The four variational principles of mechanics

Annals of Physics

Volumn 251, Issue 1, 1996, Pages 1-25

  Gray, C G ^a   Karl, G ^a   Novikov, V A ^a  

^a UNIVERSITY OF GUELPH   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030579039     PISSN: 00034916     EISSN: None     Source Type: Journal    
DOI: 10.1006/aphy.1996.0104     Document Type: Article

References (65)

1. 1. W. Yourgrau and S. Mandelstam, "Variational Principles in Dynamics and Quantum Theory," 3rd ed., Pitman, London, W. B. Saunders, Philadelphia, 1968; reprinted, Dover, New York, 1979. The notations S and W are interchanged in this reference compared to the present text, which follows current usage.
2. 2. C. Lanczos, "The Variational Principles of Mechanics," 4th ed., University of Toronto Press, Toronto, 1970; reprinted, Dover, New York, 1986; E. G. Jourdain, "The Principle of Least Action," Open Court Publishing, Chicago, 1913; W. Thomson (Lord Kelvin) and P. G. Tait, "Treatise on Natural Philosophy," Part I, Cambridge Univ. Press, Cambridge, 1879 and 1912; reprinted as "Principles of Mechanics and Dynamics," Dover, New York, 1962; A. W. Conway and A. J. McConnell (Eds.), "The Mathematical Papers of Sir William Rowan Hamilton," Vol. II, "Dynamics," Cambridge Univ. Press, Cambridge, 1940; E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55 (1983), 725.
3. 2. C. Lanczos, "The Variational Principles of Mechanics," 4th ed., University of Toronto Press, Toronto, 1970; reprinted, Dover, New York, 1986; E. G. Jourdain, "The Principle of Least Action," Open Court Publishing, Chicago, 1913; W. Thomson (Lord Kelvin) and P. G. Tait, "Treatise on Natural Philosophy," Part I, Cambridge Univ. Press, Cambridge, 1879 and 1912; reprinted as "Principles of Mechanics and Dynamics," Dover, New York, 1962; A. W. Conway and A. J. McConnell (Eds.), "The Mathematical Papers of Sir William Rowan Hamilton," Vol. II, "Dynamics," Cambridge Univ. Press, Cambridge, 1940; E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55 (1983), 725.
4. 2. C. Lanczos, "The Variational Principles of Mechanics," 4th ed., University of Toronto Press, Toronto, 1970; reprinted, Dover, New York, 1986; E. G. Jourdain, "The Principle of Least Action," Open Court Publishing, Chicago, 1913; W. Thomson (Lord Kelvin) and P. G. Tait, "Treatise on Natural Philosophy," Part I, Cambridge Univ. Press, Cambridge, 1879 and 1912; reprinted as "Principles of Mechanics and Dynamics," Dover, New York, 1962; A. W. Conway and A. J. McConnell (Eds.), "The Mathematical Papers of Sir William Rowan Hamilton," Vol. II, "Dynamics," Cambridge Univ. Press, Cambridge, 1940; E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55 (1983), 725.
5. 2. C. Lanczos, "The Variational Principles of Mechanics," 4th ed., University of Toronto Press, Toronto, 1970; reprinted, Dover, New York, 1986; E. G. Jourdain, "The Principle of Least Action," Open Court Publishing, Chicago, 1913; W. Thomson (Lord Kelvin) and P. G. Tait, "Treatise on Natural Philosophy," Part I, Cambridge Univ. Press, Cambridge, 1879 and 1912; reprinted as "Principles of Mechanics and Dynamics," Dover, New York, 1962; A. W. Conway and A. J. McConnell (Eds.), "The Mathematical Papers of Sir William Rowan Hamilton," Vol. II, "Dynamics," Cambridge Univ. Press, Cambridge, 1940; E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55 (1983), 725.
6. 2. C. Lanczos, "The Variational Principles of Mechanics," 4th ed., University of Toronto Press, Toronto, 1970; reprinted, Dover, New York, 1986; E. G. Jourdain, "The Principle of Least Action," Open Court Publishing, Chicago, 1913; W. Thomson (Lord Kelvin) and P. G. Tait, "Treatise on Natural Philosophy," Part I, Cambridge Univ. Press, Cambridge, 1879 and 1912; reprinted as "Principles of Mechanics and Dynamics," Dover, New York, 1962; A. W. Conway and A. J. McConnell (Eds.), "The Mathematical Papers of Sir William Rowan Hamilton," Vol. II, "Dynamics," Cambridge Univ. Press, Cambridge, 1940; E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55 (1983), 725.
