When action is not least

American Journal of Physics

Volumn 75, Issue 5, 2007, Pages 434-458

  Gray, C G ^a   Taylor, Edwin F ^b  

^a UNIVERSITY OF GUELPH   (Canada)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

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EID: 34547379123     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.2710480     Document Type: Article

References (186)

1. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1963), Vol. II, Chap. 19.
2. A. Bork and A. Zellweger, "Least action via computer," Am. J. Phys. 37, 386-390 (1969).
3. J. P. Provost, "Least action principle on an air table," Am. J. Phys. 43, 774-781 (1975).
4. E. F. Taylor, "A call to action," Am. J. Phys. 71, 423-425 (2003).
5. J. Hanc, S. Tuleja, and M. Hancova, "Simple derivation of Newtonian mechanics from the principle of least action," Am. J. Phys. 71, 386-391 (2003).
6. J. Hanc, S. Tuleja, and M. Hancova, "Symmetries and conservation laws: Consequences of Noether's theorem," Am. J. Phys. 72, 428-435 (2004).
7. J. Hanc, E. F. Taylor, and S. Tuleja, "Deriving Lagrange's equations using elementary calculus," Am. J. Phys. 72, 510-513 (2004).
8. J. Hanc and E. F. Taylor, "From conservation of energy to the principle of least action: A story line," Am. J. Phys. 72, 514-521 (2004).
9. T. A. Moore, "Getting the most action out of the least action: A proposal," Am. J. Phys. 72, 522-527 (2004).
10. J. Ogborn and E. F. Taylor, "Quantum physics explains Newton's laws of motion," Phys. Educ. 40, 26-34 (2005).
11. E. F. Taylor, J. Hanc, and S. Tuleja, "Variational mechanics in one and two dimensions," Am. J. Phys. 73, 603-610 (2005).
12. C. G. Gray, G. Karl, and V. A. Novikov, "Direct use of variational principles as an approximation technique in classical mechanics," Am. J. Phys. 64, 1177-1184 (1996) and references therein.
13. J. L. Lagrange, Analytical Mechanics (Mécanique Analytique) (Gauthier-Villars, Paris, 1888-89), 2nd ed. (1811) (Kluwer, Dordrecht, 1997), pp. 183 and 219.
14. The same erroneous statement occurs in work published in 1760-61, ibid, p. xxxiii.
15. We easily found about two dozen texts using the erroneous term "maximum." See, for example, E. Mach, The Science of Mechanics, 9th ed. (Open Court, La Salle, IL, 1960), p. 463;
16. A. Sommerfeld, Mechanics (Academic, New York, 1952), p. 208;
17. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, New York, 1953), Part 1, p. 281;
18. Park, Classical Dynamics and its Quantum Analogues, 2nd ed. (Springer, Berlin, 1990), p. 57;
19. H. Hand and J. D. Finch, Analytical Mechanics (Cambridge U. P., Cambridge, 1998), p. 53;
20. G. R. Fowles and G. L. Cassiday, Analytical Mechanics, 6th ed. (Saunders, Forth Worth, 1999), p. 393;
21. R. K. Cooper and C. Pellegrini, Modern Analytical Mechanics (Kluwer, New York, 1999), p. 34.
22. A similar error occurs in A. J. Hanson, Visualizing Quaternions (Elsevier, Amsterdam, 2006), p. 368, which states that geodesies on a sphere can have maximum length.
23. E. F. Taylor and J. A. Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison-Wesley Longman, San Francisco, 2000), pp. 1-7 and 3-4.
