Citation: |
[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. |