Direct use of variational principles as an approximation technique in classical mechanics

American Journal of Physics

Volumn 64, Issue 9, 1996, Pages 1177-1184

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

^a UNIVERSITY OF GUELPH   (Canada)

^b INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030510356     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18340     Document Type: Article

References (40)

1. The few primary references of which we are aware are listed in Refs. 2 and 3 below. In this journal, primary references for the direct use of Hamilton's principle in particle mechanics are: P. H. Miller, "Hamilton's principle as a computational device," Am. J. Phys. 25, 30-32 (1957); M. M. Gordon, "Study of classical scattering theory," ibid. 25, 32-37 (1957); D. W. Schlitt, "Hamilton's principle and approximate solution to problems in classical mechanics," ibid. 45, 205-207 (1977); H. Gilmartin, A. Klein, and C.-t. Li, "Application of Hamilton's principle to the study of the anharmonic oscillator in classical mechanics," ibid. 47, 636-639 (1979). In continuum mechanics and electromagnetism, direct methods (e.g., Rayleigh-Ritz) are much more common (see, e.g., Ref. 5).
2. The few primary references of which we are aware are listed in Refs. 2 and 3 below. In this journal, primary references for the direct use of Hamilton's principle in particle mechanics are: P. H. Miller, "Hamilton's principle as a computational device," Am. J. Phys. 25, 30-32 (1957); M. M. Gordon, "Study of classical scattering theory," ibid. 25, 32-37 (1957); D. W. Schlitt, "Hamilton's principle and approximate solution to problems in classical mechanics," ibid. 45, 205-207 (1977); H. Gilmartin, A. Klein, and C.-t. Li, "Application of Hamilton's principle to the study of the anharmonic oscillator in classical mechanics," ibid. 47, 636-639 (1979). In continuum mechanics and electromagnetism, direct methods (e.g., Rayleigh-Ritz) are much more common (see, e.g., Ref. 5).
3. The few primary references of which we are aware are listed in Refs. 2 and 3 below. In this journal, primary references for the direct use of Hamilton's principle in particle mechanics are: P. H. Miller, "Hamilton's principle as a computational device," Am. J. Phys. 25, 30-32 (1957); M. M. Gordon, "Study of classical scattering theory," ibid. 25, 32-37 (1957); D. W. Schlitt, "Hamilton's principle and approximate solution to problems in classical mechanics," ibid. 45, 205-207 (1977); H. Gilmartin, A. Klein, and C.-t. Li, "Application of Hamilton's principle to the study of the anharmonic oscillator in classical mechanics," ibid. 47, 636-639 (1979). In continuum mechanics and electromagnetism, direct methods (e.g., Rayleigh-Ritz) are much more common (see, e.g., Ref. 5).
4. The few primary references of which we are aware are listed in Refs. 2 and 3 below. In this journal, primary references for the direct use of Hamilton's principle in particle mechanics are: P. H. Miller, "Hamilton's principle as a computational device," Am. J. Phys. 25, 30-32 (1957); M. M. Gordon, "Study of classical scattering theory," ibid. 25, 32-37 (1957); D. W. Schlitt, "Hamilton's principle and approximate solution to problems in classical mechanics," ibid. 45, 205-207 (1977); H. Gilmartin, A. Klein, and C.-t. Li, "Application of Hamilton's principle to the study of the anharmonic oscillator in classical mechanics," ibid. 47, 636-639 (1979). In continuum mechanics and electromagnetism, direct methods (e.g., Rayleigh-Ritz) are much more common (see, e.g., Ref. 5).
5. C. G. Gray, G. Karl, and V. A. Novikov, "The four variational principles of mechanics," Ann. Phys. (in press).
6. L. A. Bunimovich, "Variational principle for periodic trajectories of hyperbolic billiards," Chaos 5, 349-355 (1995); A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of endpoints: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1942); T. L. Beck, J. D. Doll, and D. L. Freeman, "Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function," J. Chem. Phys. 90, 3181-3191 (1989); D. L. Hitzl and D. A. Levinson, "Application of Hamilton's laws of varying action to the restricted three-body problem," Celest. Mech. 22, 255-266 (1980); R. H. G. Helleman, "Variational solutions of noninte- grable systems," in Topics in Nonlinear Dynamics, edited by S. Jorna [AIP Conf. Proc. 46, 264-285 (1978)].