24. The number of authors of books and papers using "extremum" and "extremal" is endless. Some examples include R. Baierlein, Newtonian Dynamics (McGraw-Hill, New York, 1983), p. 125;
25. N. Hand and J. D. Finch, Ref. 14, p. 51;
26. J. D. Logan, Invariant Variational Principles (Academic, New York, 1977) mentions both "extremal" and "stationary," p. 8;
27. D. Landau and E. M. Lifschitz, Mechanics, 3rd ed. (Butterworth Heineman, Oxford, 2003), pp. 2 and 3;
28. see also their Classical Theory of Fields, 4th ed. (Butterworth Heineman, Oxford, 1999), p. 25;
29. S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Thomson, Brooks/Cole, Belmont. CA, 2004), p. 231;
30. R. K. Cooper and C. Pellegrini, Ref. 14.
31. F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to Optics (Pearson Prentice Hall, Upper Saddle River, NJ, 2007), p. 22;
32. J. R. Taylor, Classical Mechanics (University Science Books, Sausalito, 2005), Problem 6.5;
33. P. J. Nahin, When Least is Best (Princeton U. P., Princeton, NJ, 2004), p. 133;
34. S. Lemons, Perfect Form (Princeton U. P., Princeton, NJ, 1997), p. 8;
35. Park, Ref. 14, p. 13;
36. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw Hill, New York, 1976), p. 15;
37. R. Guenther, Modern Optics (Wiley, New York, 1990), p. 135;
38. R. W. Ditchburn, Light, 3rd ed. (Academic, London, 1976), p. 209;
39. R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Longmans, London, 1967), p. 7;
40. J. L. Synge, Geometrical Optics (Cambridge U. P., London, 1937), p. 3;
41. G. P. Sastry, "Problem on Fermat's principle," Am. J. Phys. 49, 345 (1981);
42. V. Berry, review of The Optics of Rays, Wavefronts and Caustics by O. N. Stavroudis (Academic, New York, 1972),
43. Sci. Prog. 61, 595-597 (1974);
44. Lakshminarayanan, A. K. Ghatak, and K. Thyagarajan, Lagrangian Optics (Kluwer, Boston, 2002), p. 16;
45. Perlick, Ray Optics, Fermat's Principle, and Applications to General Relativity (Springer, Berlin, 2000), pp. 149 and 152.
46. The seminal work on second variations of general functionals by Legendre, Jacobi, and Weierstrass and many others is described in the historical accounts of Refs. 19 and 20. Mayer's work (Ref. 19) was devoted specifically to the second variation of the Hamilton action. In our paper, we adapt Culverwell's work (Ref. 21) for the Maupertuis action W to the Hamilton action. Culverwell's work was preceded by that of Jacobi (Ref. 22) and Kelvin and Tait (Ref. 23).
47. H. H. Goldstine, A History of the Calculus of Variations From the 17th Through the 19th Century (Springer, New York, 1980).
48. I. Todhunter, A History of the Progress of the Calculus of Variations During the Nineteenth Century (Cambridge U. P., Cambridge, 1861) and (Dover, New York, 2005).
49. E. P. Culverwell, "The discrimination of maxima and minima values of single integrals with any number of dependent variables and any continuous restrictions of the variables, the limiting values of the variables being supposed given," Proc. London Math. Soc. 23, 241-265 (1892).
50. C. G. J. Jacobi, "Zür Theorie der Variationensrechnung und der Differential Gleichungen." J. f.Math. XVII, 68-82 (1837). An English translation is given in Ref. 20, p. 243, and a commentary is given in Ref. 19, p. 156.
51. W. Thomson (Lord Kelvin) and P. G. Tait, Treatise on Natural Philosophy (Cambridge U. P., Cambridge, 1879, 1912), Part I;
52. reprinted as Principles of Mechanics and Dynamics (Dover, New York, 1962), Part I, p. 422.
53. Gelfand and Fomin (Ref. 25) and other more recent books on calculus of variations are rigorous but rather sophisticated. A previous study (Ref. 26) of the nature of the stationarity of worldline action was based on the Jacobi-Morse eigenfunction method (Ref. 27), rather than on the more geometrical Jacobi-Culverwell-Whittaker approach.
54. I. M. Gelfand and S. V. Fomin, Calculus of Variations, translated by R. A. Silverman (Prentice Hall, Englewood Cliffs, NJ, 1963),
55. Russian edition 1961, reprinted (Dover, New York, 2000).
56. M. S. Hussein, J. G. Pereira, V. Stojanoff, and H. Takai, "The sufficient condition for an extremum in the classical action integral as an eigen-value problem," Am. J. Phys. 48, 767-770 (1980).
