Extremals of curvature energy actions on spherical closed curves

Arroyo, J.; Garay, Óscar J.; Mencía, José J.

doi:10.1016/j.geomphys.2003.10.011

Cited by 19 publications

(26 citation statements)

References 20 publications

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“…Next (6) and γ , M j = 0, we see that h ij satisfies By Lemma 3.2, the above {γ , M} defined on (t 0 − ε, t 0 + ε) is a unit-speed curve with adapted orthonormal frame, which completes the proof of Lemma 3.1.…”

Section: Existence Of Local Solutionsmentioning

confidence: 55%

“…Then the following holds. Similarly, the local solution {γ , M} on (t 0 − ε, t 0 + ε) can be continued to a solution on (−∞, t 0 + ε), and thus on the whole of R. Consequently, we have proved that there exists a unit-speed curve with adapted orthonormal frame {γ , M} defined on the whole of R satisfying (1), (6), and (5).…”

Section: Lemma 41 Let M Be a Complete Riemannian Manifold And Let Imentioning

confidence: 65%

“…Proof of Theorem 2.2. By Lemma 3.1, there exist ε > 0 and a unit-speed curve with adapted orthonormal frame {γ , M} in M defined on (t 0 − ε, t 0 + ε) satisfying (1), (6), and (5). We show that this local solution {γ , M} on (t 0 − ε, t 0 + ε) can be continued to a solution on (t 0 − ε, ∞).…”

Section: Lemma 41 Let M Be a Complete Riemannian Manifold And Let Imentioning

confidence: 87%

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Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

Kawakubo

2014

Journal of Mathematical Physics

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The Kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler-Lagrange equations associated to the energy with the effect of bending and twisting. We consider the initial-value problem for the Euler-Lagrange equations in a Riemannian manifold. In a previous paper, the author proved the existence and uniqueness of global solutions of the initial-value problem in the case where the ambient space is a space form. In the present paper, we extend this result to the case where the ambient space is a general complete Riemannian manifold. This implies that an arbitrary Kirchhoff elastic rod of finite length in a complete Riemannian manifold extends to that of infinite length.

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Section: Existence Of Local Solutionsmentioning

confidence: 55%

Section: Lemma 41 Let M Be a Complete Riemannian Manifold And Let Imentioning

confidence: 65%

Section: Lemma 41 Let M Be a Complete Riemannian Manifold And Let Imentioning

confidence: 87%

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Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

Kawakubo

2014

Journal of Mathematical Physics

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“…For instance, when f (κ, τ ) = f (κ) depends only on the curvature, it was calculated for curves in R 3 in [9], for curves in Lorentzian ambient spaces in [10], and for curves in Riemannian space forms in [7] (see also [19]). Finally, the general version in pseudo-Riemannian 3-space forms was given in [8].…”

Section: First Variation Formulamentioning

confidence: 99%

Critical curves for a Santaló problem in 3-space forms

Garay

Pauley

2013

Journal of Mathematical Analysis and Applications

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a b s t r a c t L. Santaló (1956) studied in [15] the minimization of Frenet first curvature dependent energies for spaces of curves constrained to lie in a surface of the euclidean 3-space. In this paper we analyze an extension of that problem by considering energy functionals, determined by Lagrangians which depend not only on the curvature but also on the torsion, acting on spaces of surface contained clamped curves of real 3-space forms.

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“…[18] or [27]), for critical points of elastic curves in Riemannian manifolds [6] and to describe shape transitions of carbon nanotubes under pressure (Zang et al [44]). In particular the square of curvature has appeared in energies for backbones of DNA and polymers [16,43] while in the setting of two-dimensional surfaces in three space, the square of the mean curvature appears in models of the shape of red blood cells [13,26].…”

Section: Energy Densities F = F (κ)mentioning

confidence: 99%

Helices for mathematical modelling of proteins, nucleic acids and polymers

McCoy

2008

Journal of Mathematical Analysis and Applications

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Helices occur in modelling a range of structures including ropes, fibers, polymers and biopolymers. In a recent paper with Thamwattana and Hill, the authors derived EulerLagrange equations for modelling protein structure, where the energy being minimised was assumed to depend only on the curvature and torsion of the protein backbone space curve and their first derivatives. Such a model is applicable to helices occurring in other scenarios and in this article the author considers more generally which energies will yield helices as solutions of their corresponding Euler-Lagrange equations. He finds in particular classes of energy for which all circular helices are solutions and an energy depending on curvature and its derivative which generates conical helices as solutions of the EulerLagrange equations. Also included are some new results for energies depending only on curvature, extending previous investigations by Feoli et al. [A. Feoli, V.V. Nesterenko, G. Scarpetta, Functionals linear in curvature and statistics of helical proteins, Nuclear Phys. B 705 (2005) 577-592].

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Extremals of curvature energy actions on spherical closed curves

Cited by 19 publications

References 20 publications

Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

Critical curves for a Santaló problem in 3-space forms

Helices for mathematical modelling of proteins, nucleic acids and polymers

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