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Canonical map approach to channeling stability in crystals
1.D. S. Gemell, Rev. Mod. Phys. 46, 129 (1974).
2.C. Lehman, Interaction of Radiation and Elementary Defect Production (North‐Holland, Amsterdam, 1977).
3.J. Stark, Phys. Z. 13, 973 (1912).
4.M. T. Robinson and O. S. Oen, Phys. Rev. 132, 2385 (1963).
5.A summary of earlier results of the author for nonrelativistic channeling models in two and three space dimensions was given by A. W. Sáenz, Phys. Lett. A 93, 337 (1983). The following errata are present in that paper: (a) in p. 338, left column, third line of last paragraph, replace “spectra” by “space”;
5.(b) in p. 339, right column, replace second ∞ symbol in Eq. (10) by b; and (c) in p. 339, right column, second line after Eq. (10), replace “(11)” by “(10).”
6.All statements in this paper asserting the analyticity of some real‐ or complex‐valued function with respect to a certain set of real or complex variables should be understood as analyticity in all these variables jointly.
7.C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics (Springer, Berlin, 1971), pp. 127 and 128.
8.J. K. Moser, Nachr. Akad. Wiss. Göttingen, Math. Phys. K1. II, 1 (1962).
9.H. Rüssmann, Nachr. Akad. Wiss. Gottingen, Math. Phys. K1. II, 67 (1970).
10.S. Sternberg, Celestial Mechanics, Part II (Benjamin, New York, 1969), Chap. Ill, Sec. 11, and bibliography.
11.J. K. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Ann. Math. Studies No. 77 (Princeton U.P., Princeton, NJ, 1973), Chap. II, Sec. 4a, and bibliography.
12.R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin/Cummings, Reading, MA, 1978), Sec. 8.3 and bibliography.
13.Important mathematical work on invariant curves of twist maps has been done recently by S. Aubry, M. R. Herman, A. Katok, J. N. Mather, H. Rüssmann, and others. See, e.g., the review articles by A. Katok, “Dynamical Systems and Chaos,” in Lecture Notes in Physics, Vol. 179, edited by L. Garrido (Springer, Berlin, 1983), p. 47;
13.and the review article by S. Aubry, Physica D 7, 240 (1983).
13.See also the bibliography in R. S. Mackay, Physica D 7, 283 (1983).
14.A. Nagl (Department of Physics, Catholic University, Washington, D.C.) and the present author are making a numerical surface‐of‐section study of regular and chaotic motions of fast particles in two‐dimensional crystals, in the context of a special case of the NR model considered in this paper.
15.In January 1983, J. Ellison (Department of Mathematics and Statistics, University of New Mexico, Albuquerque) informed me of unpublished work, done collaboratively with J. Su and C. Seal, on regular and chaotic motions occurring for a continuum model of axial channeling. This appears to be the earliest numerical investigation of such motions by the surface‐of‐section method.
16.A version of this generalized theorem is announced in my paper “Rigorous Results on Channeling Stability in Crystals via Canonical Maps,” in Proceedings of the Thirteenth International Colloquium on Group Theoretical Methods in Physics, edited by W. W. Zachary (World Scientific, Singapore, 1985) (in press).
17.The canonical maps referred to in this paper are obvious two‐dimensional local versions of the corresponding maps defined, e.g., in V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978), p. 239. Such canonical maps in the plane preserve oriented areas.
18.See, e.g., W. Klingenberg, Lectures on Closed Geodesies (Springer, Berlin, 1977), Lemma 3.3.2, pp. 101, 102. Needless to say, the critical comments of R. Bott, Bull. Am. Math. Soc. 7, 331 (1982) (see especially p. 347) about this book do not refer to portions of the book mentioned in this paper.
19.P. Swinnerton‐Dyer, Proc. London Math. Soc. 34, 385 (1977).
20.For the general theory, see, e.g., E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Dover, New York, 1944), 4th ed., Chap. XII, Sec. 141, and A. Wintner, The Analytical Foundations of Celestial Mechanics (Princeton U.P., Princeton, NJ, 1941), especially Sees. 180‐182.
21.See, e.g., V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968), Theorem A 31.2, p. 231. Notice that the discussion in Siegel and Moser,7 Sec. 22, ending in the sentence containing their Eq. (5), proves that the Poincaré map associated with a periodic orbit of a sufficiently smooth Hamiltonian system of two degrees of freedom is area preserving, but not that it is orientation preserving, and hence canonical in the present sense.
22.See, e.g., Siegel and Moser,7 Sec. 23 and Klingenberg,18 Sec. 3.3, pp. 100–103.
23.R. C. Churchill, M. Kummer, and D. L. Rod, J. Diff. Eq. 49, 359 (1983), Sec. 5, and especially Theorem 5.3.
24.See, e.g., Klingenberg,18 Theorem 3.3.A.1, p. 116.
25.Siegel and Moser,7 Sees. 32, 33.
26.Siegel and Moser,7 Sec. 32.
27.Siegel and Moser,7 pp. 243, 248.
28.See, e.g., Siegel and Moser,7 p. 245. See also Theorem 12.2, p. 125 of Part II of Sternberg’s celestial mechanics notes.10
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