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Geometric conditions for the existence of a rolling without twisting or slipping
Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations
1. | Department of Mathematics and Information Sciences, Northumbria University, Pandon Building, Camden Street, Newcastle upon Tyne, NE2 1XE, United Kingdom |
2. | INdAM-COFUND Marie Curie Fellow, Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07737, Germany |
References:
[1] |
W. A. Beyer and P. J. Channell, A functional equation for the embedding of a homeomorphism of the interval into a flow, Lecture Notes in Math., 1163 (1985), 7-13.
doi: 10.1007/BFb0076412. |
[2] |
L. Bianchi, "Lezioni di geometria differenziale," Vol. 1, Libraio Ed., Pisa, 1922. |
[3] |
J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57.
doi: 10.1007/BF03023575. |
[4] |
G. De Matteis, Group Analysis of the Membrane Shape Equation, in "Nonlinear Physics: Theory and Experiment, II," World Scientific, River Edge NJ, (2003), 221-226.
doi: 10.1142/9789812704467_0031. |
[5] |
M. P. do Carmo, "Differential Geometry of Curves and Surfaces," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1976. |
[6] |
M. K. Fort Jr., The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967.
doi: 10.1090/S0002-9939-1955-0080911-2. |
[7] |
R. Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications," Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1994. |
[8] |
M. Han, Conditions for a Diffeomorphism to be embedded in a $C^r$ flow, Acta Math. Sinica (N.S.), 4 (1988), 111-123.
doi: 10.1007/BF02560593. |
[9] |
W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch, 28c (1973), 693-703. |
[10] |
B. G. Konopelchenko, On solutions of the shape equation for membranes and strings, Phys. Lett. B, 414 (1997), 58-64 .
doi: 10.1016/S0370-2693(97)01137-4. |
[11] |
H. Jian-Guo and O.-Y. Zhong-can, Shape equations of the axisymmetric vesicles, Phys. Rev. E, 47 (1993), 461-467.
doi: 10.1103/PhysRevE.47.461. |
[12] |
P. F. Lam, Embedding homeomorphisms in differential flows, Colloq. Math., 35 (1976), 275-287. |
[13] |
P. F. Lam, Embedding a differentiable homeomorphism in a flow subject to a regularity condition on the derivatives of the positive transition homeomorphisms, J. Differential Equations, 30 (1978), 31-40.
doi: 10.1016/0022-0396(78)90021-9. |
[14] |
P. F. Lam, Embedding homeomorphisms in $C^1$-flows, Ann. Mat. Pura Appl., 123 (1980), 11-25.
doi: 10.1007/BF01796537. |
[15] |
R. A. Leo, L. Martina and G. Soliani, Group analysis of the three-wave resonant system in $(2+1)$-dimensions, J. Math. Phys., 27 (1986), 2623-2628.
doi: 10.1063/1.527280. |
[16] |
R. Lipowsky and E. Sackman, "Structure and Dynamics of Membranes," Elsevier Science B.V., Amsterdam, 1995. |
[17] |
G. Manno and R. Vitolo, Geometric aspects of higher order variational principles on submanifolds, Acta Appl. Math., 101 (2008), 215-229.
doi: 10.1007/s10440-008-9190-x. |
[18] |
G. Manno, On the geometry of Grassmannian equivalent connections, Adv. Geom., 8 (2008), 329-342.
doi: 10.1515/ADVGEOM.2008.021. |
[19] |
G. Manno, F. Oliveri and R. Vitolo, Differential equations uniquely determined by algebras of point symmetries, Theoret. and Math. Phys., 151 (2007), 843-850.
doi: 10.1007/s11232-007-0069-1. |
[20] |
L. Martina and P. Winternitz, Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem, Ann. Physics, 196 (1989), 231-277.
doi: 10.1016/0003-4916(89)90178-4. |
[21] |
M. A. McKiernan, On the convergence of series of iterates, Publ. Math. Debrecen, 10 (1963), 30-39. |
[22] |
M. Mutz and D. Bensimon, Observation of toroidal vesicles, Phys. Rev. A, 43 (1991), 4525-4527.
doi: 10.1103/PhysRevA.43.4525. |
[23] |
H. Naito, M. Okuda and O.-Y. Zhong-can, New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles: Beyond Delaunay's Surfaces, Phys. Rev. Lett., 74 (1995), 4345-4348.
