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1.R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1963), Vol. II, Chap. 19.
5.J. Hanc, S. Tuleja, and M. Hancova, “Simple derivation of Newtonian mechanics from the principle of least action,” Am. J. Phys.0002-9505 71, 386–391 (2003).
6.J. Hanc, S. Tuleja, and M. Hancova, “Symmetries and conservation laws: Consequences of Noether’s theorem,” Am. J. Phys.0002-9505 72, 428–435 (2004).
7.J. Hanc, E. F. Taylor, and S. Tuleja, “Deriving Lagrange’s equations using elementary calculus,” Am. J. Phys.0002-9505 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.0002-9505 72, 514–521 (2004).
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.0002-9505 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. The same erroneous statement occurs in work published in 1760–61, ibid, p. xxxiii.
14.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;
14.A. Sommerfeld, Mechanics (Academic, New York, 1952), p. 208;
14.P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, New York, 1953), Part 1, p. 281;
14.D. Park, Classical Dynamics and its Quantum Analogues, 2nd ed. (Springer, Berlin, 1990), p. 57;
14.L. H. Hand and J. D. Finch, Analytical Mechanics (Cambridge U. P., Cambridge, 1998), p. 53;
14.G. R. Fowles and G. L. Cassiday, Analytical Mechanics, 6th ed. (Saunders, Forth Worth, 1999), p. 393;
14.R. K. Cooper and C. Pellegrini, Modern Analytical Mechanics (Kluwer, New York, 1999), p. 34.
14.A similar error occurs in A. J. Hanson, Visualizing Quaternions (Elsevier, Amsterdam, 2006), p. 368, which states that geodesics on a sphere can have maximum length.
15.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.
16.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;
16.L. N. Hand and J. D. Finch, Ref. 14, p. 51;
16.J. D. Logan, Invariant Variational Principles (Academic, New York, 1977) mentions both “extremal” and “stationary,” p. 8;
16.L. D. Landau and E. M. Lifschitz, Mechanics, 3rd ed. (Butterworth Heineman, Oxford, 2003), pp. 2 and 3;
16.see also their Classical Theory of Fields, 4th ed. (Butterworth Heineman, Oxford, 1999), p. 25;
16.S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Thomson, Brooks/Cole, Belmont, CA, 2004), p. 231;
16.R. K. Cooper and C. Pellegrini, Ref. 14.
17.F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to Optics (Pearson Prentice Hall, Upper Saddle River, NJ, 2007), p. 22;
17.J. R. Taylor, Classical Mechanics (University Science Books, Sausalito, 2005), Problem 6.5;
17.P. J. Nahin, When Least is Best (Princeton U. P., Princeton, NJ, 2004), p. 133;
17.D. S. Lemons, Perfect Form (Princeton U. P., Princeton, NJ, 1997), p. 8;
17.D. Park, Ref. 14, p. 13;
17.F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw Hill, New York, 1976), p. 15;
17.R. Guenther, Modern Optics (Wiley, New York, 1990), p. 135;
17.R. W. Ditchburn, Light, 3rd ed. (Academic, London, 1976), p. 209;
17.R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Longmans, London, 1967), p. 7;
17.J. L. Synge, Geometrical Optics (Cambridge U. P., London, 1937), p. 3;
17.G. P. Sastry, “Problem on Fermat’s principle,” Am. J. Phys.0002-9505 49, 345 (1981);
17.M. V. Berry, review of The Optics of Rays, Wavefronts and Caustics by O. N. Stavroudis (Academic, New York, 1972),
17.Sci. Prog.0036-8504 61, 595–597 (1974);
17.V. Lakshminarayanan, A. K. Ghatak, and K. Thyagarajan, Lagrangian Optics (Kluwer, Boston, 2002), p. 16;
17.V. Perlick, Ray Optics, Fermat’s Principle, and Applications to General Relativity (Springer, Berlin, 2000), pp. 149 and 152.
18.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 to the Hamilton action. Culverwell’s work was preceded by that of Jacobi (Ref. 22) and Kelvin and Tait (Ref. 23).
19.H. H. Goldstine, A History of the Calculus of Variations From the 17th Through the 19th Century (Springer, New York, 1980).
20.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).
21.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).
22.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.
23.W. Thomson (Lord Kelvin) and P. G. Tait, Treatise on Natural Philosophy (Cambridge U. P., Cambridge, 1879, 1912), Part I;
23.reprinted as Principles of Mechanics and Dynamics (Dover, New York, 1962), Part I, p. 422.