7. L. A. Bunimovich, "Variational principle for periodic trajectories of hyperbolic billiards," Chaos 5, 349-355 (1995); A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of endpoints: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1942); T. L. Beck, J. D. Doll, and D. L. Freeman, "Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function," J. Chem. Phys. 90, 3181-3191 (1989); D. L. Hitzl and D. A. Levinson, "Application of Hamilton's laws of varying action to the restricted three-body problem," Celest. Mech. 22, 255-266 (1980); R. H. G. Helleman, "Variational solutions of noninte- grable systems," in Topics in Nonlinear Dynamics, edited by S. Jorna [AIP Conf. Proc. 46, 264-285 (1978)].
8. L. A. Bunimovich, "Variational principle for periodic trajectories of hyperbolic billiards," Chaos 5, 349-355 (1995); A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of endpoints: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1942); T. L. Beck, J. D. Doll, and D. L. Freeman, "Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function," J. Chem. Phys. 90, 3181-3191 (1989); D. L. Hitzl and D. A. Levinson, "Application of Hamilton's laws of varying action to the restricted three-body problem," Celest. Mech. 22, 255-266 (1980); R. H. G. Helleman, "Variational solutions of noninte- grable systems," in Topics in Nonlinear Dynamics, edited by S. Jorna [AIP Conf. Proc. 46, 264-285 (1978)].
9. L. A. Bunimovich, "Variational principle for periodic trajectories of hyperbolic billiards," Chaos 5, 349-355 (1995); A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of endpoints: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1942); T. L. Beck, J. D. Doll, and D. L. Freeman, "Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function," J. Chem. Phys. 90, 3181-3191 (1989); D. L. Hitzl and D. A. Levinson, "Application of Hamilton's laws of varying action to the restricted three-body problem," Celest. Mech. 22, 255-266 (1980); R. H. G. Helleman, "Variational solutions of noninte- grable systems," in Topics in Nonlinear Dynamics, edited by S. Jorna [AIP Conf. Proc. 46, 264-285 (1978)].
10. L. A. Bunimovich, "Variational principle for periodic trajectories of hyperbolic billiards," Chaos 5, 349-355 (1995); A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of endpoints: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1942); T. L. Beck, J. D. Doll, and D. L. Freeman, "Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function," J. Chem. Phys. 90, 3181-3191 (1989); D. L. Hitzl and D. A. Levinson, "Application of Hamilton's laws of varying action to the restricted three-body problem," Celest. Mech. 22, 255-266 (1980); R. H. G. Helleman, "Variational solutions of noninte-grable systems," in Topics in Nonlinear Dynamics, edited by S. Jorna [AIP Conf. Proc. 46, 264-285 (1978)].
11. L. A. Bunimovich, "Variational principle for periodic trajectories of hyperbolic billiards," Chaos 5, 349-355 (1995); A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of endpoints: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1942); T. L. Beck, J. D. Doll, and D. L. Freeman, "Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function," J. Chem. Phys. 90, 3181-3191 (1989); D. L. Hitzl and D. A. Levinson, "Application of Hamilton's laws of varying action to the restricted three-body problem," Celest. Mech. 22, 255-266 (1980); R. H. G. Helleman, "Variational solutions of noninte- grable systems," in Topics in Nonlinear Dynamics, edited by S. Jorna [AIP Conf. Proc. 46, 264-285 (1978)].
12. Direct methods of solving variational problems (i.e., without the use of the corresponding differential equation) are discussed in a general way in I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ, 1963), Chap. 8. The only mechanics texts of which we are aware which (briefly) mention such methods for particle mechanics are S. Timoshenko and D. H. Young, Advanced Dynamics (McGraw-Hill, New York, 1948), p. 234; S. W. Groesberg, Advanced Mechanics (Wiley, New York, 1968), p. 262.
13. Direct methods of solving variational problems (i.e., without the use of the corresponding differential equation) are discussed in a general way in I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ, 1963), Chap. 8. The only mechanics texts of which we are aware which (briefly) mention such methods for particle mechanics are S. Timoshenko and D. H. Young, Advanced Dynamics (McGraw-Hill, New York, 1948), p. 234; S. W. Groesberg, Advanced Mechanics (Wiley, New York, 1968), p. 262.