doi: 10.1103/PhysRevLett.74.4345. |
[24] |
D. Nelson, T. Piran and S. Weinberg, "Statistical Mechanics of Membranes and Surfaces," World Scientific, Teaneck, NJ, 1989. |
[25] |
F. Neuman, Solution to the Problem No. 10 of N. Kamran, in "Proceedings, 23rd International Symposium on Functional Equations" (Gargnano, Italy) Centre for Information Theory, University of Waterloo, Ontario, Canada, (1985), 60-62. |
[26] |
P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer-Verlag, New York, 1993. |
[27] |
L. V. Ovsiannikov, "Group Analysis of Differential Equations," Academic Press, New York-London, 1982. |
[28] |
L. Peliti, Amphiphilic Membranes, in "Fluctuating Geometries in Statistical Mechanics and Field Theory" (eds. F. David, P. Ginsparg and J. Zinn-Justin), Les Houches, (1994). |
[29] |
V. Pulov, M. Hadjilazova and I. M. Mladenov, Symmetries and Solutions of the Membrane Shape Equation, talk given at "XIV International Conference Geometry Integrability and Quantization'' (Varna, Bulgaria 2012), http://www.bio21.bas.bg/conference/Conference_files/sa12/slides/Pulov.pdf |
[30] |
R. Schmid, Infinite-dimensional Lie groups and algebras in mathematical physics, Adv. Math. Phys., (2010), Art ID 280362, 35 pp.
doi: 10.1155/2010/280362. |
[31] |
M. Schottenholer, "A Mathematical Introduction to Conformal Field Theory," Lecture Notes in Physics, 759, Springer-Verlag, Berlin, 2008. |
[32] |
V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Symmetry groups, conservation laws and group-invariant solutions of the membrane shape equation, Geometry, Integrability and Quantization, Softex, Sofia, (2006), 265-279. |
[33] |
V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, On the translationally-invariant solutions of the membrane shape equation, Geometry, Integrability and Quantization, Softex, Sofia, (2007), 312-321. |
[34] |
V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes, J. Phys. A, 41 (2008), 435201, 16 pp.
doi: 10.1088/1751-8113/41/43/435201. |
[35] |
V. M. Vassiliev and I. M. Mladenov, Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation, Geometry, Integrability and Quantization, Softex, Sofia, (2004), 246-265.
doi: 10.7546/giq-5-2004-246-265. |
[36] |
A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2 (1984), 21-78.
doi: 10.1007/BF01405491. |
[37] |
T. J. Willmore, "Riemannian Geometry," The Clarendon Press, Oxford University Press, New York, 1993. |
[38] |
T. J. Willmore, "Total Curvature in Riemannian Geometry," Ellis Horwood Ltd., Chichester; Halsted Press, New York, 1982. |
[39] |
L. Weigu and M. Zhang, Embedding flows and smooth conjugacy, Chinese Ann. Math. Ser. B, 18 (1997), 125-138. |
[40] |
P. Winternitz, Group Theory and exact solutions of partially integrable differential systems, in "Partially integrable evolution equations in Physics," NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 310, Kluwer Acad. Publ., Dordrecht (1990), 515-567.
doi: 10.1007/978-94-009-0591-7_20. |
[41] |
M. Zhang, Embedding problem and functional equations, Acta Math. Sinica (N.S.), 8 (1992), 148-157.
doi: 10.1007/BF02629935. |
[42] |
W.-M. Zheng and J. Liu, The Helfrich equation for axisymmetric vesicles as a first integral, Phys. Rev. E, 48 (1993), 2856-2860.
doi: 10.1103/PhysRevE.48.2856. |
[43] |
O.-Y. Zhong-can, Anchor ring-vesicle membranes, Phys. Rev. A, 41 (1990), 4517-4520.