7. 2. C. Lanczos, "The Variational Principles of Mechanics," 4th ed., University of Toronto Press, Toronto, 1970; reprinted, Dover, New York, 1986; E. G. Jourdain, "The Principle of Least Action," Open Court Publishing, Chicago, 1913; W. Thomson (Lord Kelvin) and P. G. Tait, "Treatise on Natural Philosophy," Part I, Cambridge Univ. Press, Cambridge, 1879 and 1912; reprinted as "Principles of Mechanics and Dynamics," Dover, New York, 1962; A. W. Conway and A. J. McConnell (Eds.), "The Mathematical Papers of Sir William Rowan Hamilton," Vol. II, "Dynamics," Cambridge Univ. Press, Cambridge, 1940; E. Gerjuoy, A. R. P. Rau, and L. Spruch, Rev. Mod. Phys. 55 (1983), 725.
8. 3. G. Karl and V. A. Novikov, Phys. Rev. D 51 (1995), 5069; J. Exp. Theor. Phys. 80 (1995), 783.
9. 3. G. Karl and V. A. Novikov, Phys. Rev. D 51 (1995), 5069; J. Exp. Theor. Phys. 80 (1995), 783.
10. 4. I. C. Percival, J. Phys. A: Math. Nucl. Gen. 7 (1974), 794, and Ref. [18].
11. 5. E. Schrödinger, Ann. Phys. 79 (1926), 361, 489; reprinted and translated, in "Collected Papers on Wave Mechanics," Blackie, Glasgow, 1928; reprinted by Chelsea, New York, 1982.
12. 5. E. Schrödinger, Ann. Phys. 79 (1926), 361, 489; reprinted and translated, in "Collected Papers on Wave Mechanics," Blackie, Glasgow, 1928; reprinted by Chelsea, New York, 1982.
13. 6. See, e.g., Ref. [19, p. 165].
14. 7. J. W. Gibbs, "The Scientific Papers of J. Willard Gibbs," Vol. I, "Thermodynamics," p. 56, Longmans, Green, London, 1906; reprinted by Dover, New York, 1961.
15. 8. For a general discussion of direct methods of variational calculus, see, e.g., I. M. Gelfand and S. V. Fomin, "Calculus of Variations," Chap. 8, Prentice-Hall, Englewood Cliffs, NJ, 1963.
16. 9. T. Mura and T. Koya, "Variational Methods in Mechanics," Oxford Univ. Press, New York, 1992; P. M. Morse and H. Feshbach, "Methods of Theoretical Physics," Part II, p. 1106, McGraw-Hill, New York, 1953; L. Cairo and T. Kahan, "Variational Techniques in Electromagnetism," Blackie, London, 1965.
17. 9. T. Mura and T. Koya, "Variational Methods in Mechanics," Oxford Univ. Press, New York, 1992; P. M. Morse and H. Feshbach, "Methods of Theoretical Physics," Part II, p. 1106, McGraw-Hill, New York, 1953; L. Cairo and T. Kahan, "Variational Techniques in Electromagnetism," Blackie, London, 1965.
18. 9. T. Mura and T. Koya, "Variational Methods in Mechanics," Oxford Univ. Press, New York, 1992; P. M. Morse and H. Feshbach, "Methods of Theoretical Physics," Part II, p. 1106, McGraw-Hill, New York, 1953; L. Cairo and T. Kahan, "Variational Techniques in Electromagnetism," Blackie, London, 1965.
19. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
20. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
21. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
22. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
23. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
24. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
25. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
26. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
27. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
28. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
29. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
30. 10. The few references of which we are aware which apply the direct (Ritz-type) method [8] of variational calculus to Hamilton's principle to obtain particle trajectories are listed here. Not surprisingly, Hamilton himself was aware of this method, see Hamilton Papers, Ref. [2], especially the correspondence with J. W. Lubbock (1837), pp. 250-263. A. G. Basile and C. G. Gray, J. Comp. Phys. 101 (1992), 80; T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90 (1989), 3181; D. L. Hitzl, J. Comp. Phys. 38 (1980), 185; H. Gilmartin, A. Klein, and C-t Li, Amer. J. Phys. 47 (1979), 636; R. H. G. Helleman, in "Topics in Nonlinear Dynamics" (S. Jorna, Ed.), A.I.P. Conference Proceedings, Vol. 46, p. 264, Amer. Ins. of Phys., New York, 1978; D. W. Schlitt, Amer. J. Phys. 45 (1977), 205; C. D. Bailey, Found. Phys. 5 (1975), 433; J. M. Blatt and J. N. Lyness, J. Austral. Math. Soc. 2 (1962), 257; J. M. Luttinger and R. B. Thomas, J. Math. Phys. 1 (1960), 121; P. H. Miller, Amer. J. Phys. 25 (1957), 30; J. H. Van Vleck, Phys. Rev. 22 (1923), 547; G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung," p. 130, F. Vieweg & Sohn, Braunschweig, 1918.
