Stability results for constrained calculus of variations problems: an analysis of the twisted elastic loop

Problems with a variational structure are ubiquitous throughout the physical sciences and have a distinguished scientific history. Constrained variational problems have been much less studied, particularly the theory of stability, which determines which solutions are physically realizable. In this p...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2005-05, Vol.461 (2057), p.1357-1381
Main Author: Hoffman, Kathleen A
Format: Article
Language:English
Subjects:
Bending
Boundary conditions
Calculus of variations
Constrained Variational Principles
Critical points
Distinguished Diagrams
Eigenvalues
Eigenvectors
Elastic Loop
Euler Lagrange equation
Lagrange multipliers
Pitchforks
Stability
Stiffness
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Summary:Problems with a variational structure are ubiquitous throughout the physical sciences and have a distinguished scientific history. Constrained variational problems have been much less studied, particularly the theory of stability, which determines which solutions are physically realizable. In this paper, we develop stability exchange results appropriate for parameter-dependent calculus of variations problems with two particular features: either the parameter appears in the boundary conditions, or there are isoperimetric constraints. In particular, we identify an associated distinguished bifurcation diagram, which encodes the direction of stability exchange at folds. We apply the theory to a twisted elastic loop, which can naturally be formulated as a calculus of variations problem with both isoperimetric constraints and parameter-dependent boundary conditions. In combination with a perturbation expansion that classifies certain pitchfork bifurcations as sub- or super-critical, the distinguished diagram for the twisted loop provides a classification of the stability properties of all equilibria. In particular, an unanticipated sensitive dependence of stability properties on the ratio of twisting to bending stiffness is revealed.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2004.1435