doi: 10.1103/PhysRevA.41.4517. |
[44] |
O.-Y. Zhong-can, Ji-Xing Liu and Yu-Zhang Xie, "Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases," World Scientific, Hong Kong, 1999. |
show all references
References:
[1] |
W. A. Beyer and P. J. Channell, A functional equation for the embedding of a homeomorphism of the interval into a flow, Lecture Notes in Math., 1163 (1985), 7-13.
doi: 10.1007/BFb0076412. |
[2] |
L. Bianchi, "Lezioni di geometria differenziale," Vol. 1, Libraio Ed., Pisa, 1922. |
[3] |
J. Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57.
doi: 10.1007/BF03023575. |
[4] |
G. De Matteis, Group Analysis of the Membrane Shape Equation, in "Nonlinear Physics: Theory and Experiment, II," World Scientific, River Edge NJ, (2003), 221-226.
doi: 10.1142/9789812704467_0031. |
[5] |
M. P. do Carmo, "Differential Geometry of Curves and Surfaces," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1976. |
[6] |
M. K. Fort Jr., The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967.
doi: 10.1090/S0002-9939-1955-0080911-2. |
[7] |
R. Gilmore, "Lie Groups, Lie Algebras, and Some of Their Applications," Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1994. |
[8] |
M. Han, Conditions for a Diffeomorphism to be embedded in a $C^r$ flow, Acta Math. Sinica (N.S.), 4 (1988), 111-123.
doi: 10.1007/BF02560593. |
[9] |
W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch, 28c (1973), 693-703. |
[10] |
B. G. Konopelchenko, On solutions of the shape equation for membranes and strings, Phys. Lett. B, 414 (1997), 58-64 .
doi: 10.1016/S0370-2693(97)01137-4. |
[11] |
H. Jian-Guo and O.-Y. Zhong-can, Shape equations of the axisymmetric vesicles, Phys. Rev. E, 47 (1993), 461-467.
doi: 10.1103/PhysRevE.47.461. |
[12] |
P. F. Lam, Embedding homeomorphisms in differential flows, Colloq. Math., 35 (1976), 275-287. |
[13] |
P. F. Lam, Embedding a differentiable homeomorphism in a flow subject to a regularity condition on the derivatives of the positive transition homeomorphisms, J. Differential Equations, 30 (1978), 31-40.
doi: 10.1016/0022-0396(78)90021-9. |
[14] |
P. F. Lam, Embedding homeomorphisms in $C^1$-flows, Ann. Mat. Pura Appl., 123 (1980), 11-25.
doi: 10.1007/BF01796537. |
[15] |
R. A. Leo, L. Martina and G. Soliani, Group analysis of the three-wave resonant system in $(2+1)$-dimensions, J. Math. Phys., 27 (1986), 2623-2628.
doi: 10.1063/1.527280. |
[16] |
R. Lipowsky and E. Sackman, "Structure and Dynamics of Membranes," Elsevier Science B.V., Amsterdam, 1995. |
[17] |
G. Manno and R. Vitolo, Geometric aspects of higher order variational principles on submanifolds, Acta Appl. Math., 101 (2008), 215-229.
doi: 10.1007/s10440-008-9190-x. |
[18] |
G. Manno, On the geometry of Grassmannian equivalent connections, Adv. Geom., 8 (2008), 329-342.
doi: 10.1515/ADVGEOM.2008.021. |
[19] |
G. Manno, F. Oliveri and R. Vitolo, Differential equations uniquely determined by algebras of point symmetries, Theoret. and Math. Phys., 151 (2007), 843-850.
doi: 10.1007/s11232-007-0069-1. |
[20] |
L. Martina and P. Winternitz, Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem, Ann. Physics, 196 (1989), 231-277.
doi: 10.1016/0003-4916(89)90178-4. |
[21] |
M. A. McKiernan, On the convergence of series of iterates, Publ. Math. Debrecen, 10 (1963), 30-39. |
[22] |
M. Mutz and D. Bensimon, Observation of toroidal vesicles, Phys. Rev. A, 43 (1991), 4525-4527.
doi: 10.1103/PhysRevA.43.4525. |
[23] |
H. Naito, M. Okuda and O.-Y. Zhong-can, New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles: Beyond Delaunay's Surfaces, Phys. Rev. Lett., 74 (1995), 4345-4348.