57. Hussein et al. make the common error of assuming S can be a true maximum. See also J. G. Papastavridis, "An eigenvalue criterion for the study of the Hamiltonian action's extremality," Mech. Res. Commun. 10, 171-179 (1983).
58. M. Morse, The Calculus of Variations in the Large (American Mathematical Society, Providence, RI, 1934).
59. 2W) that determines which case we are considering. Practical applications of the mechanical focal points are mentioned at the end of Ref. 37.
60. E. J. Routh, A Treatise on the Stability of a Given State of Motion (Macmillan, London, 1877), p. 103;
61. reissued as Stability of Motion, edited by A. T. Fuller (Taylor and Francis, London, 1975).
62. J. G. Papastavridis, "Toward an extremum characterization of kinetic stability," J. Sound Vib. 87, 573-587 (1983).
63. J. G. Papastavridis, "The principle of least action as a Lagrange variational problem: Stationarity and extremality conditions," Int. J. Eng. Sci. 24, 1437-1443 (1986);
64. "On a Lagrangean action based kinetic instability theorem of Kelvin and Tait," Int. J. Eng. Sci. 24, 1-17 (1986).
65. L. S, Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981), p. 143.
66. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990), p. 184.
67. In general, saddle points can be classified (or given an index (Ref. 27)) according to the number of independent directions leading to maximum-type behavior. Thus the point of zero-gradient on an ordinary horse saddle has a Morse index of unity. The Morse index for an action saddle point is equal to the number of kinetic foci between the end-points of the trajectory (Ref. 32, p. 90). Readable introductions to Morse theory are given by R. Forman, "How many equilibria are there? An introduction to Morse theory," in Six Themes on Variation, edited by R. Hardt (American Mathematical Society, Providence, RI, 2004), pp. 13-36,
68. and B. Van Brunt, The Calculus of Variations (Springer, New York, 2004), p. 254.
69. F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 1999), Chap. 14.
70. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. (originally published in 1904) (Cambridge U. P., Cambridge, 1999), p. 253.
71. The same example was treated earlier by C. G. J. Jacobi, Vorlesungen Über Dynamik (Braunschweig, Vieweg, 1884), reprinted (Chelsea, New York, 1969), p. 46.
72. A closer mechanics-optics analogy is between a kinetic focus (mechanics) and a caustic point (optics) (Ref. 38). The locus of limiting intersection points of pairs of mechanical spatial orbits is termed an envelope or caustic (see Fig. 7 for an example), just as the locus of limiting intersection points of pairs of optical rays is termed a caustic. In optics, the intersection point of a bundle of many rays is termed a focal point; a mechanical analogue occurs naturally in a few systems, for example, the sphere geodesies of Fig. 1 and the harmonic oscillator trajectories of Fig. 3, where a bundle of trajectories recrosses at a mechanical focal point. In electron microscopes (Refs. 39 and 40), mass spectrometers (Ref. 41), and particle accelerators (Ref. 42), electric and magnetic field configurations are designed to create mechanical focal points.
73. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 130 and 734;
74. J. A. Luck and J. H. Andrews, "Optical caustics in natural phenomena," Am. J. Phys. 60, 397-407 (1992).
75. M. Born and E. Wolf, Ref. 38, p. 738;
76. A. Artsimovich and S. Yu. Lukyanov. Motion of Charged Particles in Electric and Magnetic Fields (MIR, Moscow, 1980).
77. P. Grivet, Electron Optics, 2nd ed. (Pergamon, Oxford, 1972);
78. A. L. Hughes, "The magnetic electron lens," Am. J. Phys. 9, 204-207 (1941);
79. J. H. Moore, C. C. Davis, and M. A. Coplan, Building Scientific Apparatus, 3rd ed. (Perseus, Cambridge, MA, 2003), Chap. 5.