24.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.
25.I. M. Gelfand and S. V. Fomin, Calculus of Variations, translated by R. A. Silverman (Prentice Hall, Englewood Cliffs, NJ, 1963), Russian edition 1961, reprinted (Dover, New York, 2000).
26.M. S. Hussein, J. G. Pereira, V. Stojanoff, and H. Takai, “The sufficient condition for an extremum in the classical action integral as an eigenvalue problem,” Am. J. Phys.0002-9505 48, 767–770 (1980).
26.Hussein et al. make the common error of assuming 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.0093-6413 10, 171–179 (1983).
27.M. Morse, The Calculus of Variations in the Large (American Mathematical Society, Providence, RI, 1934).
28.As we shall see, the nature of the stationary value of Hamilton’s action (and also Maupertuis’ action ) depends on the sign of second variations and (defined formally in Sec. IV), which in turn depends on the existence or absence of kinetic foci (see Secs. II and VII). The same quantities (signs of the second variations and kinetic foci) are also important in classical mechanics for the question of dynamical stability of trajectories (Refs. 23 and 29–31), and in semiclassical mechanics where they determine the phase loss term in the total phase of the semiclassical propagator due to a particular classical path (Refs. 32 and 33). The phase loss depends on the Morse (or Morse-Maslov) index, which equals the number of kinetic foci between the end-points of the trajectory (see Ref. 34). Further, in devising computational algorithms to find the stationary points of the action (either or ), it is useful to know whether we are seeking a minimum or a saddle point, because different algorithms (Ref. 35) are often used for the two cases. As we discuss in this paper, it is the sign of (or ) that determines which case we are considering. Practical applications of the mechanical focal points are mentioned at the end of Ref. 37.
29.E. J. Routh, A Treatise on the Stability of a Given State of Motion (Macmillan, London, 1877), p. 103;
29.reissued as Stability of Motion, edited by A. T. Fuller (Taylor and Francis, London, 1975).
30.J. G. Papastavridis, “Toward an extremum characterization of kinetic stability,” J. Sound Vib.0022-460X 87, 573–587 (1983).
31.J. G. Papastavridis, “The principle of least action as a Lagrange variational problem: Stationarity and extremality conditions,” Int. J. Eng. Sci.0020-7225 24, 1437–1443 (1986);
31.J. G. Papastavridis,“On a Lagrangean action based kinetic instability theorem of Kelvin and Tait,” Int. J. Eng. Sci.0020-7225 24, 1–17 (1986).
32.L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981), p. 143.
33.M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990), p. 184.
34.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,
34.and B. Van Brunt, The Calculus of Variations (Springer, New York, 2004), p. 254.
35.F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 1999), Chap. 14.
36.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.
36.The same example was treated earlier by C. G. J. Jacobi, Vorlesungen Über Dynamik (Braunschweig, Vieweg, 1884), reprinted (Chelsea, New York, 1969), p. 46.
37.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 geodesics 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.
38.M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), pp. 130 and 734;
39.M. Born and E. Wolf, Ref. 38, p. 738;
39.L. A. Artsimovich and S. Yu. Lukyanov, Motion of Charged Particles in Electric and Magnetic Fields (MIR, Moscow, 1980).
40.P. Grivet, Electron Optics, 2nd ed. (Pergamon, Oxford, 1972);
40.J. H. Moore, C. C. Davis, and M. A. Coplan, Building Scientific Apparatus, 3rd ed. (Perseus, Cambridge, MA, 2003), Chap. 5.
41.P. Grivet, Ref. 40, p. 822;
41.J. H. Moore et al., Ref. 40.
42.M. S. Livingston, The Development of High-Energy Accelerators (Dover, New York, 1966);
42.L. W. Alvarez, R. Smits, and G. Senecal, “Mechanical analogue of the synchrotron, illustrating phase stability and two-dimensional focusing,” Am. J. Phys.0002-9505 43, 292–296 (1975);
43.V. G. Boltyanskii, Envelopes (MacMillan, New York, 1964).
44.P. T. Saunders, An Introduction to Catastrophe Theory (Cambridge U. P., Cambridge, 1980), p. 62.
45.Systems with subsequent kinetic foci are discussed in Secs. 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.
47.This type of variation, , , where and vanish together for , 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.
48.H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, San Francisco, 2002), p. 44.
