Elastic circles in 2-spheres

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

doi:10.1088/0305-4470/39/10/005

Cited by 16 publications

(28 citation statements)

References 4 publications

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“…If the tangent is also fixed, then both middle terms vanish as well, for δT = 0 and consequently δL = 0. 17 If T is not fixed one requires H g = 0.…”

Section: Fixed Endsmentioning

confidence: 99%

Environmental bias and elastic curves on surfaces

Guven¹,

Valencia²,

Vázquez-Montejo³

2014

J. Phys. A: Math. Theor.

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The behavior of an elastic curve bound to a surface will reflect the geometry of its environment. This may occur in an obvious way: the curve may deform freely along directions tangent to the surface, but not along the surface normal. However, even if the energy itself is symmetric in the curve's geodesic and normal curvatures, which control these modes, very distinct roles are played by the two. If the elastic curve binds preferentially on one side, or is itself assembled on the surface, not only would one expect the bending moduli associated with the two modes to differ, binding along specific directions, reflected in spontaneous values of these curvatures, may be favored. The shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of Lagrange multipliers to the Darboux frame associated with the curve. The forces transmitted to the surface along the surface normal will be determined. Features associated with a number of different energies, both of physical relevance and of mathematical interest, are described. The conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined.

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“…If the tangent is also fixed, then both middle terms vanish as well, for δT = 0 and consequently δL = 0. 17 If T is not fixed one requires H g = 0.…”

Section: Fixed Endsmentioning

confidence: 99%

Environmental bias and elastic curves on surfaces

Guven¹,

Valencia²,

Vázquez-Montejo³

2014

J. Phys. A: Math. Theor.

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“…Equation (22) can be integrated in terms of elliptic functions to give κ g as a function of s [7,9,20] κ g (s) = κ 0 cn [qs, m] , κ 0 = 2 √ mq .…”

Section: Strong Spherical Confinementmentioning

confidence: 99%

Confinement of semiflexible polymers

Guven

Vázquez-Montejo

2012

Phys. Rev. E

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A variational framework is developed to examine the equilibrium states of a semiflexible polymer that is constrained to lie on a fixed surface. As an application the confinement of a closed polymer loop of fixed length 2πR within a spherical cavity of smaller radius, R 0 , is considered. It is shown that an infinite number of distinct periodic completely attached equilibrium states exist, labeled by two integers: n = 2, 3, 4, . . . and p = 1, 2, 3, · · · , the number of periods of the polar and azimuthal angles respectively. Small loops oscillate about a geodesic circle: n = 2, p = 1 is the stable ground state; states with higher n exhibit instabilities. If R ≥ 2R 0 new states appear as oscillations about a doubly covered geodesic circle; the state n = 3, p = 2 replaces the two-fold as the ground state in a finite band of values of R. With increasing R, loop states make a transition from oscillatory and orbital behavior on crossing the poles, returning to oscillation upon collapse to a multiple cover of a geodesic circle (signaled, respectively, by an increase in p and an increase in n). The force transmitted to the surface does not increase monotonically with loop size, but does asymptotically. It behaves discontinuously where n changes. The contribution to energy from geodesic curvature is bounded. In large loops, the energy becomes dominated by a state independent contribution proportional to the loop size; the energy gap between the ground state and excited states disappears.

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“…For parameterized hypersurfaces in R 3 , earlier authors, see e.g. [10,18,19,3,5], used the surrounding space in their numerical approximations, which leads to errors in directions normal to the hypersurface. This will be avoided by the intrinsic approach used in this paper.…”

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confidence: 99%

Stable Discretizations of Elastic Flow in Riemannian Manifolds

Barrett¹,

Garcke²,

Nürnberg³

2019

SIAM J. Numer. Anal.

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The elastic flow, which is the L 2 -gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e. conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional manifold in R d , d ≥ 3. Numerical results show the robustness of the method, as well as quadratic convergence with respect to the space discretization.

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Elastic circles in 2-spheres

Cited by 16 publications

References 4 publications

Environmental bias and elastic curves on surfaces

Environmental bias and elastic curves on surfaces

Confinement of semiflexible polymers

Stable Discretizations of Elastic Flow in Riemannian Manifolds

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