American Institute of Mathematical Sciences

Advanced
June  2016, 8(2): 235-256. doi: 10.3934/jgm.2016006

Morse theory for elastica

Philip Schrader 1,

1. 

The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia

Received  December 2013 Revised  March 2016 Published  June 2016

In Riemannian manifolds the elastica are critical points of the restriction of total squared geodesic curvature to curves with fixed length which satisfy first order boundary conditions. We verify that the Palais-Smale condition holds for this variational problem, and also the related problems where the admissible curves are required to satisfy zeroth order boundary conditions, or first order periodicity conditions. We also prove a Morse index theorem for elastica and use the Morse inequalities to give lower bounds for the number of elastica of each index in terms of the Betti numbers of the path space.
Keywords: Palais-Smale condition, Euler-Bernoulli elastica, nonholonomic constraints, calculus of variations, curve straightening..
Mathematics Subject Classification: Primary: 58E05, 58E50; Secondary: 49J4.
Citation: Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences,, 2nd edition, (1988).  doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

J. Arroyo, O. J. Garay and J. Mencía, Elastic circles in 2-spheres,, J. Phys. A, 39 (2006), 2307.   Google Scholar

[3]

R. Brockett, Finite Dimensional Linear Systems,, Series in decision and control, (1970).  doi: 10.1137/1.9781611973884.  Google Scholar

[4]

R. Bryant and P. Griffiths, Reduction for constrained variational problems and $\int1/2k^2 ds$,, Amer. J. Math., 108 (1986), 525.  doi: 10.2307/2374654.  Google Scholar

[5]

M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials,, Differential Geom. Appl., 15 (2001), 107.  doi: 10.1016/S0926-2245(01)00054-7.  Google Scholar

[6]

H. I. Elíasson, Geometry of manifolds of maps,, J. Differential Geometry, 1 (1967), 169.   Google Scholar

[7]

H. I. Elíasson, Variation integrals in fiber bundles,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 67.   Google Scholar

[8]

H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds,, Bull. Amer. Math. Soc., 77 (1971), 1002.  doi: 10.1090/S0002-9904-1971-12836-7.  Google Scholar

[9]

H. I. Elíasson, Introduction to global calculus of variations,, in Global analysis and its applications (Lectures, (1972), 113.   Google Scholar

[10]

M. Golomb and J. Jerome, Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves,, SIAM J. Math. Anal., 13 (1982), 421.  doi: 10.1137/0513031.  Google Scholar

[11]

V. Jurdjevic, Non-Euclidean elastica,, Amer. J. Math., 117 (1995), 93.  doi: 10.2307/2375037.  Google Scholar

[12]

V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005).  doi: 10.1090/memo/0838.  Google Scholar

[13]

W. Klingenberg, Lectures on Closed Geodesics,, Springer-Verlag, (1978).   Google Scholar

[14]

J. Langer and D. A. Singer, The total squared curvature of closed curves,, J. Differential Geom., 20 (1984), 1.   Google Scholar

[15]

J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves,, Topology, 24 (1985), 75.  doi: 10.1016/0040-9383(85)90027-8.  Google Scholar

[16]

J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds,, Ann. Global Anal. Geom., 5 (1987), 133.  doi: 10.1007/BF00127856.  Google Scholar

[17]

E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves,, SIAM Rev., 15 (1973), 120.  doi: 10.1137/1015004.  Google Scholar

[18]

R. Levien, The Elastica: A Mathematical History,, Technical Report UCB/EECS-2008-103, (2008), 2008.   Google Scholar

[19]

A. Linnér, Existence of free nonclosed Euler-Bernoulli elastica,, Nonlinear Anal., 21 (1993), 575.  doi: 10.1016/0362-546X(93)90002-A.  Google Scholar

[20]

A. Linnér, Unified representations of nonlinear splines,, J. Approx. Theory, 84 (1996), 315.  doi: 10.1006/jath.1996.0022.  Google Scholar

[21]

A. Linnér, Curve-straightening and the Palais-Smale condition,, Trans. Amer. Math. Soc., 350 (1998), 3743.  doi: 10.1090/S0002-9947-98-01977-1.  Google Scholar

[22]

A. Linnér, Periodic geodesics generator,, Experiment. Math., 13 (2004), 199.  doi: 10.1080/10586458.2004.10504533.  Google Scholar

[23]

D. Mumford, Elastica and computer vision,, in Algebraic geometry and its applications (West Lafayette, (1990), 491.   Google Scholar

[24]

V. V. Nesterenko, A. Feoli and G. Scarpetta, Dynamics of relativistic particles with Lagrangians dependent on acceleration,, J. Math. Phys., 36 (1995), 5552.  doi: 10.1063/1.531332.  Google Scholar

[25]

R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, vol. 1353 of Lecture Notes in Mathematics,, Springer-Verlag, (1988).   Google Scholar

[26]

T. Popiel and L. Noakes, Elastica in $SO(3)$,, J. Aust. Math. Soc., 83 (2007), 105.  doi: 10.1017/S1446788700036417.  Google Scholar

[27]

P. Schrader, Existence of variationally defined curves with higher order elliptic Lagrangians,, Nonlinear Anal., 115 (2015), 1.  doi: 10.1016/j.na.2014.11.016.  Google Scholar

[28]