49.As discussed in Sec. VI, for a true worldline or , where is the (first) kinetic focus, we have for one special variation (and for all other variations) as well as for all variations. Further, we show that , etc., all vanish for the special infinitesimal variation for which vanishes, and that to second-order for larger such variations. In the latter case, typically is nonvanishing due to a single coalescing alternative worldline. In atypical (for one dimension (Ref. 58)) cases, more than one coalescing alternative worldline occurs, and the first nonvanishing term is or higher order—see Ref. 32, pp. 122–127. The harmonic oscillator is a limiting case where for all for the special variation around worldline , which reflects the infinite number of true worldlines which connect to , and which can all coalesce by varying the amplitude (see Fig. 3). In Morse theory (Refs. 27 and 34) the worldline is referred to as a degenerate critical (stationary) point.
51.This statement and the corresponding one in Ref. 52 must be qualified. It is in general not simply a matter of the time interval being short. The spatial path of the worldline must be sufficiently short. When, as usually happens, more than one actual worldline can connect a given position to a given position in the given time interval , for short time intervals only the spatially shortest worldline will have the minimum action. For example, the repulsive power-law potentials (including the limiting case of a hard-wall potential at the origin for ) and the repulsive exponential potential have been studied (Ref. 53). No matter how short the time interval , two different worldlines can connect given position to given position . The fact that two different true worldlines can connect the two points in the given time interval leads to a kinetic focus time occurring later than for the shorter of the worldlines and a (different) kinetic focus time occurring earlier than for the other worldline. For the first worldline is a minimum and for the other worldline is a saddle point. Another example is the quartic oscillator discussed in Sec. IX, where an infinite number of actual worldlines can connect given terminal events and , no matter how short the time interval . Only for the shortest of these worldlines is a minimum. The situation is different in 2D (see Appendix B).
52.E. T. Whittaker, Ref. 36, pp. 250–253. Whittaker deals with the Maupertuis action discussed in Appendix A, whereas we adapt his analysis to the Hamilton action . In more detail, our Eq. (1) corresponds to the last equation on p. 251 of Ref. 36, with , and .
53.L. I. Lolle, C. G. Gray, J. D. Poll, and A. G. Basile, “Improved short-time propagator for repulsive inverse power-law potentials,” Chem. Phys. Lett.0009-2614 177, 64–72 (1991). In Sec. X and Ref. 54 further analytical and numerical results are given for the inverse-square potential, . For given end positions and , there are two actual worldlines for given short times . There is one actual worldline for when the two worldlines have coalesced into one, and there is no actual worldline for longer times (remember and are fixed).
54.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.0021-9991 101, 80–93 (1992).
55.Better estimates can be found using the Sturm and Sturm-Liouville theories; see Papastavridis, Refs. 26 and 66.
56.This argument can be refined. In Sec. 8 we show that becomes for but is still larger in magnitude than the term, which is also . See Ref. 60 for and Eq. (40a) for .
57.There are other systems for which , etc., vanish, for example, , , . For these systems the worldlines cannot have [see Eq. (19)], so that kinetic foci do not exist.
58.In higher dimensions more than one independent variation, occurring in different directions in function space, can occur due to symmetry. In Morse theory the number of these independent variations satisfying 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 is equal to the number of kinetic foci between and , with each kinetic focus counted with its multiplicity.
59.J. M. T. Thompson and G. W. Hunt, Elastic Instability Phenomena (Wiley, Chichester, 1984), p. 20.
60.The fact that for is not surprising because is proportional to . The surprising fact is that here vanishes as as because the integral involved in the definition, Eq. (36a), of is itself . We can see directly that becomes for near for this special variation by integrating the term by parts in Eq. (36a) and using at the end-points. The result is . Because and are both true worldlines, we can apply the equation of motion to both. We then subtract these two equations of motion and expand as , giving . (If the , etc., nonlinear terms on the right-hand side are neglected in the last equation, it becomes the Jacobi-Poincaré linear variation equation used in stability studies.) If we use this result in the previous expression for , we find to lowest nonvanishing order , which is . This result and Eq. (40a) for give the desired result (42) for .
61.If we use arguments similar to those of this section and Sec. VI, we can show that vanishes again at the second kinetic focus , and that for beyond the wordline has a second, independent variation leading to , in agreement with Morse’s general theory (Ref. 34).
