×

On relaxed elastic lines of the second type on an oriented surface in the Galilean space \(G_3\). (English) Zbl 1379.53005

Summary: In the present paper, the relaxed elastic line of the second type on an oriented surface in the Galilean space \(G_3\) is defined. For the relaxed elastic lines of the second type which are lying on a given oriented surface the Euler-Lagrange equations are derived. In particular, we investigate whether they can be geodesic or curvature lines. In the last section we present some examples to confirm our claim.

MSC:

53A04 Curves in Euclidean and related spaces
53B20 Local Riemannian geometry
05A15 Exact enumeration problems, generating functions
15A18 Eigenvalues, singular values, and eigenvectors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artykbaev, A.: Recovering convex surfaces from the extrinsic curvature in Galilean space. Math. USSR Sbornik. 47(1) (1984) · Zbl 0525.53017
[2] Bryant, R.L., Griffiths, P.: Reduction for constrained variational problems and \[\int \frac{\kappa^2}{2} ds\]∫κ22ds. Am. J. Math. 108, 525-570 (1986) · Zbl 0604.58022 · doi:10.2307/2374654
[3] Brunnett, G., Crouch, P.E.: Elastic curves on the sphere. Adv. Comput. Math. 2, 23-40 (1994) · Zbl 0830.65083 · doi:10.1007/BF02519034
[4] Divjak, B.: Special curves on ruled surfaces in Galilean and pseudo-Galilean space. Acta Math. Hung. 98, 203-215 (2003) · Zbl 1026.53004 · doi:10.1023/A:1022821824927
[5] Divjak, B., Sipus, M.: Minding isometries of ruled surfaces in pseudo-Galilean space. J. Geom. 77(1), 35-47 (2003) · Zbl 1119.53009 · doi:10.1007/s00022-003-1646-6
[6] Giering, O.: Vorlesungen über höhere Geometrie. Vieweg, Wiesbaden (1982) · Zbl 0493.51001 · doi:10.1007/978-3-322-83552-9
[7] Hilbert, D., Cohn-Vossen, S.: Geometry and the Imagination. AMS Chelsea Publishing, New York (1952) · Zbl 0047.38806
[8] Hsiung, C.C.: A First Course in Differential Geometry. Wiley, New York (1981) · Zbl 0458.53001
[9] Kamenarovic, I.: Existence theorems for ruled surfaces in the Galilean space \[G_3\] G3. Rad Hazu Math. 456(10), 183-196 (1991) · Zbl 0793.53013
[10] Landau, L.D., Lifshitz, E.M.: Theory of Elasticity. Pergamon Press, Oxford (1979) · Zbl 0178.28704
[11] Langer, J., Singer, D.: Total squared curvature of closed curves. J. Differ. Geom. 20, 1-22 (1984) · Zbl 0554.53013 · doi:10.4310/jdg/1214438990
[12] Langer, J., Singer, D.: Knotted elastic curves in \[{\mathbb{R}}^3\] R3. J. Lond. Math. Soc. 2, 512-520 (1984) · Zbl 0595.53001 · doi:10.1112/jlms/s2-30.3.512
[13] Manning, G.S.: Relaxed elastic line on an curved surfaces. Q. Appl. Math. XLV(3), 515-527 (1987) · Zbl 0633.73023 · doi:10.1090/qam/910458
[14] Nickerson, H.K., Manning, G.S.: Intrinsic equations for a relaxed elastic line on an oriented surface. Geom. Dedicata XLV(3), 515-527 (1988) · Zbl 0647.53002
[15] Palman, D.: Drehzykliden des galileischen Raumes \[G_3\] G3. Math. Pannonica 2(1), 95-104 (1990) · Zbl 0747.51013
[16] Pavković, B.J., Kamenarovic, I.: The equiform differential geometry of curves in the Galilean space \[G_3\] G3. Glasnik Matematicki 22(42), 449-457 (1987) · Zbl 0652.53009
[17] Roschel, O.: Die geometrie des galileischen Raumes. Habilitationsschrift, Leoben (1984) · Zbl 0587.51014
[18] Roschel, O.: Torusflächen des galileischen Raumes \[G_3\] G3. Stud. Sci. Math. Hung. 23, 401-410 (1988) · Zbl 0579.51016
[19] Sachs, H.: Isotrope Geometrie des Raumes. Vieweg, Braunschweig (1990) · Zbl 0703.51001 · doi:10.1007/978-3-322-83785-1
[20] Sipus, Ž.M.: Ruled Weingarten surfaces in the Galilean space. Period. Math. Hung. 56(2), 213-225 (2008) · Zbl 1212.53016 · doi:10.1007/s10998-008-6213-6
[21] Strubecker, K.: Über die Flächen von Monge und Serret im Isotropen Raum. Math. Z. 8, 155-179 (1963) · Zbl 0118.37802 · doi:10.1007/BF01111660
[22] Sarioğlu \[\dot{{\rm i}}\] i˙gl, A., Tutar, A., Stachel, H.: On relaxed elastic lines of second kind on a curved hypersurface in the n-dimensional Euclidean space. J. Geom. Gr. 18(1), 81-95 (2014) · Zbl 1307.49047
[23] Tutar, A., Sarioğlu \[\dot{{\rm i}}\] i˙gl, A.: Relaxed elastic lines of second kind oriented surface in Minkowskian space. Appl. Math. Mech. 27(11), 1481-1489 (2006) · Zbl 1197.53007
[24] Ünan, Z., Yilmaz, M.: Elastic lines of second kind on an oriented surface. Ondokuz Mayıs Üniversitesi Fen Dergisi 8(1), 1-10 (1997)
[25] Yaglom, I.M.: A Simple Non-Euclidean Geometry and its Physical Basis. Springer, New York (1979) · Zbl 0393.51013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.