# 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, (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

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##### 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
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