Surface equivalent of catenary curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:30:55Z http://mathoverflow.net/feeds/question/69817 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69817/surface-equivalent-of-catenary-curve Surface equivalent of catenary curve Joseph O'Rourke 2011-07-08T19:04:38Z 2011-07-09T12:53:57Z <p>A <a href="http://en.wikipedia.org/wiki/Catenary" rel="nofollow">catenary curve</a> is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is:</p> <blockquote> <p>What is the shape taken by an idealized, thin two-dimensional sheet, pinned on a plane parallel to the ground, under the influence of gravity?</p> </blockquote> <p>The answer surely depends on how it is pinned to the plane, the boundary conditions. Natural options are:</p> <ul> <li>A disk sheet fixed to a circle.</li> <li>A square sheet fixed to a square.</li> <li>A square sheet pinned at its four corners.</li> </ul> <p>The middle option above would look something like this when inverted: <br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://cs.smith.edu/~orourke/MathOverflow/CatenaryDome.jpg" alt="CatDome" /> <sup> (Image by Tim Tyler at <a href="http://hexdome.com/essays/catenary_domes/gallery/index.html" rel="nofollow">hexdome.com</a>.) </sup></p> <p>I don't think any of these shapes is a <a href="http://en.wikipedia.org/wiki/Catenoid" rel="nofollow">catenoid</a>, which is the surface of revolution formed by a catenary curve. Is there a simple analytic description of any of these surfaces, analogous to the $\cosh$ equation for the catenary curve? I have been unsuccessful in finding anything but simulations of solutions of the differential equations.</p> <p>This question arose in imagining a higher-dimensional version of the property that an inverted catenary supports smooth rides of a square-wheeled bicycle (explored in <a href="http://mathoverflow.net/questions/29988/" rel="nofollow">this MO</a> question). Thanks for pointers!</p> http://mathoverflow.net/questions/69817/surface-equivalent-of-catenary-curve/69818#69818 Answer by Will Jagy for Surface equivalent of catenary curve Will Jagy 2011-07-08T19:12:11Z 2011-07-08T20:08:20Z <p>The thing that comes to mind is the capillary surface including gravity. See the note by Finn, available free as a pdf, as a reference at the end of:</p> <p><a href="http://en.wikipedia.org/wiki/Capillary_surface" rel="nofollow">http://en.wikipedia.org/wiki/Capillary_surface</a> </p> <p>Hmmm, <strong>maybe not</strong>. Your surface would not have a large flat region in the middle...</p> <p>A rotated catenary surface is quite simply not isometric to a flat disc. So we might, for instance, be asking about a rubber sheet, glued down along a boundary, and allowed to sag in the middle under gravity. The elastic energy less resembles the mean curvature operator in favor of the ordinary Laplacian </p> <p><a href="http://en.wikipedia.org/wiki/Elastic_energy" rel="nofollow">http://en.wikipedia.org/wiki/Elastic_energy</a> </p> <p>It appears you are looking for the biharmonic equation, as the force of gravity vector field will be considered constant and divergence free, so the displacement $u$ satisfies $\Delta^2 u = 0.$ See </p> <p><a href="http://en.wikipedia.org/wiki/Linear_elasticity#Elastostatics" rel="nofollow">http://en.wikipedia.org/wiki/Linear_elasticity#Elastostatics</a> </p> http://mathoverflow.net/questions/69817/surface-equivalent-of-catenary-curve/69837#69837 Answer by Andrey Rekalo for Surface equivalent of catenary curve Andrey Rekalo 2011-07-09T00:05:00Z 2011-07-09T12:44:53Z <p>A model equation for an inextensible, ﬂexible, heavy surface in a gravitational ﬁeld was deduced by <s>Poisson</s> Lagrange and later the problem was also studied by Poisson (see the references in the linked papers below). The equilibrium condition for a hanging heavy surface of constant mass density reads $$\sqrt{1+|\nabla u|^2}\ \nabla\cdot{}\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=\frac{1}{u+\lambda},\qquad x\in\Omega\subset\mathbb R^2,\qquad\qquad(1)$$ where $u=u(x)$ is the vertical displacement and $\lambda\in\mathbb R$ is an arbitrary constant (a Lagrange multiplier). (1) is the Euler equation of the variational integral $$I(u)=\int_{\Omega}u\sqrt{1+|\nabla u|^2}dx,$$ which can be interpreted as the vertical coordinate of the center of gravity of the surface $$\mbox{graph}(u)=\{(x,u(x)):\ x\in\Omega\}\subset\mathbb R^2\times\mathbb R.$$ </p> <p>Equation (1) is to be supplemented with the requirement that the surface has a prescribed area $A$ $$\qquad\qquad\qquad\qquad\qquad\int_{\Omega}\sqrt{1+|\nabla u|^2}dx=A,\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad(2)$$ and the Dirichlet boundary condition describing the curve from which the surface is being suspended $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\left.u\right|_{\partial \Omega}=g.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad(3)$$ One can check formally that a solution to (1)-(3) provides a graph of a heavy surface of prescribed area and boundary with the lowest center of gravity, so this is a precise 2D analogue of the classical catenary problem. </p> <p>It is known that problem (1)-(3) has no classical solutions for the values of area $A$ outside of some bounded interval $[A_{\min},A_{\max}]$. Moreover, the corresponding variational problem has no global solutions for all $A\in\mathbb R$. A short survey of some old and relatively new results concerning well-posedness of (1)-(3) and its multidimensional analogues can be found in the paper by Dierkes and Huisken, "The N-dimensional analogue of the catenary: Prescribed area", in J. Jost (ed) <em>Calculus of Variations and Geometric Analysis</em>, Int. Press (1996), pp. 1-13.</p> <p><strong>Addendum.</strong> Here is a more recent survey by Dierkes: <a href="http://books.google.com/books?hl=en&amp;lr=&amp;id=Q6K-FIsH6RQC&amp;oi=fnd&amp;pg=PA177&amp;ots=9DdAcwoC_Q&amp;sig=cW8HIdpH7fT09i2SskuHl3S64SE#v=onepage&amp;q&amp;f=false" rel="nofollow">"Singular Minimal Surfaces"</a> (in <em>Geometric Analysis and Nonlinear Partial Differential Equations</em>, Springer (2003), pp. 177-194). </p>