62.If we use , we can rewrite the Hamilton action as a phase-space integral, that is, . We set and vary and independently and find (Ref. 63) the Hamilton equations of motion , . We can then show (Ref. 64) that in phase space, the trajectories , that satisfy the Hamilton equations are always saddle points of , that is, never a true maximum or a true minimum. In the proof it is assumed that has the normal form .
64.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.
65.O. Bottema, “Beisipiele zum Hamiltonschen Prinzip,” Monatsh. Math.0026-9255 66, 97–104 (1962).
66.J. G. Papastavridis, “On the extremal properties of Hamilton’s action integral,” J. Appl. Mech.0021-8936 47, 955–956 (1980).
67.C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), p. 318. These authors use the Rayleigh-Ritz direct variational method (see Ref. 12 for a detailed discussion of this method) with a two-term trial trajectory , where and and are variational parameters, to study the half-cycle and one-cycle trajectories. Because the kinetic focus time for this oscillator, they find, in agreement with our results, that is a minimum for the half-cycle trajectory (with , ) and a saddle point for the one-cycle trajectory (with , ). However, in the figure accompanying their calculation, which shows the stationary points in space, they label the origin a maximum. The point represents the equilibrium trajectory . As we have seen, a true maximum in cannot occur, so that other “directions” in function space not considered by the authors must give minimum-type behavior of , leading to an overall saddle point.
68.The two-incline oscillator potential has the form , with and the angle of inclination. Here 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.
69.Other constant force or linear potential systems include the 1D Coulomb model (Ref. 70) , the bouncing ball (Ref. 71), and for the constant force spring (Ref. 72).
70.I. R. Lapidus, “One- and two-dimensional Hydrogen atoms,” Am. J. Phys.0002-9505 49, 807 (1981).
72.A. Capecelatro and L. Salzarulo, Quantitative Physics for Scientists and Engineers: Mechanics (Aurie Associates, Newark, NJ, 1977), p. 162;
72.C.-Y. Wang and L. T. Watson, “Theory of the constant force spring,” Trans. ASME, J. Appl. Mech.0021-8936 47, 956–958 (1980);
72.H. Helm, “Comment on ‘A constant force generator for the demonstration of Newton’s second law’,” Am. J. Phys.0002-9505 52, 268 (1984).
73.C. G. Gray, G. Karl, and V. A. Novikov, “Progress in classical and quantum variational principles,” Rep. Prog. Phys.0034-4885 67, 159–208 (2004).
74.Nearly pure quartic potentials have been found in molecular physics for ring-puckering vibrational modes (Ref. 75) and for the caged motion of the potassium ion in the endohedral fullerene complex (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).
75.R. P. Bell, “The occurrence and properties of molecular vibrations with ,” Proc. R. Soc. London, Ser. A1364-5021 183, 328–337 (1945);
75.J. Laane, “Origin of the ring-puckering potential energy function for four-membered rings and spiro compounds. A possibility of pseudorotation,” J. Phys. Chem.0022-3654 95, 9246–9249 (1991).
76.C. G. Joslin, J. Yang, C. G. Gray, S. Goldman, and J. D. Poll, “Infrared rotation and vibration-rotation bands of endohedral Fullerene complexes. .” Chem. Phys. Lett.0009-2614 211, 587–594 (1993).
77.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.
78.J. Thomchick and J. P. McKelvey, “Anharmonic vibrations of an ‘ideal’ Hooke’s law oscillator,” Am. J. Phys.0002-9505 46, 40–45 (1978).
79.R. Baierlein, Ref. 16, p. 73.
80.In two dimensions with and , our analytic condition (4) for the kinetic focus of worldline becomes , where denotes the determinant of matrix . The generalization to other dimensions is obvious. This condition (in slightly different form) is due to Mayer, Ref. 19, p. 269. For a clear discussion, see J. G. Papastavridis, Analytical Mechanics (Oxford U. P., Oxford, 2002), p. 1061.
80.For multidimensions a caustic becomes in general a surface in space-time. The analogous theory for multidimensional spatial caustics, relevant for the action , 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 to . 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.
81.A dynamics problem can be formulated as an initial value problem. For example, find from Newton’s equation of motion with initial conditions . It can also be formulated as a boundary value problem; for example, find from Hamilton’s principle with boundary conditions and . 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 particles, see A. Nusser and E. Branchini, “On the least action principle in cosmology,” Mon. Not. R. Astron. Soc.0035-8711 313, 587–595 (2000).
82.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.