80. P. Grivet, Ref. 40, p. 822;
81. J. H. Moore et al., Ref. 40.
82. M. S. Livingston, The Development of High-Energy Accelerators (Dover, New York, 1966);
83. L. Bullock, "Electrostatic strong-focusing lens," Am. J. Phys. 23, 264-268 (1955);
84. W. Alvarez, R. Smits, and G. Senecal, "Mechanical analogue of the synchrotron, illustrating phase stability and two-dimensional focusing," Am. J. Phys. 43, 292-296 (1975);
85. A. Chao et al., "Experimental investigation of nonlinear dynamics in the Fermilab Tevatron," Phys. Rev. Lett. 61, 2752-2755 (1988).
86. For recent texts, see A. W. Chao, "Resource Letter PBA-1; Particle beams and accelerators," Am. J. Phys. 74, 855-862 (2006).
87. V. G. Boltyanskii, Envelopes (MacMillan, New York, 1964).
88. P. T. Saunders, An Introduction to Catastrophe Theory (Cambridge U. P., Cambridge, 1980), p. 62.
89. Systems with subsequent kinetic foci are discussed in Sees. VIII and IX. For examples with only a single kinetic focus, see Figs. 6 and 7. 46 M. C. Gutzwiller, "The origins of the trace formula," in Classical, Semiclassical and Quantum Dynamics in Atoms, edited by H. Friedrich and B. Eckhardt (Springer, New York, 1997), pp. 8-28.
90. This type of variation, δx = αφ, δẋ = αφ, where δx and δẋ vanish together for α → 0, is termed a weak variation. See, for example, C. Fox, An Introduction to the Calculus of Variations (Oxford U. P., Oxford, 1950), reprinted (Dover, New York, 1987), p. 3.
91. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, San Francisco, 2002), p. 44.
92. k5=0 for all k for the special variation around worldline PQ, which reflects the infinite number of true worldlines which connect P to Q, and which can all coalesce by varying the amplitude (see Fig. 3). In Morse theory (Refs. 27 and 34) the worldline PQ is referred to as a degenerate critical (stationary) point.
93. D. Morin, Introductory Classical Mechanics, (www.courses.fas. harvard.edu/-phys16/Textbook/) Chap. 5, p. V-8.
94. P). Only for the shortest of these worldlines is S a minimum. The situation is different in 2D (see Appendix B).
95. 2 → ẋ.
96. P and x are fixed).
97. A. G. Basile and C. G. Gray, "A relaxation algorithm for classical paths as a function of end points: Application to the semiclassical propagator for far-from-caustic and near-caustic conditions," J. Comput. Phys. 101, 80-93 (1992).
98. Better estimates can be found using the Sturm and Sturm-Liouville theories; see Papastavridis, Refs. 26 and 66.
99. 3S.
100. 2=0 [see Eq. (19)], so that kinetic foci do not exist.
101. 2S =0 is called the multiplicity of the kinetic focus (Van Brunt, Ref. 34, p. 254; Ref. 32, p. 90). If the mulitplicity is different from unity, Morse's theorem is modified from the statement in Ref. 34 to read as follows; The Morse index of the saddle point in action of worldline PR is equal to the number of kinetic foci between P and R, with each kinetic focus counted with its multiplicity.
102. J. M. T. Thompson and G. W. Hunt, Elastic Instability Phenomena (Wiley, Chichester, 1984), p. 20.
103. 0.
104. 2S<0, in agreement with Morse's general theory (Ref. 34).
105. 2/2m+U(x).
106. Reference 48, p. 353.
107. M. R. Hestenes, "Elements of the calculus of variations," in Modern Mathematics for the Engineer, edited by E. F. Beckenbach (McGraw Hill, New York, 1956), pp. 59-91.
108. O. Bottema, "Beisipiele zum Hamiltonschen Prinzip," Monatsh. Math. 66, 97-104 (1962).
109. J. G. Papastavridis, "On the extremal properties of Hamilton's action integral," J. Appl. Mech. 47, 955-956 (1980).
110. 2)=(0,0) represents the equilibrium trajectory x(t)=0. As we have seen, a true maximum in S cannot occur, so that other "directions" in function space not considered by the authors must give minimum-type behavior of S, leading to an overall saddle point.
