Self-Bäcklund curves in centroaffine geometry and Lamé’s equation

Bialy, Misha; Bor, Gil; Tabachnikov, Serge

doi:10.1090/cams/9

Self-Bäcklund curves in centroaffine geometry and Lamé’s equation
HTML articles powered by AMS MathViewer

by Misha Bialy, Gil Bor and Serge Tabachnikov

Comm. Amer. Math. Soc. 2 (2022), 232-282

DOI: https://doi.org/10.1090/cams/9

Published electronically: August 24, 2022

HTML | PDF

Abstract:

Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves.

Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam’s problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics.

We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.

References

N. I. Akhiezer, Elements of the theory of elliptic functions, Translations of Mathematical Monographs, vol. 79, American Mathematical Society, Providence, RI, 1990. Translated from the second Russian edition by H. H. McFaden. MR 1054205, DOI 10.1090/mmono/079
M. Arnold, D. Fuchs, and S. Tabachnikov, A family of integrable transformations of centroaffine polygons: geometrical aspects, arXiv:2112.08124, 2021.
Tarik Aougab, Xidian Sun, Serge Tabachnikov, and Yuwen Wang, On curves and polygons with the equiangular chord property, Pacific J. Math. 274 (2015), no. 2, 305–324. MR 3332906, DOI 10.2140/pjm.2015.274.305
V. I. Arnol′d, Huygens and Barrow, Newton and Hooke, Birkhäuser Verlag, Basel, 1990. Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals; Translated from the Russian by Eric J. F. Primrose. MR 1078625, DOI 10.1007/978-3-0348-9129-5
V. Arnold, Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane, J. Geom. Phys. 53 (2005), no. 4, 421–427. MR 2125401, DOI 10.1016/j.geomphys.2004.07.008
Maxim Arnold, Dmitry Fuchs, Ivan Izmestiev, and Serge Tabachnikov, Cross-ratio dynamics on ideal polygons, Int. Math. Res. Not. IMRN 9 (2022), 6770–6853. MR 4411469, DOI 10.1093/imrn/rnaa289
H. Auerbach, Sur un problème de M. Ulam concernant l’èquilibre des corps flottants, Studia Math. 7 (1938), 121–142.
Misha Bialy, Andrey E. Mironov, and Lior Shalom, Magnetic billiards: non-integrability for strong magnetic field; Gutkin type examples, J. Geom. Phys. 154 (2020), 103716, 14. MR 4099906, DOI 10.1016/j.geomphys.2020.103716
Misha Bialy, Andrey E. Mironov, and Lior Shalom, Outer billiards with the dynamics of a standard shift on a finite number of invariant curves, Exp. Math. 30 (2021), no. 4, 469–474. MR 4346647, DOI 10.1080/10586458.2018.1563514
Misha Bialy, Andrey E. Mironov, and Serge Tabachnikov, Wire billiards, the first steps, Adv. Math. 368 (2020), 107154, 27. MR 4085879, DOI 10.1016/j.aim.2020.107154
Gil Bor, Mark Levi, Ron Perline, and Sergei Tabachnikov, Tire tracks and integrable curve evolution, Int. Math. Res. Not. IMRN 9 (2020), 2698–2768. MR 4095423, DOI 10.1093/imrn/rny087
G. Bor, Centroaffine curves, https://www.cimat.mx/~gil/centro-affine/, 2017.
J. Bracho, L. Montejano, and D. Oliveros, A classification theorem for Zindler carrousels, J. Dynam. Control Systems 7 (2001), no. 3, 367–384. MR 1848363, DOI 10.1023/A:1013099830164
J. Bracho, L. Montejano, and D. Oliveros, Carousels, Zindler curves and the floating body problem, Period. Math. Hungar. 49 (2004), no. 2, 9–23. MR 2106462, DOI 10.1007/s10998-004-0519-6
Annalisa Calini, Thomas Ivey, and Gloria Marí-Beffa, Remarks on KdV-type flows on star-shaped curves, Phys. D 238 (2009), no. 8, 788–797. MR 2522973, DOI 10.1016/j.physd.2009.01.007
Ildefonso Castro, Ildefonso Castro-Infantes, and Jesús Castro-Infantes, New plane curves with curvature depending on distance from the origin, Mediterr. J. Math. 14 (2017), no. 3, Paper No. 108, 19. MR 3633368, DOI 10.1007/s00009-017-0912-z
Robert Connelly and Balázs Csikós, Classification of first-order flexible regular bicycle polygons, Studia Sci. Math. Hungar. 46 (2009), no. 1, 37–46. MR 2656480, DOI 10.1556/SScMath.2008.1074
Van Cyr, A number theoretic question arising in the geometry of plane curves and in billiard dynamics, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3035–3040. MR 2917076, DOI 10.1090/S0002-9939-2012-11258-4
Philip J. Davis, Circulant matrices, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 543191
Serge Tabachnikov and Filiz Dogru, Dual billiards, Math. Intelligencer 27 (2005), no. 4, 18–25. MR 2183864, DOI 10.1007/BF02985854
P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322, DOI 10.1017/CBO9781139172059
M. S. P. Eastham, The spectral theory of periodic differential equations, Texts in Mathematics (Edinburgh), Scottish Academic Press, Edinburgh; Hafner Press, New York, 1973. MR 3075381
