Spinor representation of curves and complex forces on filaments
Keywords:Geometric variational principles, Spinor, Filaments, Bending Energy, Euler Elastica
We present a theoretical framework to study equilibrium configurations of filaments within a spinor representation of curves. The curve representing the filament is described by a unit two-component spinor field and its charge conjugate satisfying two-dimensional equations coupled by the curvature and torsion. The spinor field replaces the Frenet-Serret frame, whereas its structure equations replace the Frenet-Serret equations. Employing this spinorial description of curves, we derive the Euler-Lagrange equations of curves whose energies depend on their curvature and torsion. We analyze the conservation laws of the spinors representing the balance of the forces and torques along the filament. We illustrate this framework by applying these results to the Euler Elastica, whose bending energy is quadratic in the curvature.
R. D. Kamien, The geometry of soft materials: a primer, Rev. Mod. Phys. 74 (2002) 953.
T. R. Powers, Dynamics of filaments and membranes in a viscous fluid, Rev. Mod. Phys. 82 (2010) 1607.
R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Month. 82 (1975) 246.
H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972) 477.
Y. Shi and J. E. Hearst, The kirchhoff elastic rod, the nonlinear schrodinger equation, and dna supercoiling ¨ , J. Chem. Phys. 101 (1994) 5186.
Y. Takahashi, A spinor reconstruction theorem, Prog. Theor. Phys. 69 (1983) 369.
Y. Takahashi, A spinorization of the frenet-serret equation, Prog. Theor. Phys. 70 (1983) 1466.
Y. Takahashi, The gauss and the weingarten equations for extended objects and their spinorization J. Math. Phys. 25 (1984) 2728.
G. F. Torres del Castillo and G. Sanchez Barrales, Spinor formulation of the differential geometry of curves Rev. Col. Mat., 38 (2004) 27.
T. Ioannidou, Y. Jiang, and A. J. Niemi, Spinors, strings, integrable models, and decomposed yang-mills theory Phys. Rev. D 90 (2014) 025012.
J. Guven and P. Vazquez-Montejo, ´ Spinor representation of surfaces and complex stresses on membranes and interfaces, Phys. Rev. E 82 (2010) 041604.
J. Guven Membrane geometry with auxiliary variables and quadratic constraints J. Phys. A: Math. Gen., 37 (2004) L313.
J. Guven and P. Vazquez-Montejo, Confinement of semiflexible polymers Phys. Rev. E 85 (2012) 026603.
J. Guven, D. M. Valencia, and P. Vazquez-Montejo, Environmental bias and elastic curves on surfaces J. Phys. A: Math. Theor. 47 (2014) 355201.
J. Guven and P. Vazquez-Montejo, ´ The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws, 1st ed. (Springer International Publishing, Cham, 2018) pp. 167–219.
D. A. Solis and P. Vazquez-Montejo, ´ First integrals for elastic curves: twisting instabilities of helices, J. Phys. A: Math. Theor. 54 (2021) 305702.
R. Capovilla, C. Chryssomalakos, and J. Guven, Hamiltonians for curves, J. Phys. A: Math. Gen. 35 (2002) 6571.
Z. C. Tu and Z. C. Ou-Yang, Elastic theory of low-dimensional continua and its applications in bio- and nano-structures, J. Comput. Theor. Nanosci. 5 (2008) 422.
E. L. Starostin and G. H. M. van der Heijden. Force and moment balance equations for geometric variational problems on curves Phys. Rev. E 79 (2009) 066602.
M. P. do Carmo, Differential Geometry of Curves and Surfaces, 2nd ed. (Dover Publications, New York, 2016).
T. Frankel, The Geometry of Physics: An Introduction, 2nd. ed. (Cambridge University Press, Cambridge, 2004).
E. Cartan, The Theory of Spinors, 1st. ed. (Dover Publications, New York, 2012).
P. Vazquez-Montejo, Z. McDargh, M. Deserno, and J. Guven, Cylindrical confinement of semiflexible polymers, Phys. Rev. E, 91 (2015) 063203.
L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity, 3rd ed. (Butterworth-Heinemann, Oxford, 1986).
J. Langer and D. A. Singer. The total squared curvature of closed curves, J. Differential Geom. 20 (1984) 1.
J. Langer and D. A. Singer. Knotted elastic curves in R3, J. London Math. Soc. 30 (1984) 512.
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover Publications, New York, 1965).
I. Tadjbakhsh and F. Odeh. Equilibrium states of elastic rings, J. Math. Anal. Appl. 18 (1967) 59.
G. Arreaga, R. Capovilla, C. Chryssomalakos, and J. Guven, Area-constrained planar elastica Phys. Rev. E 65 (2002) 031801.
A. Goriely and M. Tabor, The nonlinear dynamics of filaments Nonlinear Dyn. 21 (2000) 101.
C. Goubault, P. Jop, M. Fermigier, J. Baudry, E. Bertrand, and J. Bibette, Flexible magnetic filaments as micromechanical sensors, Phys. Rev. Lett. 91 (2003) 260802.
A. Cebers, Dynamics of a chain of magnetic particles connected with elastic linkers, J. Phys.: Condens. Matter 15 (2003) S1335.
P. Vazquez-Montejo, J. M. Dempster, and M. Olvera de la Cruz, Paramagnetic filaments in a fast precessing field: Planar versus helical conformations, Phys. Rev. Materials 1 (2017) 064402.
How to Cite
Copyright (c) 2022 Didier A. Solis, Pablo Vázquez-Montejo
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.