111. The two-incline oscillator potential has the form U(x)=C|x|, with C = mg sin α cos α and α the angle of inclination. Here x is a horizontal direction. A detailed discussion of this oscillator is given by B. A. Sherwood, Notes on Classical Mechanics (Stipes, Champaign, Il, 1982), p. 157.
112. 2|x|, the bouncing ball (Ref. 71), and for x ≥ 0 the constant force spring (Ref. 72).
113. I. R. Lapidus, "One- and two-dimensional Hydrogen atoms," Am. J. Phys. 49, 807 (1981).
114. K. Andrew and J. Supplee, "A Hydrogen atom in d-dimensions," Am. J. Phys. 58, 1177-1183 (1990).
115. I. R. Gatland, "Theory of a nonharmonic oscillator," Am. J. Phys. 59, 155-158 (1991) and references therein;
116. W. M. Hartmann, "The dynamically shifted oscillator," Am. J. Phys. 54, 28-32 (1986).
117. A. Capecelatro and L. Salzarulo, Quantitative Physics for Scientists and Engineers: Mechanics (Aurie Associates, Newark, NJ, 1977), p. 162;
118. C.-Y. Wang and L. T. Watson, "Theory of the constant force spring," Trans. ASME, J. Appl. Mech. 47, 956-958 (1980);
119. H. Helm, "Comment on 'A constant force generator for the demonstration of Newton's second law'," Am. J. Phys. 52, 268 (1984).
120. C. G. Gray, G. Karl, and V. A. Novikov, "Progress in classical and quantum variational principles," Rep. Prog. Phys. 67, 159-208 (2004).
121. 60 (Ref. 76), where the quadratic terms in the potential are small. Ferroelectric soft modes in solids are also sometimes approximately represented by quartic potentials (Refs. 77 and 78).
122. 4," Proc. R. Soc. London, Ser. A 183, 328-337 (1945);
123. J. Laane, "Origin of the ring-puckering potential energy function for four-membered rings and spiro compounds. A possibility of pseudorotation," J. Phys. Chem. 95, 9246-9249 (1991).
124. 60." Chem. Phys. Lett. 211, 587-594 (1993).
125. A. S. Barker, "Far infrared dispersion and the Raman spectra of ferroelectric crystals," in Far-Infrared Properties of Solids, edited by S. S. Mitra and S. Nudelman (Plenum, New York, 1970), pp. 247-296.
126. J. Thomchick and J. P. McKelvey, "Anharmonic vibrations of an 'ideal' Hooke's law oscillator," Am. J. Phys. 46, 40-45 (1978).
127. R. Baierlein, Ref. 16, p. 73.
128. ij. The generalization to other dimensions is obvious. This condition (in slightly different form) is due to Mayer, Ref. 19, p. 269.
129. For a clear discussion, see J. G. Papastavridis, Analytical Mechanics (Oxford U. P., Oxford, 2002), p. 1061.
130. For multidimensions a caustic becomes in general a surface in space-time. The analogous theory for multidimensional spatial caustics, relevant for the action W, is discussed in Ref. 105. A simple example of a 2D surface spatial caustic is obtained by revolving the pattern of Fig. 7 about the vertical axis, thereby generating a paraboloid of revolution surface caustic/envelope. Due to axial symmetry, the caustic has a second (linear) branch, that is, the symmetry axis from y=0 to y=Y. An analogous optical example is discussed by M. V. Berry, "Singularities in waves and rays," in Physics of Defects, Les Houches Lectures XXXIV, edited by R. D. Balian, M. Kleman, and J.-P. Poirier (North Holland, Amsterdam, 1981). pp. 453-543.
131. R). Solving a boundary value problem with initial value problem methods (for example, the shooting method) is standard (Ref. 82). Solving an initial value problem with boundary value problem methods is much less common (Ref. 83). For an example of a boundary value problem with mixed conditions (prescribed initial velocities and final positions) for about 107 particles, see A. Nusser and E. Branchini, "On the least action principle in cosmology," Mon. Not. R. Astron. Soc. 313, 587-595 (2000).