Robert Foote, Mark Levi, and Serge Tabachnikov, Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture, Amer. Math. Monthly 120 (2013), no. 3, 199–216. MR 3030293, DOI 10.4169/amer.math.monthly.120.03.199
A. P. Fordy, A historical introduction to solitons and Bäcklund transformations, Harmonic maps and integrable systems, Aspects Math., E23, Friedr. Vieweg, Braunschweig, 1994, pp. 7–28. MR 1264180, DOI 10.1007/978-3-663-14092-4_{2}
Atsushi Fujioka and Takashi Kurose, Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves, Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 1, 165–175. MR 2647776, DOI 10.1142/S0219887810003987
Atsushi Fujioka and Takashi Kurose, Multi-Hamiltonian structures on spaces of closed equicentroaffine plane curves associated to higher KdV flows, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 048, 11. MR 3210587, DOI 10.3842/SIGMA.2014.048
Eugene Gutkin, Capillary floating and the billiard ball problem, J. Math. Fluid Mech. 14 (2012), no. 2, 363–382. MR 2925114, DOI 10.1007/s00021-011-0071-0
H. Guggenheimer, Hill equations with coexisting periodic solutions, J. Differential Equations 5 (1969), 159–166. MR 239193, DOI 10.1016/0022-0396(69)90109-0
Eberhard Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 47–51. MR 23591, DOI 10.1073/pnas.34.2.47
Mark Levi and Serge Tabachnikov, On bicycle tire tracks geometry, hatchet planimeter, Menzin’s conjecture, and oscillation of unicycle tracks, Experiment. Math. 18 (2009), no. 2, 173–186. MR 2549686, DOI 10.1080/10586458.2009.10128894
Wilhelm Magnus and Stanley Winkler, Hill’s equation, Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0197830
Horst Martini and Konrad J. Swanepoel, Antinorms and Radon curves, Aequationes Math. 72 (2006), no. 1-2, 110–138. MR 2258811, DOI 10.1007/s00010-006-2825-y
Nozomu Matsuura, Discrete KdV and discrete modified KdV equations arising from motions of planar discrete curves, Int. Math. Res. Not. IMRN 8 (2012), 1681–1698. MR 2920827, DOI 10.1093/imrn/rnr080
R. Daniel Mauldin (ed.), The Scottish Book, 2nd ed., Birkhäuser/Springer, Cham, 2015. Mathematics from the Scottish Café with selected problems from the new Scottish Book; Including selected papers presented at the Scottish Book Conference held at North Texas University, Denton, TX, May 1979. MR 3242261, DOI 10.1007/978-3-319-22897-6
Sophie Morier-Genoud, Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc. 47 (2015), no. 6, 895–938. MR 3431573, DOI 10.1112/blms/bdv070
V. Ovsienko and S. Tabachnikov, Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and flattening of Legendrian curves, Selecta Math. (N.S.) 2 (1996), no. 2, 297–307. MR 1414890, DOI 10.1007/BF01587937
G. Pastras, Four lectures on Weierstrass elliptic function and applications in classical and quantum mechanics, arXiv:1706.07371.
Ulrich Pinkall, Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), no. 3-4, 328–332. MR 1331105, DOI 10.1007/BF03322836
P. Reinhardt and P. L. Walker, Weierstrass elliptic and modular functions, NIST Handbook Math. Functions, chap. 23, Cambridge Univ. Press, Cambridge, 2010, https://dlmf.nist.gov/23.
David A. Singer, Curves whose curvature depends on distance from the origin, Amer. Math. Monthly 106 (1999), no. 9, 835–841. MR 1732664, DOI 10.2307/2589616
Serge Tabachnikov, Tire track geometry: variations on a theme, Israel J. Math. 151 (2006), 1–28. MR 2214115, DOI 10.1007/BF02777353
Serge Tabachnikov, Variations on R. Schwartz’s inequality for the Schwarzian derivative, Discrete Comput. Geom. 46 (2011), no. 4, 724–742. MR 2846176, DOI 10.1007/s00454-011-9371-7
Serge Tabachnikov, On the bicycle transformation and the filament equation: results and conjectures, J. Geom. Phys. 115 (2017), 116–123. MR 3623617, DOI 10.1016/j.geomphys.2016.05.013
Serge Tabachnikov, On centro-affine curves and Bäcklund transformations of the KdV equation, Arnold Math. J. 4 (2018), no. 3-4, 445–458. MR 3949812, DOI 10.1007/s40598-019-00105-y
S. Tabachnikov and E. Tsukerman, On the discrete bicycle transformation, Publ. Mat. Urug. 14 (2013), 201–219. MR 3235356
Chuu-Lian Terng and Zhiwei Wu, Central affine curve flow on the plane, J. Fixed Point Theory Appl. 14 (2013), no. 2, 375–396. MR 3248564, DOI 10.1007/s11784-014-0161-8
F. Wegner, Floating bodies of equilibrium in 2D, the tire track problem and electrons in a parabolic magnetic field, arXiv:physics/0701241.
Franz Wegner, Floating bodies of equilibrium, Stud. Appl. Math. 111 (2003), no. 2, 167–183. MR 1990237, DOI 10.1111/1467-9590.t01-1-00231
F. Wegner, Three problems – one solution, http://www.tphys. uni-heidelberg.de/~wegner/Fl2mvs/Movies.html#float.
Konrad Von Zindler, Über konvexe Gebilde, Monatsh. Math. Phys. 31 (1921), no. 1, 25–56 (German). MR 1549095, DOI 10.1007/BF01702711