Abstract
The points of view of Hamilton, Lagrange, Gauss, Hertz, and Lord Rayleigh emphasized the concept of energy rather than force. The equations of motion of the system are not derived by consideration of equilibrium of forces acting on the system, possibly including the Newtonian and d’Alambert inertia forces. Instead, the primary role is played by energy considerations. As an example of such an approach, a condition of stable equilibrium under static loads is replaced by the condition of a local minimum for the potential energy of a mechanical system. Instead of solving the equations of motion of a vibrating system to find its natural frequencies, it is possible to consider the mean values of potential and kinetic energies, or to minimize an appropriate energy functional.
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References for Chapter 1
H.C. Corben, P. Stehle, Classical mechanics, J. Wiley and Sons, New York, 1960.
H. Goldstein, Classical mechanics, Addison Wesley Reading, Mass., 1953.
R. Abraham, Foundations of Classical mechanics, Benjamin, New York, 1967.
L.D. Landau, E.M. Lifshitz, Mechanics, Addison Wesley, Reading Mass., 1960.
Lord Rayleigh, The theory of sound, Dover Publications, New York.
E.J. Routh, Treatise on the dynamics of rigid bodies, Macmillan, London, 1905 (also reprinted by Dover Publications, New York).
Ernst Mach, The science of mechanics, 5th English edition, Open Court Publishing Co., La Salle, 111., 1942.
W. von Kleinschmidt, H.K. Schulze, Brachistochronen in einem zentral—symmetrischen Schwerefeld, ZAMM, 50, 1970, p. 234–236.
A.V. Russalovskaya, G.I. Ivanov, A.I. Ivanov, On the brachistochrone for a particle with variable mass subjected to motion with friction and an exponential law of mass variation, Dopovidi Akad, Nauk Ukrain. S.S.R. 11, (1975), p. 97–112.
N.W. Ashby, E. Brittain, W.F. Love, W. Wyss, Brachistochrone with Coulomb friction, American J. Physics, 43, (1975), p. 902–906.
D.S. Djukic, On the brachistochronic motion of a non— conservative dynamic system, Zbornik Rad. Matem. Instit. Beograd, (N.S.), 3, #11 (1979) p. 39–46.
J.E. Drummond, G.L. Downes, The brachistochrone with acceleration: A running track, J. Optimization Theory and Applications, 7, (1971), p. 444–449.
Isaac Newton, Mathematical papers, 7 volumes, edited by D.T. Whiteside (1967–1976), Cambridge University Press, Cambridge.
F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound,Comm.Pure Appl.Math.,vol. 13, (1960),p. 551–585.
L.E. Payne, On stabilization of ill—posed Cauchy problems in nonlinear elasticity Contemporary Mathematics, vol. IV, Problems in elastic stability and vibrations, Vadim Komkov editor, (1981) American Math. Society, Providence, R. I.
H. Hertz, Prinzipien der Mechanik in neuen Zusammenhang dargestellt, Barth Verlag, Leipzig, 1910.
G.D. Birkho ff, Dynamical systems, American Math, Society, New York, 1928.
H. Poincaré, Oeuvres de Henri Poincaré, Gauthier Villars, Paris, 1952.
V. Komkov, Continuability and estimates of solutions of (a(t)-e (x) • x’)’ + c(t)f(x) = 0 Annales Pol. Math., XXX, (1974), p. 125–137.
J T.Burton, R.Grimmer, On solvability of solutions of second order differential equations,Proc.Amer. Math.Society, 29,#2, (1971),p. 277–283.
H.Anton, L.Y.Bahar,On the use of Schwartz distributions in the Lagrange variational problem,Hadronic Journal, 1.,(1978),p.1215–1226.
K. Weierstrass, Gesammelte Werke, Meier und Müller Verlag, Berlin, 1894.
E.Cartan,Leçons sur les invariants,Hermann et Cie., Paris,1922.
C. Caratheodory, Variationsrechnung, Teubner Verlag, Leipzig, 1935. English translation by Chelsea Publ. Co., London, 1939.
A. Kneser, Lehrbuch der Variationsrechnung, Vieweg, Brunswick, 1925.
The Karman, M.A. Biot Mathematical methods in engineering, McGraw Hill, New York, 1940.
K.C. Valanis, Vadim Komkov, Irreversi— sible thermodynamics from the point of view of internal variable theory, Archives of Mechanics, Vol. 32, #1, (1978), p. 33. 58.
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© 1986 D. Reidel Publishing Company
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Komkov, V. (1986). Energy Methods, Classical Calculus of Variations Approach — Selected Topics and Applications. In: Variational Principles of Continuum Mechanics with Engineering Applications. Mathematics and Its Applications, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4564-7_2
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DOI: https://doi.org/10.1007/978-94-009-4564-7_2
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