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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, 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.

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