83.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);
83.D. Greenspan, “Approximate solution of initial value problems for ordinary differential equations by boundary value techniques,” J. Math. Phys. Sci. 15, 261–274 (1967).
84.The converse effect cannot occur: a time-dependent potential with at all times always has as seen from Eq. (19). If is such that 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 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 focusing) in particle accelerators and storage rings is based on the same idea (Ref. 42).
85.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.0002-9505 50, 924–931 (1982).
86.P. K. Gosh, Ion Traps (Oxford U. P., Oxford, 1995), p. 7.
87.J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley, New York, 1950), p. 112.
87.Stoker’s statements on series convergence need amendment in light of the Kolmogorov-Arnold-Moser (KAM) theory (Ref. 89). See J. Moser, “Combination tones for Duffing’s equation,” Commun. Pure Appl. Math.0010-3640 18, 167–181 (1965);
87.T. Kapitaniak, J. Awrejcewicz and W.-H. Steeb, “Chaotic behaviour in an anharmonic oscillator with almost periodic excitation,” J. Phys. A0305-4470 20, L355–L358 (1987);
87.A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981), p. 216;
87.A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics (Wiley, New York, 1995), p. 234;
87.S. Wiggins, “Chaos in the quasiperiodically forced Duffing oscillator,” Phys. Lett. A0375-9601 124, 138–142 (1987).
88.G. Seifert, “On almost periodic solutions for undamped systems with almost periodic forcing,” Proc. Am. Math. Soc.0002-9939 31, 104–108 (1972);
88.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;
88.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;
88.M. S. Berger, “Two new approaches to large amplitude quasi-periodic motions of certain nonlinear Hamiltonian systems,” Contemp. Math.0271-4132 108, 11–18 (1990).
89.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.
90.See, for example, M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, New York, 1989), p. 35;
90.J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, 2nd ed. (Wiley, Chichester, 2002), pp. 310.
91.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;
91.N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, Redwood City, CA, 1992), p. 23;
91.A. B. Pippard, The Physics of Vibration (Cambridge U. P., Cambridge, 1978), Vol. 1, pp. 253–271.
92.The forced Duffing oscillator with is studied in Ref. 30. For the Duffing oscillator reduces to the quartic oscillator.
93.R. H. G. Helleman, “Variational solutions of non-integrable systems,” in Topics in Nonlinear Dynamics, edited by S. Jorna (AIP, New York, 1978), pp. 264–285. This author studies the forced Duffing oscillator with (note the sign change in compared to Ref. 92), and the Henon-Heiles oscillator with the potential in Eq. (83).
94.In Ref. 54 the harmonic potential with an oscillating equilibrium position is studied. The worldlines for this system are all nonchaotic.
95.M. Henon and C. Heiles, “The applicability of the third integral of the motion: Some numerical experiments,” Astron. J.0004-6256 69, 73–79 (1964).
96.There have been a few formal studies of action for chaotic systems, but few concrete examples seem to be available. See, for example, S. Bolotin, “Variational criteria for nonintegrability and chaos in Hamiltonian systems,” in Hamiltonian Mechanics, edited by J. Seimenis (Plenum, New York, 1994), pp. 173–179.
98.H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste (Gauthier-Villars, Paris, 1899), Vol. 3;
98.New Methods of Celestial Mechanics (AIP, New York, 1993), Part 3, p–958.
99.The situation is complicated because, as Eq. (A1) shows, there are two forms for , 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 . Space-time kinetic foci occur for the time-dependent form of , as for . Typically the kinetic foci for the two forms for differ from each other (Refs. 30 and 98) and from those for .
100.E. J. Routh, A Treatise on Dynamics of a Particle (Cambridge U. P., Cambridge, 1898), reprinted (Dover, New York, 1960), p. 400.
101.A. P. French, “The envelopes of some families of fixed-energy trajectories,” Am. J. Phys.0002-9505 61, 805–811 (1993);
101.E. I. Butikov, “Families of Keplerian orbits,” Eur. J. Phys.0143-0807 24, 175–183 (2003).
