Extremals of the generalized Euler-Bernoulli energy and applications

Journal of Geometry and Symmetry in Physics

Volumn 12, Issue , 2008, Pages 27-61

  Garay, Óscar J ^a  

^a UNIVERSITY OF THE BASQUE COUNTRY UPV EHU   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84855854783     PISSN: 13125192     EISSN: 13145673     Source Type: Journal    
DOI: 10.7546/jgsp-12-2008-27-61     Document Type: Review

References (62)

1. Arreaga G., Capovilla R. and Guven J., Frenet-Serret Dynamics, Class. Quantun Grav. 18 (2001) 5065-5083. (Pubitemid 33588712)
2. Arroyo J., Presión Calibrada Total, Estudio Variacional y Aplicaciones al Problema de Chen-Willmore, PhD Thesis. Univ. del País Vasco, Leioa, Spain, 2001.
3. Arroyo J., Barros M. and Garay O., Willmore-Chen tubes on homogeneous spaces in warped product spaces, Pacific J. Math. 188 (1999) 201-207.
4. Arroyo J., Barros M. and Garay O., Some examples of critical points for the total mean curvature functional, Proc. Edinburgh Math. Soc. 43 (2000) 587-603.
5. Arroyo J., Barros M. and Garay O., Holography and total charge, Journal of Geometry and Physics 742 (2001) 1-8.
6. Arroyo J., Barros M. and Garay O., Relativistic Particles with Rigidity Generating Non-Standard Examples of Willmore-Chen Hyper surf aces, J. Phys. A: Math. Gen. 35 (2002) 6815-6824. (Pubitemid 37049329)
7. Arroyo J., Barros M. and Garay O., Models of Relativistic Particles with Curvature and Torsion Revisited, Gen. Rel. Gravit. 36 2004 1441-1451. (Pubitemid 39015060)
8. Arroyo J., Barros M. and Garay O., Closed Free Hyperelastic Curves in the Hyperbolic Plane and Chen-Willmore Rotational Hypersurfaces, Israel J. Math. 138 (2003) 171-187. (Pubitemid 38191658)
9. 3(1), Contemporary Math. 288 (2001) 258-262.
10. 2(1), J. Geom. Phys. 48 (2003) 339-353.
11. Arroyo J., Garay O. and Meneia J., Extremals of Curvature Energy Actions on Spherical Closed Curves, J. Geom. Phys. 51 (2004) 101-125. (Pubitemid 38457736)
12. Arroyo J., Garay O. and Meneía J., Elastic Circles in 2-Spheres, J. Phys. A: Math. Gen. 39 (2006) 2307-2324. (Pubitemid 43296271)
13. Arroyo J., Garay O. and Meneía J., Elastic Curves in the Hyperbolic Plane, Preprint, 2008.
14. Barros M., General Helices and a Theorem of Laneret, Proc. A.M.S. 125 (1997) 1503-1509. (Pubitemid 127743962)
15. Barros M., Simple geometric models with applications to physics, InCurvature and Variational Modeling in Physics and Biophysics, O. Garay, E. García-Río and R. Vázquez-Lorenzo (Eds), AIP Conference Proceedings 1002, Melville-New York, 2008, pp 71-113.
16. Barros M., Ferrandez A., Javaloyes M. and Lucas P., Relativistic Particles with Rigidity and Torsion in D = 3 Spacetimes, Class. Quantun Grav. 35 (2005) 489-513. (Pubitemid 40239714)
17. Barros M., Ferrández A. and Lucas P., Conformal tension in string theories and M-Theory, Nucl. Phys. B 584 (2000) 719-748.
18. 2 ds, Amer. J. Math. 108 (1986) 525-570.
19. Canham P., The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biol. 26 (1970) 61-81.
20. Chen B. Y., Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. 6 (1974) 380-385.
21. Chen B. Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984.
22. Djondjorov P., Hadzhilazova M., Mladenov I. and Vassilev V., Explicit parametrization of the Euler's Elastica, In Geometry, Integrability and Quantization, I. Mladenov (Ed), Softex, Sofia, 2008, pp 175-186.
23. Euler L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattisimo sensu accepti, Bousquet, Lausannae et Genevae E 65A., O.O. Ser. I, vol 24, 1744.
24. Goldstein R. and Langer S., Nonlinnear dynamics of stiff polymers, Phys.Rev. Lett. 75 (1995) 1094-1097.
25. Guo Z., Li H. and Wang Ch., The Second Variational Formula for Willmore Submanifolds in Sspn, Results Math., 40 (2001) 205-225. (Pubitemid 33345717)
26. Helfrich W., Elastic properties of Lipid Bilayers: Theory and possible experiments, Z. Naturforsch 28c (1973) 693-703.
27. Hertrich-Jeromin U., Introduction to moebius differential geometry, London Mathematical Soc. Lecture Note Series 300, Cambridge Univ. Press., Cambridge, 2003.
28. Hopf H., Über Die Abbildungen der Dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931) 637-665.
29. Jülicher F., The Morphology of Vesicles of higher Topological Genus: Conformal Degeneracy and Conformal Modes, J. Phys. II France 6 (1996) 1797-1824. (Pubitemid 126795506)
30. Jurdjevic V., Non-Euclidean elastica, Amer. J. Math. 117 (1995) 93-124.
