American Institute of Mathematical Sciences

May  2013, 33(5): 2065-2083. doi: 10.3934/dcds.2013.33.2065

An $H^1$ model for inextensible strings

 1 Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, United States 2 Department of Mathematics, University of New Orleans, Lakefront, New Orleans, LA 70148, United States

Received  November 2011 Revised  February 2012 Published  December 2012

We study geodesics of the $H^1$ Riemannian metric $$« u,v » = ∫_0^1 ‹ u(s), v(s)› + α^2 ‹ u'(s), v'(s)› ds$$ on the space of inextensible curves $\gamma\colon [0,1]\to\mathbb{R}^2$ with $| γ'|≡ 1$. This metric is a regularization of the usual $L^2$ metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The $H^1$ geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is $C^{\infty}$ in the Banach topology $C^1([0,1], \mathbb{R}^2)$, and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one endpoint of the curves fixed, we have global-in-time solutions. We conclude with some surprising features in the periodic case, along with an analogy to the equations of incompressible fluid mechanics.
Citation: Stephen C. Preston, Ralph Saxton. An $H^1$ model for inextensible strings. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2065-2083. doi: 10.3934/dcds.2013.33.2065
References:
 [1] V. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. [2] M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group, Preprint, arXiv:1105.0327. [3] L. Biliotti, The exponential map of a weak Riemannian Hilbert manifold, Illinois J. Math., 48 (2004), 1191-1206. [4] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. [5] M. P. do Carmo, "Riemannian Geometry," Birkhäuser, Boston, 1992. [6] D. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. Math. (2), 105 (1977), 141-200. [7] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. (2), 92 (1970), 102-163. [8] J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Comm. Pure Appl. Analysis, 11 (2012), 1407-1419. doi: 10.3934/cpaa.2012.11.1407. [9] P. Hartman, "Ordinary Differential Equations," Wiley, New York, 1964. [10] T. Hou and C. Li, On global well-posedness of the Lagrangian averaged Euler equations, SIAM J. Math. Anal., 38 (2006), 782-794. doi: 10.1137/050625783. [11] S. Lang, "Fundamentals of Differential Geometry," Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8. [12] J. E. Marsden, D. G. Ebin and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, Proc. Can. Math. Congress, 1 (1972), 135-279. [13] P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc., 8 (2006), 1-48. doi: 10.4171/JEMS/37. [14] G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [15] S. C. Preston, The motion of whips and chains, J. Diff. Eq., 251 (2011), 504-550. doi: 10.1016/j.jde.2011.05.005. [16] S. C. Preston, The geometry of whips, Ann. Global Anal. Geom., 41 (2012), 281-305. doi: 10.1007/s10455-011-9283-z. [17] R. Saxton, Existence of solutions for a finite nonlinearly hyperelastic rod, J. Math. Anal. Appl., 105 (1985), 59-75. doi: 10.1016/0022-247X(85)90096-4. [18] S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335. [19] A. Shnirelman, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4 (1994), 586-620. doi: 10.1007/BF01896409. [20] A. Thess, O. Zikanov and A. Nepomnyashchy, Finite-time singularity in the vortex dynamics of a string, Phys. Rev. E, 59 (1999), 3637-3640. doi: 10.1103/PhysRevE.59.3637.

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References:
 [1] V. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. [2] M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group, Preprint, arXiv:1105.0327. [3] L. Biliotti, The exponential map of a weak Riemannian Hilbert manifold, Illinois J. Math., 48 (2004), 1191-1206. [4] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. [5] M. P. do Carmo, "Riemannian Geometry," Birkhäuser, Boston, 1992. [6] D. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. Math. (2), 105 (1977), 141-200. [7] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. (2), 92 (1970), 102-163. [8] J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Comm. Pure Appl. Analysis, 11 (2012), 1407-1419. doi: 10.3934/cpaa.2012.11.1407. [9] P. Hartman, "Ordinary Differential Equations," Wiley, New York, 1964. [10] T. Hou and C. Li, On global well-posedness of the Lagrangian averaged Euler equations, SIAM J. Math. Anal., 38 (2006), 782-794. doi: 10.1137/050625783. [11] S. Lang, "Fundamentals of Differential Geometry," Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8. [12] J. E. Marsden, D. G. Ebin and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, Proc. Can. Math. Congress, 1 (1972), 135-279. [13] P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc., 8 (2006), 1-48. doi: 10.4171/JEMS/37. [14] G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [15] S. C. Preston, The motion of whips and chains, J. Diff. Eq., 251 (2011), 504-550. doi: 10.1016/j.jde.2011.05.005. [16] S. C. Preston, The geometry of whips, Ann. Global Anal. Geom., 41 (2012), 281-305. doi: 10.1007/s10455-011-9283-z. [17] R. Saxton, Existence of solutions for a finite nonlinearly hyperelastic rod, J. Math. Anal. Appl., 105 (1985), 59-75. doi: 10.1016/0022-247X(85)90096-4. [18] S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335. [19] A. Shnirelman, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4 (1994), 586-620. doi: 10.1007/BF01896409. [20] A. Thess, O. Zikanov and A. Nepomnyashchy, Finite-time singularity in the vortex dynamics of a string, Phys. Rev. E, 59 (1999), 3637-3640. doi: 10.1103/PhysRevE.59.3637.
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