On the Cauchy Problem for a Dynamical Euler's Elastica

Burchard, Almut; Thomas, Lawrence E.

doi:10.1081/pde-120019382

Cited by 17 publications

(12 citation statements)

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“…Burchard and Thomas [5] obtained a local well-posedness result for the related problem of inextensible elastica, in which there is a potential energy term reflecting a resistance to bending; however it is not clear whether the solutions are preserved in the limit as the potential term goes to zero, so this result does not help in the present situation.…”

Section: Introductionmentioning

confidence: 86%

The motion of whips and chains

Preston

2011

Journal of Differential Equations

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We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations $$ \eta_{tt} = \partial_s(\sigma \eta_s), \qquad \sigma_{ss}-\lvert \eta_{ss}\rvert^2 = -\lvert \eta_{st}\rvert^2, \qquad \lvert \eta_s\rvert^2 \equiv 1 $$ with boundary conditions $\eta(t,1)=0$ and $\sigma(t,0)=0$. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy $$ \sum_{\ell=0}^m \int_0^1 s^{\ell} \lvert \partial_s^{\ell}\eta_t\rvert^2 \, ds + \int_0^1 s^{\ell+1} \lvert \partial_s^{\ell+1}\eta\rvert^2 \, ds, $$ when $m\ge 3$. In addition we show persistence of smooth solutions as long as the energy for $m=3$ remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.Comment: 47 pages, 8 figure

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Section: Introductionmentioning

confidence: 86%

The motion of whips and chains

Preston

2011

Journal of Differential Equations

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“…Similar multipliers can be found in Burchard and Thomas[14], Singer[15], Tornberg and Shelley[16], who do not employ a variational derivation, and Guven and Vázquez-Montejo[17], who use multiple redundant multipliers. The multipliers in[14] and[16] are misidentified as the tension. For seemingly similar but qualitatively different multipliers, see the following footnote.…”

mentioning

confidence: 60%

On the Planar Elastica, Stress, and Material Stress

Singh

Hanna

2018

J Elast

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We revisit the classical problem of the planar Euler elastica with applied forces and moments, and present a classification of the shapes in terms of tangentially conserved quantities associated with spatial and material symmetries. We compare commonly used director, variational, and dynamical systems representations, and present several illustrative physical examples. We remark that an approach that employs only the shape equation for the tangential angle obscures physical information about the tension in the body. * harmeet@vt.edu † hannaj@vt.edu 1 Material forces and closely related quantities, which in our one-dimensional system have just one component, appear under many names in the literature, including Eshelbian force, quasimomentum, pseudomomentum, (Kelvin) impulse, and configurational force [3][4][5][6][7][8][9][10][11][12]. In the present case, one can derive everything from consideration of conventional force balance and its projection onto the tangents of the body, but the concept of material force is useful as a descriptive term, and also corresponds to an important symmetry of the Lagrangian. We generally prefer the term "spatial" to "conventional", but try to avoid it in this note because of its alternative meaning as "non-planar" in the context of rods embedded in three dimensions rather than two.arXiv:1706.03047v4 [physics.class-ph]

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“…The next problem we will consider is a conjectured isoperimetric inequality for closed, smooth curves in the plane. It has attracted considerable attention in the literature during the last decade (see, e.g., [60,77,63,75,40]), and it has many interesting connections in geometry and physics. In particular, Benguria and Loss [31] have shown a connection between this problem and a special case of the Lieb-Thirring inequalities [120,117], inequalities which play a fundamental role in Lieb and Thirring's proof of the stability of matter (see, in particular, [119] and the review article [115]).…”

Section: An Isoperimetric Inequality For Ovals In the Planementioning

confidence: 99%