Journal ArticleDOI
Conjugate Points Revisited and Neumann-Neumann Problems
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It is shown how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann-Neumann boundary conditions.Abstract:
The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on the classic result that “stability requires the lack of conjugate points.” Furthermore, we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann-Neumann boundary conditions.read more
Citations
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Journal ArticleDOI
Elastic Stability of Concentric Tube Robots: A Stability Measure and Design Test
TL;DR: In this article, the authors use bifurcation and elastic stability theory to determine whether a given robot is snap-free (i.e., whether snapping can occur anywhere in the robot's workspace).
Book
A First Course in the Calculus of Variations
TL;DR: The homogeneous problem Variable-endpoint conditions Variable extremals Strong variations Sufficient conditions Bibliography Index as mentioned in this paper The first variation Cases and examples Basic generalizations Constraints The second variation Review and preview The homogeneous Problem
Journal ArticleDOI
On the stability of a rod adhering to a rigid surface: Shear-induced stable adhesion and the instability of peeling
TL;DR: In this paper, the authors used variational methods to establish conditions for the nonlinear stability of adhesive states between an elastica and a rigid halfspace, which can be used to explain how microfiber array adhesives can be activated by shearing and deactivated by peeling.
Journal ArticleDOI
A generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints
Ajeet Kumar,Timothy J. Healey +1 more
TL;DR: In this article, a generalized approach to stability of static equilibria of nonlinearly elastic rods, subjected to general loading, boundary conditions and constraints (of both point-wise and integral type), based upon the linearized dynamics stability criterion is presented.
Journal ArticleDOI
On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut
TL;DR: In this article, the authors explore Legendre's treatment with the aid of three classical buckling problems for elastic struts and show that the conclusions from Legendre and Jacobi's treatments are equivalent for some sets of boundary conditions.
References
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Book
Methods of Mathematical Physics
Richard Courant,David Hilbert +1 more
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Book
Nonlinear problems of elasticity
TL;DR: This book discusses the theory and applications of Bifurcation Theory and its applications to Elasticity, as well as problems in Nonlinear Elasticity and Dynamical Problems.
Book
Linear Operator Theory in Engineering and Science
Arch W. Naylor,George R. Sell +1 more
TL;DR: Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.