# American Institute of Mathematical Sciences

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

## Morse theory for elastica

 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.
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, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0. [2] J. Arroyo, O. J. Garay and J. Mencía, Elastic circles in 2-spheres, J. Phys. A, 39 (2006), 2307-2324. [3] R. Brockett, Finite Dimensional Linear Systems, Series in decision and control, Wiley, 1970. doi: 10.1137/1.9781611973884. [4] R. Bryant and P. Griffiths, Reduction for constrained variational problems and $\int1/2k^2 ds$, Amer. J. Math., 108 (1986), 525-570. doi: 10.2307/2374654. [5] M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geom. Appl., 15 (2001), 107-135. doi: 10.1016/S0926-2245(01)00054-7. [6] H. I. Elíasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194. [7] H. I. Elíasson, Variation integrals in fiber bundles, in Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 67-89. [8] H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds, Bull. Amer. Math. Soc., 77 (1971), 1002-1005. doi: 10.1090/S0002-9904-1971-12836-7. [9] H. I. Elíasson, Introduction to global calculus of variations, in Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II, Internat. Atomic Energy Agency, Vienna, 1974, 113-135. [10] M. Golomb and J. Jerome, Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves, SIAM J. Math. Anal., 13 (1982), 421-458. doi: 10.1137/0513031. [11] V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math., 117 (1995), 93-124. doi: 10.2307/2375037. [12] V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem. Amer. Math. Soc., 178 (2005), viii+133pp. doi: 10.1090/memo/0838. [13] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, 1978, Grundlehren der Mathematischen Wissenschaften, Vol. 230. [14] J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), 1-22. [15] J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology, 24 (1985), 75-88. doi: 10.1016/0040-9383(85)90027-8. [16] J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds, Ann. Global Anal. Geom., 5 (1987), 133-150. doi: 10.1007/BF00127856. [17] E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves, SIAM Rev., 15 (1973), 120-133. doi: 10.1137/1015004. [18] R. Levien, The Elastica: A Mathematical History, Technical Report UCB/EECS-2008-103, EECS Department, University of California, Berkeley, 2008, http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html. [19] A. Linnér, Existence of free nonclosed Euler-Bernoulli elastica, Nonlinear Anal., 21 (1993), 575-593. doi: 10.1016/0362-546X(93)90002-A. [20] A. Linnér, Unified representations of nonlinear splines, J. Approx. Theory, 84 (1996), 315-350. doi: 10.1006/jath.1996.0022. [21] A. Linnér, Curve-straightening and the Palais-Smale condition, Trans. Amer. Math. Soc., 350 (1998), 3743-3765. doi: 10.1090/S0002-9947-98-01977-1. [22] A. Linnér, Periodic geodesics generator, Experiment. Math., 13 (2004), 199-206, http://projecteuclid.org/getRecord?id=euclid.em/1090350934. doi: 10.1080/10586458.2004.10504533. [23] D. Mumford, Elastica and computer vision, in Algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, 1994, 491-506. [24] V. V. Nesterenko, A. Feoli and G. Scarpetta, Dynamics of relativistic particles with Lagrangians dependent on acceleration, J. Math. Phys., 36 (1995), 5552-5564. doi: 10.1063/1.531332. [25] R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, vol. 1353 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. [26] T. Popiel and L. Noakes, Elastica in $SO(3)$, J. Aust. Math. Soc., 83 (2007), 105-124. doi: 10.1017/S1446788700036417. [27] P. Schrader, Existence of variationally defined curves with higher order elliptic Lagrangians, Nonlinear Anal., 115 (2015), 1-11. doi: 10.1016/j.na.2014.11.016. [28] C. Truesdell, The influence of elasticity on analysis: The classic heritage, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 293-310. doi: 10.1090/S0273-0979-1983-15187-X. [29] K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geometry, 8 (1973), 555-564.

