Hamiltonians for curves

Capovilla, Riccardo; Chryssomalakos, Chryssomalis; Guven, Jemal

doi:10.1088/0305-4470/35/31/304

Cited by 68 publications

(121 citation statements)

References 23 publications

Supporting

Mentioning

120

Contrasting

Order By: Relevance

“…Our formulation may help to choose boundary conditions that avoid unwanted configurations. Alternatively, one can re-use the variations already calculated in Capovilla et al (2002) to yield the same results, without having to compute new variations. There, for example, the variation is given for H 1 = f 1 (k, t) ds, stating that it can be written down from the variations of H 2 = f 2 (k) ds and H 3 = f 3 (t) ds, which are previously calculated.…”

Section: Discussionmentioning

confidence: 99%

“…Their explicit expressions are not required; their contributions can be read off from Capovilla et al (2002). The additional terms generated by the additional dependence of f on k and t are therefore easily managed.…”

Section: Discussionmentioning

confidence: 99%

“…This is done in the appendix, where it is shown that it is straightforward to accommodate the additional functional dependence on k , t without computing any new variations, by simply reusing the results already given in Langer & Perline (1991) and Capovilla et al (2002). The additional terms generated by the dependence of g on k and t are easily managed.…”

Section: The Boundary-value Problemmentioning

confidence: 99%

See 2 more Smart Citations

Triangular buckling patterns of twisted inextensible strips

2010

View full text Add to dashboard Cite

When twisting a strip of paper or acetate under high longitudinal tension, one observes, at some critical load, a buckling of the strip into a regular triangular pattern. Very similar triangular facets have recently been found in solutions to a new set of geometrically exact equations describing the equilibrium shape of thin inextensible elastic strips. Here, we formulate a modified boundary-value problem for these equations and construct post-buckling solutions in good agreement with the observed pattern in twisted strips. We also study the force-extension and moment-twist behaviour of these strips by varying the mode number n of triangular facets and find critical loads with jumps to higher modes.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: The Boundary-value Problemmentioning

confidence: 99%

See 1 more Smart Citation

Triangular buckling patterns of twisted inextensible strips

2010

View full text Add to dashboard Cite

show abstract

“…7), using the isometry between the two geometries, with the replacements R = (r 2 + p 2 ) 1/2 and α = −β. 29 Geodesics, as before, satisfy a Clairaut type relation…”

Section: Curves On the Helicoidmentioning

confidence: 91%

Environmental bias and elastic curves on surfaces

Guven¹,

Valencia²,

Vázquez-Montejo³

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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.

show abstract

“…(For an analogous treatment of elastic curves in space see [11].) In particular, translational invariance implies the proportionality of the divergence of the membrane stress tensor to any applied external forces; if the membrane is closed and the enclosed volume is fixed, this source will be the hydrostatic pressure enforcing the constraint.…”

Section: Introductionmentioning

confidence: 99%

Stresses in lipid membranes

Capovilla

Guven

2002

J. Phys. A: Math. Gen.

Self Cite

171

298

View full text Add to dashboard Cite

Abstract. The stresses in a closed lipid membrane described by the Helfrich hamiltonian, quadratic in the extrinsic curvature, are identified using Noether's theorem. Three equations describe the conservation of the stress tensor: the normal projection is identified as the shape equation describing equilibrium configurations; the tangential projections are consistency conditions on the stresses which capture the fluid character of such membranes. The corresponding torque tensor is also identified. The use of the stress tensor as a basis for perturbation theory is discussed. The conservation laws are cast in terms of the forces and torques on closed curves. As an application, the first integral of the shape equation for axially symmetric configurations is derived by examining the forces which are balanced along circles of constant latitude.

show abstract

Hamiltonians for curves

Cited by 68 publications

References 23 publications

Triangular buckling patterns of twisted inextensible strips

Triangular buckling patterns of twisted inextensible strips

Environmental bias and elastic curves on surfaces

Stresses in lipid membranes

Contact Info

Product

Resources

About