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Spatially complex equilibria of buckled rods. (English) Zbl 0655.73029

The authors explore in a very innovative way the well known Kirchhoff analogy of the mathematical structure of the equations of a rod loaded at its end and the equations of a heavy rigid body pivoted at a fixed point. In this analogy the arc length along the axis of the rod plays the role of a time like coordinate. By emphasizing the Hamiltonian structure of the rod equations, from the existence of transverse homoclinic and heteroclinic orbits the existence of spatially chaotic equilibrium states of the rod are concluded.
From a physical (experimental) point of view, probably the problem is more complicated because rod problems are boundary value problems and hence chaotic deformations will lead to points of contact of the different parts of the rod.
Reviewer: H.Troger

MSC:

74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] R. Abraham & J. E. Marsden [1978] Foundations of Mechanics, 2nd edition, Benjamin/Cummings, Reading, MA. · Zbl 0393.70001
[2] S. S. Antman [1984] Large lateral buckling of nonlinearly elastic beams, Ar · Zbl 0533.73042
[3] S. S. Antman & C. S. Kenney [1981] Large buckled states of nonlinearly elastic rods under torsion, thrust and gravity, Ar · Zbl 0472.73036
[4] V. I. Arnold [1964] Instability of dynamical systems with several degrees of f
[5] V. I. Arnold [1978] Mathematical Methods of Classical Mechanics Springer Verlag, N.Y., Heidelberg, Berlin (Russian original, Moscow, 1974).
[6] G. D. Birkhoff [1927] Dynamical Systems, A.M.S. Publications, Providence, R.I.
[7] E. Buzano, G. Geymonat, & T. Poston [1985] Post-buckling behavior of a nonlinearly hyperelastic thin rod with cross section invariant un · Zbl 0568.73048
[8] L. Euler [1744] Additamentum I de Curvis Elasticis, Methodus Inveniendi Lineas Curvas Maximi Minimivi Proprietate Gaudentes, La
[9] B. D. Greenspan & P. J. Holmes [1983] Homoclinic orbits, subharmonics, and global bifurcations in forced oscillations, in Nonlinear Dynamics and Turbulence (editors: B. Barenblatt, G. Iooss & D. D. Joseph) Pitman, London. Chapter 10, pp. 172–214. · Zbl 0532.58019
[10] H. Goldstein [1980] Classical Mechanics, Addison-Wesley, Reading, MA.
[11] J. Guckenheimer & P. J. Holmes [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, New York (corrected second printing, 1986). · Zbl 0515.34001
[12] P. J. Holmes & J. E. Marsden [1982] Horseshoes in perturbations of Hamiltonian systems with two degrees of fr · Zbl 0489.58013
[13] P. J. Holmes & J. E. Marsden [1983] Horseshoes and Arnold diffusion for Hamiltonian system on Lie groups, Ind · Zbl 0488.70006
[14] G. Kirchhoff [1859] Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, Journal · ERAM 056.1494cj
[15] P. S. Krishnaprasad, J. E. Marsden & J. C. Simo [1986], The Hamiltonian structure of nonlinear elasticity: the convective representation of solids, rods and plates, (preprint). · Zbl 0668.73014
[16] J. Larmor [1884] On the direct application of the principle of least action to the dynamics of solid and fluid systems, and analogous elastic problems, Pr · JFM 16.0755.03
[17] A. E. H. Love [1927] A Treatise on the Mathematical Theory of Elasticity Cambridge University Press (4th edition).
[18] V. K. Melnikov [1963] On the stability of the center for time period perturbations,
[19] A. Mielke [1987] Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity, MSI Tech. Rep. ’87-25, Arch. Rational Mech. Anal., to appear.
[20] J. Moser [1973] Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, N.J.
[21] S. Smale [1963] Diffeomorphisms with many periodic points, in Differential and Combinational Topology, ed. S. S. Cairns, pp. 63–80, Princeton University Press, Princeton, N.J.
[22] S. Smale [1967] Differentiable Dynamical Systems · Zbl 0202.55202
[23] E. T. Whittaker [1937] A treatise on the Analytical Dynamics of Particles and Rigid Bodies (4th edition) Cambridge University Press, Cambridge. · JFM 63.1286.03
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