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, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0. [2] J. Arroyo, O. J. Garay and J. Mencía, Elastic circles in 2-spheres, J. Phys. A, 39 (2006), 2307-2324. [3] R. Brockett, Finite Dimensional Linear Systems, Series in decision and control, Wiley, 1970. doi: 10.1137/1.9781611973884. [4] R. Bryant and P. Griffiths, Reduction for constrained variational problems and $\int1/2k^2 ds$, Amer. J. Math., 108 (1986), 525-570. doi: 10.2307/2374654. [5] M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian cubic polynomials, Differential Geom. Appl., 15 (2001), 107-135. doi: 10.1016/S0926-2245(01)00054-7. [6] H. I. Elíasson, Geometry of manifolds of maps, J. Differential Geometry, 1 (1967), 169-194. [7] H. I. Elíasson, Variation integrals in fiber bundles, in Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 67-89. [8] H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds, Bull. Amer. Math. Soc., 77 (1971), 1002-1005. doi: 10.1090/S0002-9904-1971-12836-7. [9] H. I. Elíasson, Introduction to global calculus of variations, in Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II, Internat. Atomic Energy Agency, Vienna, 1974, 113-135. [10] M. Golomb and J. Jerome, Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves, SIAM J. Math. Anal., 13 (1982), 421-458. doi: 10.1137/0513031. [11] V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math., 117 (1995), 93-124. doi: 10.2307/2375037. [12] V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem. Amer. Math. Soc., 178 (2005), viii+133pp. doi: 10.1090/memo/0838. [13] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, 1978, Grundlehren der Mathematischen Wissenschaften, Vol. 230. [14] J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), 1-22. [15] J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology, 24 (1985), 75-88. doi: 10.1016/0040-9383(85)90027-8. [16] J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds, Ann. Global Anal. Geom., 5 (1987), 133-150. doi: 10.1007/BF00127856. [17] E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves, SIAM Rev., 15 (1973), 120-133. doi: 10.1137/1015004. [18] R. Levien, The Elastica: A Mathematical History, Technical Report UCB/EECS-2008-103, EECS Department, University of California, Berkeley, 2008, http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html. [19] A. Linnér, Existence of free nonclosed Euler-Bernoulli elastica, Nonlinear Anal., 21 (1993), 575-593. doi: 10.1016/0362-546X(93)90002-A. [20] A. Linnér, Unified representations of nonlinear splines, J. Approx. Theory, 84 (1996), 315-350. doi: 10.1006/jath.1996.0022. [21] A. Linnér, Curve-straightening and the Palais-Smale condition, Trans. Amer. Math. Soc., 350 (1998), 3743-3765. doi: 10.1090/S0002-9947-98-01977-1. [22] A. Linnér, Periodic geodesics generator, Experiment. Math., 13 (2004), 199-206, http://projecteuclid.org/getRecord?id=euclid.em/1090350934. doi: 10.1080/10586458.2004.10504533. [23] D. Mumford, Elastica and computer vision, in Algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, 1994, 491-506. [24] V. V. Nesterenko, A. Feoli and G. Scarpetta, Dynamics of relativistic particles with Lagrangians dependent on acceleration, J. Math. Phys., 36 (1995), 5552-5564. doi: 10.1063/1.531332. [25] R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, vol. 1353 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. [26] T. Popiel and L. Noakes, Elastica in $SO(3)$, J. Aust. Math. Soc., 83 (2007), 105-124. doi: 10.1017/S1446788700036417. [27] P. Schrader, Existence of variationally defined curves with higher order elliptic Lagrangians, Nonlinear Anal., 115 (2015), 1-11. doi: 10.1016/j.na.2014.11.016. [28] C. Truesdell, The influence of elasticity on analysis: The classic heritage, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 293-310. doi: 10.1090/S0273-0979-1983-15187-X. [29] K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geometry, 8 (1973), 555-564.
 [1] Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829 [2] Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17 [3] A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315 [12] Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control and Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45 [13] Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks and Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723 [14] Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675 [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 and Continuous Dynamical Systems, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152 [16] Mohammad Akil, Ibtissam Issa, Ali Wehbe. Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021059 [17] Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961 [18] Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473 [19] 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 [20] 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

2020 Impact Factor: 0.857