31. Kamien R., The Geometry of Soft Materials: A Primer, Rev. Mod. Phy. 74 (2002) 953-971. (Pubitemid 36116694)
32. Kawakubo S., Kirchhoff elastic rods in three-dimensional space forms, J. Math. Soc. Japan 60 (2008) 551-582.
33. Kholodenko A. and Nesterenko V., Clasical dynamics of the rigid string from the willmore funcional, J. Geom. Phys. 16 (1995) 15-26.
34. Kierfeld J., Baczinsky K., Gutjahr P and Lypowsky R., Semiflexible polymers and filaments: From variational problems to fluctuations, In Curvature and Variational Modeling in Physics and Biophysics, O. Garay, E. García-Río and R. Vázquez-Lorenzo (Eds), AIP Conference Proceedings 1002, Melville-New York, 2008, pp 151-185.
35. Kuznetsov Y. and Plyushchay M., Tachyonless models of relativistic particles with curvature and torsion, Phys. Lett. B 297 (1992) 49-54.
36. Kuznetsov Y. and Plyushchay M., The model of the relativistic particle with curvature and torsion, Nucl. Phys. B 389 (1993) 181-208.
37. Landau L. and Lifshitz E., Theory of elasticity, (Course of Theoretical Physics, Vol 7, Butterworth-Heinemann, Boston, MA, 1986.
38. Langer J. and Singer D., The total squared curvature of closed curves, J. Diff. Geom. 20 (1984) 1-22.
39. 3, J. London Math. Soc. 30 (1984) 512-520
40. Langer J. and Singer D., Curves in the hyperbolic plane and mean curvature of Tori in 3-Space, Bull. London Math. Soc. Ann. 16 (1984) 531-534.
41. Langer J. and Singer D., Lagrangian Aspects of the Kirchhoff Elastic Rod, SIAM Rev. 38 (1996) 605-618. (Pubitemid 126619775)
42. Lipowsky R., The Conformation of Membranes, Nature 349 (1991) 475-481. (Pubitemid 21912083)
43. Michalet X. and Bensimon D., Vesicles of toroidal topology: Observed morphology and shape transformations, J. Phys. II France 5 (1995) 263-287.
44. Michalet X. and Bensimon D., Observation of stable shapes and conformal diffusion in Genus 2 Vesicles, Science 269 (1995) 666-668.
45. Mladenov I., New Solutions of the Shape Equation, Eur. Phys. J. B 29 (2002) 327-330. (Pubitemid 135695480)
46. Nersessian A., Massive 4d particle with Torsion and conformal mechanics, Phys. Lett. B 473 (2000) 94-101.
47. Nesterenko V., On the model of the relativistic particle with curvature and Torsion, J. Math. Phys. 34 (1993) 5589-5595.
48. Nesterenko V., Feoli A. and Scarpetta G., Dynamics of relativistic particles with lagrangians dependent on acceleration, J. Math. Phys. 36 (1995) 5552-5564.
49. Nitsche J., Boundary value problems for variational integrals involving surface curvatures, Q. Appl. Math. 60 (1993) 363-387.
50. Palais R., Critical point theory and the minimax principle, global analysis, Proc. Sympos. Pure Math. 15, 1970, pp. 185-212
51. Palmer B., Variational problems which are quadratic in the surface curvatures, In Curvature and Variational Modeling in Physics and Biophysics, O. Garay, E. García-Río and R. Vázquez-Lorenzo (Eds), AIP Conference Proceedings 1002, Melville-New York, 2008, pp 71-113.
52. 3, Invent. Math. 81 (1995) 379-386.
53. Plyushchay M., Massless point particle with rigidity, Mod. Phys. Lett. A 4 (1989) 837-847.
54. Plyushchay M., Massless particle with rigidity as a model for the description of Bosons and Fermions, Phys. Lett. B 243 (1990) 383-388.
55. Plyushchay M., The model of the relativistic particle with Torsion, Nuclear Phys. B 362 (1991) 54-72.
56. Santangelo C., The geometry and topology of liquid crystals, InCurvature and Variational Modeling in Physics and Biophysics, O. Garay, E. García-Río and R. Vázquez-Lorenzo (Eds), AIP Conference Proceedings 1002, Melville-New York, 2008, pp 114-150.
57. Seifert U., Berndl K. and Lipowsky R., Shape transformations of vesicles: Phase diagram for spontaneous curvature and bilayer coupling model, Phys. Rev. A. 44 (1991) 1182-1202.
58. Seifert U., Configurations of Fluid Membranes and Vesicles, Adv. Phys. 46 (1997) 13-137. (Pubitemid 127535649)
59. Singer D., Lectures on elastic curves and rods, In Curvature and Variational Modeling in Physics and Biophysics, O. Garay, E. García-Río and R. Vázquez-Lorenzo (Eds), AIP Conference Proceedings 1002, Melville-New York, 2008, pp 3-32.
60. Weiner J., On a problem of Chen, Willmore, et al., Indiana Univ. Math. J. 27 (1978) 19-35.
61. Willmore T., Total Curvature in Riemannian Geometry, E Horwood Limited, Chicester, 1982.
62. Zhong-Can O-Y., Ji-Xing L. and Yu-Zhang X., Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, World Scientific, Singapore, 1999.