14. Direct methods of solving variational problems (i.e., without the use of the corresponding differential equation) are discussed in a general way in I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ, 1963), Chap. 8. The only mechanics texts of which we are aware which (briefly) mention such methods for particle mechanics are S. Timoshenko and D. H. Young, Advanced Dynamics (McGraw-Hill, New York, 1948), p. 234; S. W. Groesberg, Advanced Mechanics (Wiley, New York, 1968), p. 262.
15. T. Mura and T. Koya, Variational Methods in Mechanics (Oxford U.P., Oxford, 1992). P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pt. II, p. 106. L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Blackie, London, 1965).
16. T. Mura and T. Koya, Variational Methods in Mechanics (Oxford U.P., Oxford, 1992). P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pt. II, p. 106. L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Blackie, London, 1965).
17. T. Mura and T. Koya, Variational Methods in Mechanics (Oxford U.P., Oxford, 1992). P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pt. II, p. 106. L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Blackie, London, 1965).
18. E. A. Hylleraas, "Über den grundzustand des heliumatoms," Z. Phys. 48, 469-494 (1928); "Die energie des heliumatoms im grundzustande," Phys. Z. 30, 249-250 (1929).
19. E. A. Hylleraas, "Über den grundzustand des heliumatoms," Z. Phys. 48, 469-494 (1928); "Die energie des heliumatoms im grundzustande," Phys. Z. 30, 249-250 (1929).
20. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of superconductivity," Phys. Rev. 108, 1175-1204 (1957).
21. R. P. Feynman, "Atomic theory of the two-fluid model of liquid helium," Phys. Rev. 94, 262-277 (1954).
22. G. Karl and V. A. Novikov, "Variational estimates for excited states," Phys. Rev. D 51, 5069-5078 (1995), and references therein.
23. See, for example, E. Gerjuoy, A. R. P. Rau, and L. Spruch, "A unified formulation of the construction of variational principles," Rev. Mod. Phys. 55, 725-774 (1983).
24. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), 2nd ed.
25. E=0 to the 1D quartic oscillator, by using trial trajectories which do not conserve the energy E.
26. I. C. Percival, "Variational principles for the invariant toroids of classical dynamics," J. Phys. A 7, 794-802 (1974).
27. L. D. Landau and I. M. Lifshitz, Mechanics (Addison-Wesley, Reading, MA, 1969), 2nd ed., p. 26; A. Sommerfeld, Mechanics (Academic, New York, 1964), p. 90.
28. L. D. Landau and I. M. Lifshitz, Mechanics (Addison-Wesley, Reading, MA, 1969), 2nd ed., p. 26; A. Sommerfeld, Mechanics (Academic, New York, 1964), p. 90.
29. E. O. Schulz-Dubois, "Foucault pendulum experiment by Kamerlingh Onnes and degenerate perturbation theory," Am. J. Phys. 38, 173-188 (1970). H. R. Crane, "Foucault pendulum 'wall clock'," Am. J. Phys. 63, 33-39 (1995).
30. E. O. Schulz-Dubois, "Foucault pendulum experiment by Kamerlingh Onnes and degenerate perturbation theory," Am. J. Phys. 38, 173-188 (1970). H. R. Crane, "Foucault pendulum 'wall clock'," Am. J. Phys. 63, 33-39 (1995).
31. M. G. Olsson, "The precessing spherical pendulum," Am. J. Phys. 46, 1118-1119 (1978); "Spherical pendulum revisited," ibid. 49, 531-534 (1981).
32. M. G. Olsson, "The precessing spherical pendulum," Am. J. Phys. 46, 1118-1119 (1978); "Spherical pendulum revisited," ibid. 49, 531-534 (1981).
33. M. M. Gordon, Ref. 1.
34. 2 potential," Phys. Rev. Lett. 65, 2837-2838 (1990).
35. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
36. C. C. Martens, R. L. Waterland, and W. P. Reinhardt, "Classical, semiclassical and quantum mechanics of a globally chaotic system: Integrability in the adiabatic approximation," J. Chem. Phys. 90, 2328-2337 (1989).
37. See, e.g., Gelfand and Fomin, Ref. 4, p. 54.
38. This is part of the Kolmogorov-Arnold-Moser (KAM) theorem; see, e.g., Gutzwiller, Ref. 19, p. 132.
39. I. C. Percival, "A variational principle for invariant tori of fixed frequency," J. Phys. A 12, L57-60 (1979). This result has been derived independently by A. Klein and C.-t. Li, "Semiclassical quantization of nonseparabie systems," J. Math. Phys. 20, 572-578 (1979).
40. I. C. Percival, "A variational principle for invariant tori of fixed frequency," J. Phys. A 12, L57-60 (1979). This result has been derived independently by A. Klein and C.-t. Li, "Semiclassical quantization of nonseparabie systems," J. Math. Phys. 20, 572-578 (1979).