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Motion of elastic wire with thickness

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  • Physics


Koiso, N. and Sugimoto, M. Osaka J. Math. 47 (2010), 787–815 MOTION OF ELASTIC WIRE WITH THICKNESS NORIHITO KOISO and MITSURU SUGIMOTO (Received January 6, 2009, revised April 3, 2009) Abstract There are several types of equation of motion of elastic wires. In this paper, we treat an equation taking account of the thickness of wire. The equation was in- troduced by Caflisch and Maddocks on plane curves, and they proved the existence of solutions. We will prove the existence of solutions for any dimensional Euclidean space. Note that, in the case of plane, the equation can be explicitly written in terms of polar coordinates. For higher dimensional case, we use covariant differentiation on the unit sphere. 1. Introduction and results In the previous work [9], the first author considered the motion of a fixed length elas- tic wire, governed by the elastic energy. Let D (x) be a curve in the N -dimensional Euclidean space RN with arc length parameter x 2 [0, 1], i.e., j 0(x)j � 1. We denote its motion by D (x , t). We denote by ( � j � ) (resp. j � j, h � j � i, k � k) the pointwise inner product (resp. pointwise norm, the L2-inner product with respect to the variable x , the L2-norm with respect to the variable x). The potential energy of the elastic wire (x) is defined by the square integral U D k xxk2 of the curvature. Assuming that the wire is infinitely thin, the kinetic energy of the motion of the wire is defined by E D k tk2. By Hamilton’s principle, the equation of motion is given as critical points of the variational problem defined by the functional (1.1) F D ∫ T 0 E �U dt D ∫ T 0 k tk 2 � k xxk 2 dt . The equation turns out to be a coupled system of semi-linear 1-dimensional plate equa- tion, where the derivatives of unknown functions up to fourth order are involved. In [9] and also in A. Burchard and L.E. Thomas [2], the existence of a unique short-time solution satisfying some initial data was proved. The former used a per- turbation to a compos

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