102.We assume that we are dealing with bound orbits. Similar comments apply to scattering orbits (hyperbolas). Just as for the orbits in the linear gravitational potential discussed in the preceding paragraph, here too there are restrictions and special cases (Ref. 19, p. 164; Ref. 103, p. 122). If the second point lies within the “ellipse of safety” (the envelope (French, Ref. 101)) of the elliptical trajectories of energy originating at , then two ellipses with energy can connect to . If lies on the ellipse of safety, then one ellipse of energy can connect to , and if lies outside the ellipse of safety, then no ellipse of energy can connect the two points. Usually the initial and final points and together with the center of force at (0,0) (one focus of the elliptical path) define the plane of the orbit. If , , and (0,0) lie on a straight line, the plane of the orbit is not uniquely defined, and there is almost always an infinite number of paths of energy in three dimensions that can connect to . A particular case of the latter is a periodic orbit where . Because the orbit can now be brought into coalescence with an alternative true orbit by a rotation around the line joining to (0,0), a third kinetic focus arises for elliptical periodic orbits in three dimensions (see Ref. 33, p. 29).
103.N. G. Chetaev, Theoretical Mechanics (Springer, Berlin, 1989).
104.For the repulsive potential, the hyperbolic spatial orbits have a (parabolic shaped) caustic/envelope (French, Ref. 101).
105.If the orbit equation has the explicit form , or the implicit form , the spatial kinetic focus is found from or , respectively. Here is the launch angle at (see Fig. 15 for an example). The derivation of these spatial kinetic focus conditions is similar to the derivation of the space-time kinetic focus condition of Eq. (4) (see Ref. 106, p. 59). In contrast, if the orbit equation is defined parametrically by the trajectory equations and , the spatial kinetic focus condition is . This Jacobian determinant condition is similar to that of Ref. 80 for the space-time kinetic focus (see Ref. 106, p. 73 for a derivation). As an example, consider a family of figure-eight-like harmonic oscillator orbits of Fig. 16, launched from the origin at time , all with speed (and therefore the same energy ), at various angles . The trajectory equations are and , where . The determinant condition for the spatial kinetic focus reduces to , which locates the kinetic focus (in time) for the orbit with launch angle . Elimination of and from these three equations leads to the locus of the (first) spatial kinetic foci, the spatial caustic/envelope equation , which is a parabolic shaped curve with two cusps on the -axis.
106.R. H. Fowler, The Elementary Differential Geometry of Plane Curves (Cambridge U. P., Cambridge, 1920).
107.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).
108.R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics (Dover, New York, 1971), p. 227;
108.H. Pollard, Celestial Mechanics (Mathematical Association of America, Washington, 1976), p. 28;
108.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).
109.R. H. Gooding, “A procedure for the solution of Lambert’s orbital boundary-value problem,” Celest. Mech. Dyn. Astron.0923-2958 48, 145–165 (1990).
110.It is clear from Fig. 13 that a kinetic focus occurs after time . To show rigorously that this focus is the first kinetic focus (unlike for where it is the second), we can use a result of Gordon (Ref. 111) that the action is a minimum for time . If one revolution corresponds to the second kinetic focus, the trajectory would correspond to a saddle point. The result can also be obtained algebraically by applying the general relation (4) to the relation for the radial distance, where we use angular momentum as the parameter labeling the various members of the family in Fig. 13. We obtain from . The latter equation implies that , because . At fixed energy (or fixed major axis ), the period is independent of for the attractive potential, so that the solution of occurs for , which is therefore the kinetic focus time .
111.W. B. Gordon, “A minimizing property of Keplerian orbits,” Am. J. Math.0002-9327 99, 961–971 (1977).
112.Note that for the actual 2D trajectories in the potential , kinetic foci exist for the spatial paths of the Maupertuis action , but do not exist for the space-time trajectories of the Hamilton action . This result illustrates the general result stated in Appendix A that the kinetic foci for and differ in general.
113.A. P. French, Vibrations and Waves (Norton, New York, 1966), p. 36.
114.J. C. Slater and N. H. Frank, Introduction to Theoretical Physics (McGraw Hill, New York, 1933), p. 85.
116.C. G. Gray, G. Karl, and V. A. Novikov, “From Maupertuis to Schrödinger. Quantization of classical variational principles,” Am. J. Phys.0002-9505 67, 959–961 (1999).
117.E. Schrödinger, “Quantisierung als eigenwert problem I,” Ann. Phys.0003-4916 79, 361–376 (1926),
117.translated in E. Schrödinger, Collected Papers on Wave Mechanics (Blackie, London, 1928), Chelsea reprint 1982.
117.For modern discussions and applications, see, for example, E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1998), p. 135;
117.S. T. Epstein, The Variation Method in Quantum Chemistry (Academic, New York, 1974).
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