dorsal/arxiv
View SchemaA note on the motion of surfaces
| Authors | Robert I. McLachlan, Harvey Segur |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9306003 |
| URL | https://arxiv.org/abs/solv-int/9306003 |
| DOI | 10.1016/0375-9601(94)91278-5 |
Abstract
We study the motion of surfaces in an intrinsic formulation in which the surface is described by its metric and curvature tensors. The evolution equations for the six quantities contained in these tensors are reduced in number in two cases: (i) for arbitrary surfaces, we use principal coordinates to obtain two equations for the two principal curvatures, highlighting the similarity with the equations of motion of a plane curve; and (ii) for surfaces with spatially constant negative curvature, we use parameterization by Tchebyshev nets to reduce to a single evolution equation. We also obtain necessary and sufficient conditions for a surface to maintain spatially constant negative curvature as it moves. One choice for the surface's normal motion leads to the modified-Korteweg de Vries equation,the appearance of which is explained by connections to the AKNS hierarchy and the motion of space curves.
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"abstract": "We study the motion of surfaces in an intrinsic formulation in which the\nsurface is described by its metric and curvature tensors. The evolution\nequations for the six quantities contained in these tensors are reduced in\nnumber in two cases: (i) for arbitrary surfaces, we use principal coordinates\nto obtain two equations for the two principal curvatures, highlighting the\nsimilarity with the equations of motion of a plane curve; and (ii) for surfaces\nwith spatially constant negative curvature, we use parameterization by\nTchebyshev nets to reduce to a single evolution equation. We also obtain\nnecessary and sufficient conditions for a surface to maintain spatially\nconstant negative curvature as it moves. One choice for the surface\u0027s normal\nmotion leads to the modified-Korteweg de Vries equation,the appearance of which\nis explained by connections to the AKNS hierarchy and the motion of space\ncurves.",
"arxiv_id": "solv-int/9306003",
"authors": [
"Robert I. McLachlan",
"Harvey Segur"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1016/0375-9601(94)91278-5",
"title": "A note on the motion of surfaces",
"url": "https://arxiv.org/abs/solv-int/9306003"
},
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