132. See, for example, W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. P., Cambridge, 1992), p. 749.
133. H. R. Lewis and P. J. Kostelec, "The use of Hamilton's principle to derive time-advance algorithms for ordinary differential equations," Computer Phys. Commun. 96, 129-151 (1996);
134. Greenspan, "Approximate solution of initial value problems for ordinary differential equations by boundary value techniques," J. Math. Phys. Sci. 15, 261-274 (1967).
135. 2>0 as seen from Eq. (19). If U(x,t) is such that U″ alternates in sign with time, kinetic foci (and hence trajectory stability) may occur. An example is a pendulum with a rapidly vertically oscillating support point. In effect the gravitational field is oscillating. The pendulum can oscillate stably about the (normally unstable) upward vertical direction (Ref. 85). Two- and three-dimensional examples of this type are Paul traps (Ref. 86) and quadrupole mass filters (Ref. 85), which use oscillating quadrupole electric fields to trap ions. The equilibrium trajectory x(f)=0 at the center of the trap is unstable for purely electrostatic fields but is stabilized by using time-dependent electric fields. Focusing by alternating-gradients (also known as strong focus ing) in particle accelerators and storage rings is based on the same idea (Ref. 42).
136. M. H. Friedman, J. E. Campana, L. Kelner, E. H. Seeliger, and A. L. Yergey, "The inverted pendulum: A mechanical analog of the quadrupole mass filter," Am. J. Phys. 50, 924-931 (1982).
137. P. K. Gosh, Ion Traps (Oxford U. P., Oxford, 1995), p. 7.
138. J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley, New York, 1950), p. 112. Stoker's statements on series convergence need amendment in light of the Kolmogorov-Amold-Moser (KAM) theory (Ref. 89).
139. See J. Moser, "Combination tones for Duffing's equation," Commun. Pure Appl. Math. 18, 167-181 (1965);
140. T. Kapitaniak, J. Awrejcewicz and W.-H. Steeb, "Chaotic behaviour in an anharmonic oscillator with almost periodic excitation," J. Phys. A 20, L355-L358 (1987);
141. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981), p. 216;
142. A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics (Wiley, New York, 1995), p. 234;
143. S. Wiggins, "Chaos in the quasiperiodically forced Duffing oscillator," Phys. Lett. A 124, 138-142 (1987).
144. G. Seifert, "On almost periodic solutions for undamped systems with almost periodic forcing," Proc. Am. Math. Soc. 31, 104-108 (1972);
145. J. Moser, "Perturbation theory of quasiperiodic solutions and differential equations," in Bifurcation Theory and Nonlinear Eigenvalue Problems, edited by J. B. Keller and S. Antman (Benjamin, New York, 1969), pp. 283-308;
146. J. Moser, "Perturbation theory for almost periodic solutions for undamped nonlinear differential equations," in International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, edited by J. P. Lasalle and S. Lefschetz (Academic, New York, 1963), pp. 71-79;
147. S. Berger, "Two new approaches to large amplitude quasi-periodic motions of certain nonlinear Hamiltonian systems," Contemp. Math. 108, 11-18 (1990).
148. G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Weak Chaos and QuasiRegular Patterns (Cambridge U. P., Cambridge, 1991). p. 30.
149. See, for example, M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, New York, 1989), p. 35;
150. J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, 2nd ed. (Wiley, Chichester, 2002), pp. 310.
151. For example, the equilibrium position can be modulated. A somewhat similar system is a ball bouncing on a vertically oscillating table. The motion can be chaotic. See, for example, J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1983), p. 102;
152. N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, Redwood City, CA, 1992), p. 23;
153. A. B. Pippard, The Physics of Vibration (Cambridge U. P., Cambridge, 1978), Vol. 1, pp. 253, 271.
154. 0 cos ωt is studied in Ref. 30. For k=0 the Duffing oscillator reduces to the quartic oscillator.
155. 0 cos ωt (note the sign change in C compared to Ref. 92), and the Henon-Heiles oscillator with the potential in Eq. (83).
156. c(t) is studied. The worldlines for this system are all nonchaotic.
