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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, (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. Google Scholar |
[3] |
R. Brockett, Finite Dimensional Linear Systems,, Series in decision and control, (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.
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.
doi: 10.1016/S0926-2245(01)00054-7. |
[6] |
H. I. Elíasson, Geometry of manifolds of maps,, J. Differential Geometry, 1 (1967), 169.
|
[7] |
H. I. Elíasson, Variation integrals in fiber bundles,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 67.
|
[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. |
[9] |
H. I. Elíasson, Introduction to global calculus of variations,, in Global analysis and its applications (Lectures, (1972), 113.
|
[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. |
[11] |
V. Jurdjevic, Non-Euclidean elastica,, Amer. J. Math., 117 (1995), 93.
doi: 10.2307/2375037. |
[12] |
V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005).
doi: 10.1090/memo/0838. |
[13] |
W. Klingenberg, Lectures on Closed Geodesics,, Springer-Verlag, (1978).
|
[14] |
J. Langer and D. A. Singer, The total squared curvature of closed curves,, J. Differential Geom., 20 (1984), 1.
|
[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. |
[16] |
J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds,, Ann. Global Anal. Geom., 5 (1987), 133.
doi: 10.1007/BF00127856. |
[17] |
E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves,, SIAM Rev., 15 (1973), 120.
doi: 10.1137/1015004. |
[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. |
[20] |
A. Linnér, Unified representations of nonlinear splines,, J. Approx. Theory, 84 (1996), 315.
doi: 10.1006/jath.1996.0022. |
[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. |
[22] |
A. Linnér, Periodic geodesics generator,, Experiment. Math., 13 (2004), 199.
doi: 10.1080/10586458.2004.10504533. |
[23] |
D. Mumford, Elastica and computer vision,, in Algebraic geometry and its applications (West Lafayette, (1990), 491.
|
[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. |
[25] |
R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, vol. 1353 of Lecture Notes in Mathematics,, Springer-Verlag, (1988).
|
[26] |
T. Popiel and L. Noakes, Elastica in $SO(3)$,, J. Aust. Math. Soc., 83 (2007), 105.
doi: 10.1017/S1446788700036417. |
[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. |
[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. |
[29] |
K. Uhlenbeck, The Morse index theorem in Hilbert space,, J. Differential Geometry, 8 (1973), 555.
|
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, (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. Google Scholar |
[3] |
R. Brockett, Finite Dimensional Linear Systems,, Series in decision and control, (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.
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.
doi: 10.1016/S0926-2245(01)00054-7. |
[6] |
H. I. Elíasson, Geometry of manifolds of maps,, J. Differential Geometry, 1 (1967), 169.
|
[7] |
H. I. Elíasson, Variation integrals in fiber bundles,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 67.
|
[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. |
[9] |
H. I. Elíasson, Introduction to global calculus of variations,, in Global analysis and its applications (Lectures, (1972), 113.
|
[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. |
[11] |
V. Jurdjevic, Non-Euclidean elastica,, Amer. J. Math., 117 (1995), 93.
doi: 10.2307/2375037. |
[12] |
V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005).
doi: 10.1090/memo/0838. |
[13] |
W. Klingenberg, Lectures on Closed Geodesics,, Springer-Verlag, (1978).
|
[14] |
J. Langer and D. A. Singer, The total squared curvature of closed curves,, J. Differential Geom., 20 (1984), 1.
|
[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. |
[16] |
J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds,, Ann. Global Anal. Geom., 5 (1987), 133.
doi: 10.1007/BF00127856. |
[17] |
E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves,, SIAM Rev., 15 (1973), 120.
doi: 10.1137/1015004. |
[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. |
[20] |
A. Linnér, Unified representations of nonlinear splines,, J. Approx. Theory, 84 (1996), 315.
doi: 10.1006/jath.1996.0022. |
[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. |
[22] |
A. Linnér, Periodic geodesics generator,, Experiment. Math., 13 (2004), 199.
doi: 10.1080/10586458.2004.10504533. |
[23] |
D. Mumford, Elastica and computer vision,, in Algebraic geometry and its applications (West Lafayette, (1990), 491.
|
[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. |
[25] |
R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, vol. 1353 of Lecture Notes in Mathematics,, Springer-Verlag, (1988).
|
[26] |
T. Popiel and L. Noakes, Elastica in $SO(3)$,, J. Aust. Math. Soc., 83 (2007), 105.
doi: 10.1017/S1446788700036417. |
[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. |
[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. |
[29] |
K. Uhlenbeck, The Morse index theorem in Hilbert space,, J. Differential Geometry, 8 (1973), 555.
|
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