Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler–Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values σ ≤ 1 at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and σ. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter λ, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of λ of interest. Here we present an alternative approach that takes advantage of the presence of a parameter λ. Rather than search for conjugate points σ ≤ 1 at a fixed value of λ, we search for a set of special parameter values λm (with corresponding Jacobi solution ζ ) for which σ = 1 is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any λ equals the number of these ζ for which 〈ζm,Sζm〉 < 0, where S is the Jacobi differential operator at λ. This computation is particularly simple when λ appears linearly in S. We apply this approach to the elastic strut, in which the force appears linearly in S, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE.