Morse theory for elastica

Philip  Schrader

doi:10.3934/jgm.2016006

Article Contents

2016, Volume 8, Issue 2: 235-256. Doi: 10.3934/jgm.2016006

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Morse theory for elastica

Philip Schrader^1,

1.
The University of Western Australia, 35 Stirling Highway, Crawley WA 6009

Received: November 30, 2013

Revised: February 29, 2016

Published: May 2016

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Abstract

In Riemannian manifolds the elastica are critical points of the restriction of total squared geodesic curvature to curves with fixed length which satisfy first order boundary conditions. We verify that the Palais-Smale condition holds for this variational problem, and also the related problems where the admissible curves are required to satisfy zeroth order boundary conditions, or first order periodicity conditions. We also prove a Morse index theorem for elastica and use the Morse inequalities to give lower bounds for the number of elastica of each index in terms of the Betti numbers of the path space.

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Mathematics Subject Classification: Primary: 58E05, 58E50; Secondary: 49J40.

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