doi: 10.1103/PhysRevLett.74.4345. |
[24] |
D. Nelson, T. Piran and S. Weinberg, "Statistical Mechanics of Membranes and Surfaces," World Scientific, Teaneck, NJ, 1989. |
[25] |
F. Neuman, Solution to the Problem No. 10 of N. Kamran, in "Proceedings, 23rd International Symposium on Functional Equations" (Gargnano, Italy) Centre for Information Theory, University of Waterloo, Ontario, Canada, (1985), 60-62. |
[26] |
P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer-Verlag, New York, 1993. |
[27] |
L. V. Ovsiannikov, "Group Analysis of Differential Equations," Academic Press, New York-London, 1982. |
[28] |
L. Peliti, Amphiphilic Membranes, in "Fluctuating Geometries in Statistical Mechanics and Field Theory" (eds. F. David, P. Ginsparg and J. Zinn-Justin), Les Houches, (1994). |
[29] |
V. Pulov, M. Hadjilazova and I. M. Mladenov, Symmetries and Solutions of the Membrane Shape Equation, talk given at "XIV International Conference Geometry Integrability and Quantization'' (Varna, Bulgaria 2012), http://www.bio21.bas.bg/conference/Conference_files/sa12/slides/Pulov.pdf |
[30] |
R. Schmid, Infinite-dimensional Lie groups and algebras in mathematical physics, Adv. Math. Phys., (2010), Art ID 280362, 35 pp.
doi: 10.1155/2010/280362. |
[31] |
M. Schottenholer, "A Mathematical Introduction to Conformal Field Theory," Lecture Notes in Physics, 759, Springer-Verlag, Berlin, 2008. |
[32] |
V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Symmetry groups, conservation laws and group-invariant solutions of the membrane shape equation, Geometry, Integrability and Quantization, Softex, Sofia, (2006), 265-279. |
[33] |
V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, On the translationally-invariant solutions of the membrane shape equation, Geometry, Integrability and Quantization, Softex, Sofia, (2007), 312-321. |
[34] |
V. M. Vassilev, P. A. Djondjorov and I. M. Mladenov, Cylindrical equilibrium shapes of fluid membranes, J. Phys. A, 41 (2008), 435201, 16 pp.
doi: 10.1088/1751-8113/41/43/435201. |
[35] |
V. M. Vassiliev and I. M. Mladenov, Geometric symmetry groups, conservation laws and group-invariant solutions of the Willmore equation, Geometry, Integrability and Quantization, Softex, Sofia, (2004), 246-265.
doi: 10.7546/giq-5-2004-246-265. |
[36] |
A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 2 (1984), 21-78.
doi: 10.1007/BF01405491. |
[37] |
T. J. Willmore, "Riemannian Geometry," The Clarendon Press, Oxford University Press, New York, 1993. |
[38] |
T. J. Willmore, "Total Curvature in Riemannian Geometry," Ellis Horwood Ltd., Chichester; Halsted Press, New York, 1982. |
[39] |
L. Weigu and M. Zhang, Embedding flows and smooth conjugacy, Chinese Ann. Math. Ser. B, 18 (1997), 125-138. |
[40] |
P. Winternitz, Group Theory and exact solutions of partially integrable differential systems, in "Partially integrable evolution equations in Physics," NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 310, Kluwer Acad. Publ., Dordrecht (1990), 515-567.
doi: 10.1007/978-94-009-0591-7_20. |
[41] |
M. Zhang, Embedding problem and functional equations, Acta Math. Sinica (N.S.), 8 (1992), 148-157.
doi: 10.1007/BF02629935. |
[42] |
W.-M. Zheng and J. Liu, The Helfrich equation for axisymmetric vesicles as a first integral, Phys. Rev. E, 48 (1993), 2856-2860.
doi: 10.1103/PhysRevE.48.2856. |
[43] |
O.-Y. Zhong-can, Anchor ring-vesicle membranes, Phys. Rev. A, 41 (1990), 4517-4520.
doi: 10.1103/PhysRevA.41.4517. |
[44] |
O.-Y. Zhong-can, Ji-Xing Liu and Yu-Zhang Xie, "Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases," World Scientific, Hong Kong, 1999. |
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