C. Truesdell, The influence of elasticity on analysis: The classic heritage,, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 293.  doi: 10.1090/S0273-0979-1983-15187-X.  Google Scholar

[29]

K. Uhlenbeck, The Morse index theorem in Hilbert space,, J. Differential Geometry, 8 (1973), 555.   Google Scholar

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences,, 2nd edition, (1988).  doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

J. Arroyo, O. J. Garay and J. Mencía, Elastic circles in 2-spheres,, J. Phys. A, 39 (2006), 2307.   Google Scholar

[3]

R. Brockett, Finite Dimensional Linear Systems,, Series in decision and control, (1970).  doi: 10.1137/1.9781611973884.  Google Scholar

[4]

R. Bryant and P. Griffiths, Reduction for constrained variational problems and $\int1/2k^2 ds$,, Amer. J. Math., 108 (1986), 525.  doi: 10.2307/2374654.  Google Scholar

[5]

M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials,, Differential Geom. Appl., 15 (2001), 107.  doi: 10.1016/S0926-2245(01)00054-7.  Google Scholar

[6]

H. I. Elíasson, Geometry of manifolds of maps,, J. Differential Geometry, 1 (1967), 169.   Google Scholar

[7]

H. I. Elíasson, Variation integrals in fiber bundles,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 67.   Google Scholar

[8]

H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds,, Bull. Amer. Math. Soc., 77 (1971), 1002.  doi: 10.1090/S0002-9904-1971-12836-7.  Google Scholar

[9]

H. I. Elíasson, Introduction to global calculus of variations,, in Global analysis and its applications (Lectures, (1972), 113.   Google Scholar

[10]

M. Golomb and J. Jerome, Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves,, SIAM J. Math. Anal., 13 (1982), 421.  doi: 10.1137/0513031.  Google Scholar

[11]

V. Jurdjevic, Non-Euclidean elastica,, Amer. J. Math., 117 (1995), 93.  doi: 10.2307/2375037.  Google Scholar

[12]

V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005).  doi: 10.1090/memo/0838.  Google Scholar

[13]

W. Klingenberg, Lectures on Closed Geodesics,, Springer-Verlag, (1978).   Google Scholar

[14]

J. Langer and D. A. Singer, The total squared curvature of closed curves,, J. Differential Geom., 20 (1984), 1.   Google Scholar

[15]

J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves,, Topology, 24 (1985), 75.  doi: 10.1016/0040-9383(85)90027-8.  Google Scholar

[16]

J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds,, Ann. Global Anal. Geom., 5 (1987), 133.  doi: 10.1007/BF00127856.  Google Scholar

[17]

E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves,, SIAM Rev., 15 (1973), 120.  doi: 10.1137/1015004.  Google Scholar

[18]

R. Levien, The Elastica: A Mathematical History,, Technical Report UCB/EECS-2008-103, (2008), 2008.   Google Scholar

[19]

A. Linnér, Existence of free nonclosed Euler-Bernoulli elastica,, Nonlinear Anal., 21 (1993), 575.  doi: 10.1016/0362-546X(93)90002-A.  Google Scholar

[20]

A. Linnér, Unified representations of nonlinear splines,, J. Approx. Theory, 84 (1996), 315.  doi: 10.1006/jath.1996.0022.  Google Scholar

[21]

A. Linnér, Curve-straightening and the Palais-Smale condition,, Trans. Amer. Math. Soc., 350 (1998), 3743.  doi: 10.1090/S0002-9947-98-01977-1.  Google Scholar

[22]

A. Linnér, Periodic geodesics generator,, Experiment. Math., 13 (2004), 199.  doi: 10.1080/10586458.2004.10504533.  Google Scholar

[23]

D. Mumford, Elastica and computer vision,, in Algebraic geometry and its applications (West Lafayette, (1990), 491.   Google Scholar

[24]

V. V. Nesterenko, A. Feoli and G. Scarpetta, Dynamics of relativistic particles with Lagrangians dependent on acceleration,, J. Math. Phys., 36 (1995), 5552.  doi: 10.1063/1.531332.  Google Scholar

[25]

R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, vol. 1353 of Lecture Notes in Mathematics,, Springer-Verlag, (1988).   Google Scholar

[26]

T. Popiel and L. Noakes, Elastica in $SO(3)$,, J. Aust. Math. Soc., 83 (2007), 105.  doi: 10.1017/S1446788700036417.  Google Scholar

[27]

P. Schrader, Existence of variationally defined curves with higher order elliptic Lagrangians,, Nonlinear Anal., 115 (2015), 1.  doi: 10.1016/j.na.2014.11.016.  Google Scholar

[28]

C. Truesdell, The influence of elasticity on analysis: The classic heritage,, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 293.  doi: 10.1090/S0273-0979-1983-15187-X.  Google Scholar

[29]

K. Uhlenbeck, The Morse index theorem in Hilbert space,, J. Differential Geometry, 8 (1973), 555.   Google Scholar

[1]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
[2]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829
[3]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987
[4]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709
[5]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425
[6]

Ammar Khemmoudj, Imane Djaidja. General decay for a viscoelastic rotating Euler-Bernoulli beam. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3531-3557. doi: 10.3934/cpaa.2020154
[7]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[8]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[9]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[10]

Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure & Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785
[11]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315
[12]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675
[13]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[14]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723
[15]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152
[16]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961
[17]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473
[18]

Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
[19]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[20]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences