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

Abstract

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.

New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

“…Thus, it is also an important question to determine the optimal value γ in these types of variational problems. Clearly, (x , ζ ) satisfies ( 27), (28), and (30), proving that the pair (x , ζ ) is a candidate to be a solution of the problem. Since L is jointly convex, we can conclude by Theorem 7 that (x , ζ ) is a solution of the proposed problem.…”

“…The first works in this scientific area are due to Riewe [3,19]. Since then, many papers were published on different topics of the fractional calculus of variations for different types of fractional operators (see [2,[20][21][22][23][24][25][26][27][28][29] and the references therein). For more details, we recommend the works [30][31][32].…”

General quantum variational calculus

“…The motivation to study generalized variational problems where the Lagrangian depends on the state values can be found in economics problems ( [24]); the dependence of a real parameter is important, for example, in physical problems ( [15]). …”

A non-standard class of variational problems of Herglotz type

“…Remark 1. Obviously, in equation (11) we can omit λ(a), because λ(a) = 1. However, we choose to include λ(a) to establish a relationship between the natural boundary conditions (11) and (12).…”

“…The motivation to study non-standard classical variational problems where the Lagrangian depends on the final state value can be found in economics problems ( [3], [24]); for the importance of the Lagrangian dependence of a real parameter we refer to physical problems ( [11]).…”