Topological Ideas and Fluid Mechanics

DEC 01, 1996

New mathematical techniques and greater computational power have made it possible to apply knot theory and braid theory to fluid flows.

DOI: 10.1063/1.881574

Renzo L. Ricca

Mitchell A. Berger

The use of topological ideas in physics and fluid mechanics dates back to the very origin of topology as an independent science. In a brief note in 1833 Karl Gauss, while lamenting the lack of progress in the “geometry of position” (or Geometria Situs, as topology was then known I, gives a remarkable example of the relationship between topology and measurable physical quantities such as electric currents. He considers two inseparably linked circuits, each of them a copper wire with ends joined, and flowing electric current. Without comment he puts forward a formula that gives the relationship between the magnetic action induced by the currents and a pure number that depends only on the type of link, and not on the geometry. This number is a topological invariant now known as the linking number. The formula, as well as the very first studies in topology done by Johann Benedict Listing in 1847, became known to Kelvin (then William Thomson), James Clerk Maxwell and Peter Guthrie Tait in Britain.

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More about the authors

Renzo L. Ricca, University College, London.

Mitchell A. Berger, University College, London.