31. 11. See, e.g., L. D. Landau and E. M. Lifshitz, "Mechanics," 2nd ed., p. 26, Addison-Wesley, Reading, MA, 1969.
32. 12. J. H. Van Vleck, Ref. [10].
33. 13. C. G. Gray, G. Karl, and V. A. Novikov, Amer. J. Phys., in press.
34. 14. M. M. Gordon, Amer. J. Phys. 25 (1957), 32.
35. 2 potential.
36. 16. C. C. Martens, R. L. Waterland, and W. P. Reinhardt, J. Chem. Phys. 90 (1989), 2328.
37. 17. R. P. Feynman, Rev. Mod. Phys. 20 (1948), 267; see also P. A. M. Dirac, Physik. Z. Sowjetunion 3 (1933), 64.
38. 17. R. P. Feynman, Rev. Mod. Phys. 20 (1948), 267; see also P. A. M. Dirac, Physik. Z. Sowjetunion 3 (1933), 64.
39. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
40. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
41. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
42. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
43. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
44. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
45. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
46. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
47. 18. M. V. Berry and K. E. Mount, Rep. Progr. Phys. 35 (1972), 315; I. C. Percival, Adv. Chem. Phys. 36 (1977), 1; M. V. Berry, in "Chaotic Behaviour of Deterministic Systems" (G. Iooss, R. H. G. Helleman, and R. Stora, Ed.), p. 171, North-Holland, Amsterdam, 1983; B. Eckhardt, Phys. Rep. 163 (1988), 205; M. C. Gutzwiller; "Chaos in Classical and Quantum Mechanics," Springer-Verlag, New York 1990; M. V. Berry, in "Chaos and Quantum Physics" (M. J. Giannoni, A. Voros, and J. Zinn-Justin, Ed.), p. 252; Elsevier, Amsterdam, 1991; M. S. Child, "Semiclassical Mechanics with Molecular Applications," Oxford Univ. Press, Oxford, 1991; O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223 (1993), 43.
48. 19. See, e.g.; L. D. Landau and I. M. Lifshitz, "Quantum Mechanics (Nonrelativistic Theory)," 3rd ed., Pergamon, New York, 1977.
49. 20. See, e.g., H. A. Kramers, "Quantum Mechanics," p. 207, North-Holland, Amsterdam, 1957; reprinted, Dover, New York, 1964.
50. 21. F. Haake, "Quantum Signatures of Chaos," Springer-Verlag, New York, 1991.
51. 22. This is part of the Kolmogorov-Arnold-Moser (KAM) theorem; see, e.g., Gutzwiller, [18, p. 132].
52. 23. I. C. Percival, J. Phys. A 12 (1979), L57.
53. 24. A. Klein and C-t Li, J. Math. Phys. 20 (1979), 572.
54. 25. See, e.g., Gelfand and Fomin, [8, p. 79, 87].
55. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
56. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
57. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
58. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
59. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
60. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
61. 26. L. Hulthén, Fysiogr. Sällsk. Lund Förh. 14 (1944), 2; J. Schwinger, Phys. Rev. 72 (1947), 742; Phys. Rev. 78 (1950), 135; W. Kohn, Phys. Rev. 74 (1948), 1763; I. E. Tamm, Zh. Eksp. Teor. Fiz. 18 (1948), 337; B. A. Lippman and J. Schwinger, Phys. Rev. 79 (1950), 469; H. Feshbach and S. I. Rubinow, Phys. Rev. 88 (1952), 484; for a review, see, e.g., Gerjuoy et al., [2].
62. 27. See, e.g., Gelfand and Fomin, [8, Chap. 3].
63. 28. V. Trkal, Proc. Cambridge Philos. Soc. 21 (1922), 80.
64. 29. G. Temple and W. G. Bickley, "Rayleigh's Principle," Oxford Univ. Press, Oxford, 1933; reprinted, Dover, New York, 1956.
65. 30. W. R. Greenberg, A. Klein, and C-t Li, Phys. Rev. Lett. 75 (1995), 1244.