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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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, (). Google Scholar

[3]

L. Biliotti, The exponential map of a weak Riemannian Hilbert manifold,, Illinois J. Math., 48 (2004), 1191. Google Scholar

[4]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002). doi: 10.1088/0305-4470/35/32/201. Google Scholar

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M. P. do Carmo, "Riemannian Geometry,", Birkhäuser, (1992). Google Scholar

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D. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force,, Ann. Math. (2), 105 (1977), 141. Google Scholar

[7]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math. (2), 92 (1970), 102. Google Scholar

[8]

J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Comm. Pure Appl. Analysis, 11 (2012), 1407. doi: 10.3934/cpaa.2012.11.1407. Google Scholar

[9]

P. Hartman, "Ordinary Differential Equations,", Wiley, (1964). Google Scholar

[10]

T. Hou and C. Li, On global well-posedness of the Lagrangian averaged Euler equations,, SIAM J. Math. Anal., 38 (2006), 782. doi: 10.1137/050625783. Google Scholar

[11]

S. Lang, "Fundamentals of Differential Geometry,", Springer-Verlag, (1999). doi: 10.1007/978-1-4612-0541-8. Google Scholar

[12]

J. E. Marsden, D. G. Ebin and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity,, Proc. Can. Math. Congress, 1 (1972), 135. Google Scholar

[13]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc., 8 (2006), 1. doi: 10.4171/JEMS/37. Google Scholar

[14]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar

[15]

S. C. Preston, The motion of whips and chains,, J. Diff. Eq., 251 (2011), 504. doi: 10.1016/j.jde.2011.05.005. Google Scholar

[16]

S. C. Preston, The geometry of whips,, Ann. Global Anal. Geom., 41 (2012), 281. doi: 10.1007/s10455-011-9283-z. Google Scholar

[17]

R. Saxton, Existence of solutions for a finite nonlinearly hyperelastic rod,, J. Math. Anal. Appl., 105 (1985), 59. doi: 10.1016/0022-247X(85)90096-4. Google Scholar

[18]

S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337. doi: 10.1006/jfan.1998.3335. Google Scholar

[19]

A. Shnirelman, Generalized fluid flows, their approximation and applications,, Geom. Funct. Anal., 4 (1994), 586. doi: 10.1007/BF01896409. Google Scholar

[20]

A. Thess, O. Zikanov and A. Nepomnyashchy, Finite-time singularity in the vortex dynamics of a string,, Phys. Rev. E, 59 (1999), 3637. doi: 10.1103/PhysRevE.59.3637. Google Scholar

show all references

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. Google Scholar

[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, (). Google Scholar

[3]

L. Biliotti, The exponential map of a weak Riemannian Hilbert manifold,, Illinois J. Math., 48 (2004), 1191. Google Scholar

[4]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002). doi: 10.1088/0305-4470/35/32/201. Google Scholar

[5]

M. P. do Carmo, "Riemannian Geometry,", Birkhäuser, (1992). Google Scholar

[6]

D. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force,, Ann. Math. (2), 105 (1977), 141. Google Scholar

[7]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math. (2), 92 (1970), 102. Google Scholar

[8]

J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Comm. Pure Appl. Analysis, 11 (2012), 1407. doi: 10.3934/cpaa.2012.11.1407. Google Scholar

[9]

P. Hartman, "Ordinary Differential Equations,", Wiley, (1964). Google Scholar

[10]

T. Hou and C. Li, On global well-posedness of the Lagrangian averaged Euler equations,, SIAM J. Math. Anal., 38 (2006), 782. doi: 10.1137/050625783. Google Scholar

[11]

S. Lang, "Fundamentals of Differential Geometry,", Springer-Verlag, (1999). doi: 10.1007/978-1-4612-0541-8. Google Scholar

[12]

J. E. Marsden, D. G. Ebin and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity,, Proc. Can. Math. Congress, 1 (1972), 135. Google Scholar

[13]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc., 8 (2006), 1. doi: 10.4171/JEMS/37. Google Scholar

[14]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar

[15]

S. C. Preston, The motion of whips and chains,, J. Diff. Eq., 251 (2011), 504. doi: 10.1016/j.jde.2011.05.005. Google Scholar

[16]

S. C. Preston, The geometry of whips,, Ann. Global Anal. Geom., 41 (2012), 281. doi: 10.1007/s10455-011-9283-z. Google Scholar

[17]

R. Saxton, Existence of solutions for a finite nonlinearly hyperelastic rod,, J. Math. Anal. Appl., 105 (1985), 59. doi: 10.1016/0022-247X(85)90096-4. Google Scholar

[18]

S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337. doi: 10.1006/jfan.1998.3335. Google Scholar

[19]

A. Shnirelman, Generalized fluid flows, their approximation and applications,, Geom. Funct. Anal., 4 (1994), 586. doi: 10.1007/BF01896409. Google Scholar

[20]

A. Thess, O. Zikanov and A. Nepomnyashchy, Finite-time singularity in the vortex dynamics of a string,, Phys. Rev. E, 59 (1999), 3637. doi: 10.1103/PhysRevE.59.3637. Google Scholar

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