Isoperimetric conjugate points with application to the stability of DNA minicircles

Manning, Robert S.; Rogers, Kathleen A.; Maddocks, John H.

doi:10.1098/rspa.1998.0291

Cited by 61 publications

(52 citation statements)

References 19 publications

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“…3. Furthermore, one of the necessary conditions for optimality is identical to the Riccati equation (17) associated with the condition L1 discussed in Sect. 3.1.…”

Section: Optimal Control Formulationmentioning

confidence: 99%

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

O’Reilly

Peters

2011

J Elast

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It is common practice in analyses of the configurations of an elastica to use Jacobi's necessary condition to establish conditions for stability. Analyses of this type date to Born's seminal work on the elastica in 1906 and continue to the present day. Legendre developed a treatment of the second variation which predates Jacobi's. The purpose of this paper is to explore Legendre's treatment with the aid of three classical buckling problems for elastic struts. Central to this treatment is the issue of existence of solutions to a Riccati differential equation. We present two different variational formulations for the buckling problems, both of which lead to the same Riccati equation, and we demonstrate that the conclusions from Legendre and Jacobi's treatments are equivalent for some sets of boundary conditions. In addition, the failure of both treatments to classify stable configurations of a free-free strut are contrasted.

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“…3. Furthermore, one of the necessary conditions for optimality is identical to the Riccati equation (17) associated with the condition L1 discussed in Sect. 3.1.…”

Section: Optimal Control Formulationmentioning

confidence: 99%

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

O’Reilly

Peters

2011

J Elast

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“…Further, the arguments in [19], involving a proof of the monotonicity of the eigenvalues of QSQ as a function of σ, demonstrate the analogue to Jacobi's strengthened condition: the lack of an isoperimetric conjugate point implies the existence of a local minimum. In addition, similar to Morse's theory in the unconstrained case, the number of isoperimetric conjugate points equals the maximal dimension of a subspace of H cons d on which the second variation is negative.…”

Section: Constrained Conjugate Pointsmentioning

confidence: 99%

“…In addition, we compute sheets of buckled solutions using a family of one-dimensional branches generated by the parameter continuation package AUTO [5,6]. We determine the index of configurations on these sheets via a numerical implementation of the conjugate point test developed in [19] since a closed-form solution of the differential equations is not feasible in this case.…”

Section: An Elastic Strut Clamped At Each End With a Relative Twismentioning

confidence: 99%

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

Hoffman¹,

Manning²,

Paffenroth³

2002

SIAM J. Appl. Dyn. Syst.

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Abstract. 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 m ) 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 m for which ζ m , Sζ m < 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.

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“…It has been applied to model the configuration and the stability of super-helically constrained DNA [31][32][33][34][35][36]. Because of the special slender and superdeformation characteristics of the elastic rod model, its equation of motion is strongly nonlinear, which makes its solution difficult to be found.…”

Section: Introductionmentioning

confidence: 99%

Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

Wang

Feng

2017

Acta Phys. Pol. A

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Basing on the analytical mechanics methods, the Lagrangian equations of thin elastic rod is constructed. The definition of conformal invariance for the Lagrange mechanics of elastic rod is given. The criterions that conformal invariance of elastic rod is the Lie symmetry are obtained based on the Lie point transformation group. The structure equation and conserved quantity deduced from conformal invariance of elastic rod are constructed. Take twist rod as an example to illustrate the application of the results got in this paper.

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Isoperimetric conjugate points with application to the stability of DNA minicircles

Cited by 61 publications

References 19 publications

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

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

Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

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