple de Mouvement d’un Point Assujeti a une Liason Exprimée par une Relation Non Lineare Entre les Composantes de la Vitesse, Comptes Rendus, 1911, pp. 48–50
C. F. Gauss, Über ein neues allgemeines Grundgesetz der Mechanik, Journal für die Reine und Angewandte Mathematik 4, 232–235 (1829)
W. J. Gibbs, On the fundamental formulae of dynamics, Amer. Jour. Math. 2, 49–64 (1879)
- Robert E. Kalaba and Firdaus E. Udwadia, Lagrangian mechanics, Gauss’ principle, quadratic programming, and generalized inverses: new equations for non-holonomically constrained discrete mechanical systems, Quart. Appl. Math. 52 (1994), no. 2, 229–241. MR 1276235, DOI https://doi.org/10.1090/qam/1276235
Y. Neimark and N. Fufaev, Dynamics of Nonholonomic Systems, Amer. Math. Soc. Translations, vol. 33, 1972
L. Pars, A Treatise on Analytical Dynamics, Ox Bow Press, Connecticut, Second Printing, 1972
- R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413. MR 69793
- C. Radhakrishna Rao, Linear statistical inference and its applications, 2nd ed., John Wiley & Sons, New York-London-Sydney, 1973. Wiley Series in Probability and Mathematical Statistics. MR 0346957
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- Firdaus E. Udwadia and Robert E. Kalaba, Analytical dynamics, Cambridge University Press, Cambridge, 1996. A new approach. MR 1447192
- Firdaus E. Udwadia, Robert E. Kalaba, and Hee-Chang Eun, Equations of motion for constrained mechanical systems and the extended d’Alembert’s principle, Quart. Appl. Math. 55 (1997), no. 2, 321–331. MR 1447580, DOI https://doi.org/10.1090/qam/1447580
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P. Appell, Traité de Mécanique Rationelle, Third edition, Paris, 1911
P. Appell, Exemple de Mouvement d’un Point Assujeti a une Liason Exprimée par une Relation Non Lineare Entre les Composantes de la Vitesse, Comptes Rendus, 1911, pp. 48–50
C. F. Gauss, Über ein neues allgemeines Grundgesetz der Mechanik, Journal für die Reine und Angewandte Mathematik 4, 232–235 (1829)
W. J. Gibbs, On the fundamental formulae of dynamics, Amer. Jour. Math. 2, 49–64 (1879)
R. E. Kalaba and F. E. Udwadia, Lagrangian mechanics, Gauss’s principle, quadratic programming, and generalized inverses: New equations for nonholonomically constrained discrete mechanical systems, Quart. Appl. Math. 52, 229–241 (1994)
Y. Neimark and N. Fufaev, Dynamics of Nonholonomic Systems, Amer. Math. Soc. Translations, vol. 33, 1972
L. Pars, A Treatise on Analytical Dynamics, Ox Bow Press, Connecticut, Second Printing, 1972
R. Penrose, A generalized inverse of matrices, Proc. Camb. Philos. Soc. 51, 406–313 (1955)
C. R. Rao, Linear Statistical Inference and Applications, Wiley, 1973
C. Rao and S. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley, New York, 1972
F. E. Udwadia and R. E. Kalaba, A New Perspective on Constrained Motion, Proc. Roy. Soc. London 439, 407–410 (1992)
F. E. Udwadia and R. E. Kalaba, Analytical Dynamics: A New Approach, Cambridge University Press, London, 1996
F. E. Udwadia, R. E. Kalaba, and E. Hee-Chang, Equations of motion for constrained mechanical systems and the extended D’Alembert’s principle, Quart. Appl. Math. 55, 321–331 (1997)
E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge, Fourth Edition, 1937
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© Copyright 1998
American Mathematical Society
