- Previous Article
- JGM Home
- This Issue
-
Next Article
A weak approach to the stochastic deformation of classical mechanics
Morse theory for elastica
1. | The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia |
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 |
2021 Impact Factor: 0.737
Tools
Metrics
Other articles
by authors
[Back to Top]