157. M. Henon and C. Heiles, "The applicability of the third integral of the motion: Some numerical experiments," Astron. J. 69, 73-79 (1964).
158. There have been a few formal studies of action for chaotic systems, but
159. Reference 48, p. 434.
160. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste (Gauthier-Villars, Paris, 1899), Vol. 3;
161. New Methods of Celestial Mechanics (AIP, New York, 1993), Part 3, p. 958.
162. The situation is complicated because, as Eq. (A1) shows, there are two forms for W, that is, the time-independent (first) form and the time-dependent (last) form. Spatial kinetic foci (discussed in Appendix B) occur for the time-independent form of W. Space-time kinetic foci occur for the time-dependent form of W, as for S. Typically the kinetic foci for the two forms for W differ from each other (Refs. 30 and 98) and from those for S.
163. E. J. Routh, A Treatise on Dynamics of a Particle (Cambridge U. P., Cambridge, 1898), reprinted (Dover, New York, 1960), p. 400.
164. A. P. French, "The envelopes of some families of fixed-energy trajectories," Am. J. Phys. 61, 805-811 (1993);
165. E. I. Butikov, "Families of Keplerian orbits," Bur. J. Phys. 24, 175-183 (2003).
166. P) to (0,0), a third kinetic focus arises for elliptical periodic orbits in three dimensions (see Ref. 33, p. 29).
167. N. G. Chetaev, Theoretical Mechanics (Springer, Berlin, 1989).
168. For the repulsive 1/r potential, the hyperbolic spatial orbits have a (parabolic shaped) caustic/envelope (French, Ref. 101).
169. 2)|x|, which is a parabolic shaped curve with two cusps on the y-axis.
170. R. H. Fowler, The Elementary Differential Geometry of Plane Curves (Cambridge U. P., Cambridge, 1920).
171. The finding of the two elliptical (or hyperbolic or parabolic) shaped trajectories from observations giving the two end-positions and the time interval is a famous problem of astronomy and celestial mechanics, solved by Lambert (1761), Gauss (1801-1809), and others (Ref. 108).
172. R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics (Dover, New York, 1971), p. 227;
173. H. Pollard, Celestial Mechanics (Mathematical Association of America, Washington, 1976), p. 28;
174. P. R. Escobal, Methods of Orbit Determination (Wiley, New York, 1965), p. 187. For the elliptical orbits, more than two trajectories typically become possible at sufficiently large time intervals; these additional trajectories correspond to more than one complete revolution along the orbit (Ref. 109).
175. R. H. Gooding, "A procedure for the solution of Lambert's orbital boundary-value problem," Celest. Mech. Dyn. Astron. 48, 145-165 (1990).
176. Q.
177. W. B. Gordon, "A minimizing property of Keplerian orbits," Am. J. Math. 99, 961-971 (1977).
178. Note that for the actual 2D trajectories in the potential U(x, y)=mgy, kinetic foci exist for the spatial paths of the Maupertuis action W, but do not exist for the space-time trajectories of the Hamilton action S. This result illustrates the general result stated in Appendix A that the kinetic foci for W and S differ in general.
179. A. P. French, Vibrations and Waves (Norton, New York, 1966), p. 36.
180. J. C. Slater and N. H. Frank, Introduction to Theoretical Physics (McGraw Hill, New York, 1933), p. 85.
181. C. G. Gray, G. Karl, and V. A. Novikov, "The four variational principles of mechanics," Ann. Phys. (N.Y.) 251, 1-25 (1996).
182. C. G. Gray, G. Karl, and V. A. Novikov, "From Maupertuis to Schrödinger. Quantization of classical variational principles," Am. J. Phys. 67, 959-961 (1999).
183. E. Schrödinger, "Quantisierung als eigenwert problem I," Ann. Phys. 79, 361-376 (1926),
184. translated in E. Schrödinger, Collected Papers on Wave Mechanics (Blackie, London, 1928),
185. Chelsea reprint 1982. For modern discussions and applications, see, for example, E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1998), p. 135;
186. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic, New York, 1974).