Geodesic
Curvature functionals for curves in the equi-affine plane
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 419-435
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After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Notice. An extended version of this article is available on arXiv:0912.4075.
After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Notice. An extended version of this article is available on arXiv:0912.4075.
DOI : 10.1007/s10587-011-0064-4
Classification : 49K05, 49K15, 49Q05, 53A15
Keywords: curvature functionals; variational problems; affine curves 
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Verpoort, Steven. Curvature functionals for curves in the equi-affine plane. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 419-435. doi: 10.1007/s10587-011-0064-4

[1] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Tenth Printing). National Bureau of Standards, Washington (1972). | MR

[2] Arreaga, G., Capovilla, R., Chryssomalakos, C., Guven, J.: Area-Constrained Planar Elastica. Physical Review E 65 (2002), 14 pages. | DOI

[3] Blaschke, W.: Vorlesungen über Differentialgeometrie. I: Elementare Differentialgeometrie. Springer, Berlin (1923).

[4] Blaschke, W.: Vorlesungen über Differentialgeometrie. II: Affine Differentialgeometrie. Springer, Berlin (1923).

[5] Blaschke, W.: Vorlesungen über Differentialgeometrie. III: Differentialgeometrie der Kreise und Kugeln. Springer, Berlin (1929).

[6] Blaschke, W.: Ebene Kinematik (Eine Vorlesung). Teubner, Leipzig (1938).

[7] Bol, G.: Beantwoording van Prijsvraag 13, 1934. Nieuw Archief voor Wiskunde 20 (1940), 113-162. | MR

[8] Bottema, O.: Vlakke Krommen met de Affiene Natuurlijke Vergelijking $k = c\wp(s)$. Nieuw Archief voor Wiskunde 21 (1941), 89-100. | MR

[9] Cartan, É.: Sur un Problème du Calcul des Variations en Géométrie Projective Plane. Matematicheskii Sbornik 34 (1927), 349-364.

[10] Gălugăreanu, G., Gheorghiu, G. T.: Sur l'Interprétation Géométrique des Invariants Différentiels Fondamentaux en Géométrie Affine et Projective des Courbes Planes. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie 43 (1941), 69-83. | MR

[11] Huang, Rongpei: A Note on the Generalized Subaffine Elastica in $\mathbb{R}^2$. Chinese Quarterly Journal of Mathematics. Shuxue Jikan 18 (2003), 88-92. | MR

[12] Heil, E.: Abschätzungen für einige Affininvarianten konvexer Kurven. Monatshefte für Mathematik {71} (1967), 405-423. | MR | Zbl

[13] Leichtweiss, K.: The Affine Geometry of Convex Bodies. Johann Ambrosius Barth, Heidelberg (1998). | MR

[14] Li, An-Min: Variational Formulae for Higher Affine Mean Curvatures. Results in Mathematics 13 (1988), 318-326. | DOI | MR

[15] Li, An-Min, Simon, U., Zhao, Guo-Song: Global Affine Differential Geometry of Hypersurfaces. De Gruyter Expositions in Mathematics 11. Walter de Gruyter, Berlin (1993). | MR

[16] Mihăilescu, T.: Géométrie Différentielle Affine des Courbes Planes. Czech. Math. J. 9 (1959), 84 265-288. | MR

[17] Mihăilescu, T.: Sobre la Variación del Arco Afín de las Curvas Planas. Mathematicae Notae 17 (1959/1961), 59-81. | MR

[18] Schirokow, A. P., Schirokow, P. A.: Affine Differentialgeometrie. B.G. Teubner, Leipzig (1962). | MR | Zbl

[19] Tutaev, L. K.: Lines and Surfaces in Three-Dimensional Affine Space. Israel Program for Scientific Translations, Jerusalem (1964). | MR

[20] Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis: an Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Third Edition, Cambridge University Press (1920). | MR

[21] Verpoort, S.: Curvature Functionals for Curves in the Equi-Affine Plane (an extended version of the current article). arXiv:0912.4075. | MR

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