Abstract
It has been recognized in the literature of the calculus of variations that the classical statement of the principle of least action (Hamilton's principle for conservative systems) is not strictly correct. Recently, mathematical proofs have been offered for what is claimed to be a more precise statement of Hamilton's principle for conservative systems. According to a widely publicized version of this more precise statement, the action integral for conservative systems is a minimum for discrete systems for small time intervals only and is never minimum for continuous systems. In this paper, two contradictions to this “more precise” statement are demonstrated, one for a discrete system and one for a continuous system.
Similar content being viewed by others
References
W. Yourgrau and S. Mandelstam,Variational Principles in Dynamics and Quantum Theory, 3rd ed. (Pitman & Sons, 1968), pp. 16, 175, 177.
W. F. Osgood,Mechanics (MacMillan, 1937), p. 356.
M. R. Hestenes,Modern Mathematics for the Engineer (McGraw-Hill, 1956), p. 75.
I. M. Gelfand and S. V. Fomin,Calculus of Variations (Prentice Hall, 1963), p. 159.
D. R. Smith,Variational Methods in Optimization (Prentice Hall, 1974), p. 195.
D. R. Smith and C. V. Smith, Jr.,AIAA J. 12, 1573 (1974).
C. D. Bailey,ACTA Mechanica 36, 63 (1980).
C. D. Bailey,J. Sound Vibration 44, 15 (1976).
C. D. Bailey,Computer Methods in Applied Mechanics and Engineering 7, 235 (1976).
C. D. Bailey,Found. Phys. 5, 433 (1975).
C. D. Bailey,ASME J. Applied Mechanics 43, 684 (1976).
C. D. Bailey and J. L. Haines,Computer Methods in Applied Mechanics and Engineering, to appear.
D. H. Hodges,AIAA J. 17, 924 (1979).
D. H. Hodges,American Helicopter Society J. 24, 43 (1979).
D. L. Hitzl and D. A. Levinson,Celestial Mechanics,22, 255 (1980).
D. L. Hitzl,J. Computational Phys.,38, 185 (1980).
M. Baruch and R. Riff, Technion Israel Institute of Technology, TAE 403 (1980).
T. E. Simkins,AIAA J. 16, 559 (1978).
W. R. Hamilton,Phil. Trans. R. Soc. Lond. 124, 247 (1834).
H. Goldstein,Classical Mechanics (Addison-Wesley, 1950).
E. T. Whittaker,Analytical Dynamics (Cambridge University Press, 1st ed., 1904; 4th ed., 1937), p. 245.
A. G. Webster,The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies (G. E. Stechert & Co., 1st ed., 1904; 3rd ed., 1942), p. 97.
C. D. Bailey,AIAA J. 17, 541 (1979).
C. V. Smith, Jr.,ASME J. Applied Mechanics 44, 796 (1977).
C. D. Bailey,ASME J. Applied Mechanics 44, 796 (1977).
C. D. Bailey,J. Sound Vibration 54, 454 (1977).
C. D. Bailey,AIAA J. 18, 347 (1980).
R. D. Witchey, The Ohio State University, Thesis (1980).
X. Culverwell,Proc. London Math. Soc. XXIII, 241 (1892) [cited in Ref. 21].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bailey, C.D. On a more precise statement of Hamilton's principle. Found Phys 11, 279–296 (1981). https://doi.org/10.1007/BF00726269
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00726269