Abstract
This article provides a survey of contemporary rod mechanics, including both dynamic and static theories. Much of what we discuss is regarded as classic material within the mechanics community, but the objective here is to provide a self-contained account accessible to workers interested in modelling DNA. We also describe a number of recent results and computations involving rod mechanics that have been obtained by our group at the University of Maryland. This work was largely motivated by applications to modelling DNA, but our approach reflects a background of research in continuum mechanics. In particular, we emphasize the role that Hamiltonian formulations and symmetries play in the effective computation of special solutions, conservation laws of dynamics and integrals of statics.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by AFOSR, NSF and ONR
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Antman, S.S. Nonlinear Problems of Elasticity. (Springer-Verlag, New York, 1994).
Bauer, W.R., R.A. Lund & J.H. White. Twist and Writhe of a DNA Loop Containing Intrinsic Bends. Proc. Natl. Acad. Sci. USA 90 (1990) 833–837.
Beliaev, A. & A. Il’ichev. Conditional Stability of Solitary Waves Propagating in Elastic Rods. Physica D 90 (1996) 107.
Benham, C.J. Onset of writhing in circular elastic polymers. Phys. Rev. A 39 (1989) 2582–2586.
Bottema, O. & B. Roth. Theoretical Mechanics (Dover, New York, 1979).
Caflisch, R.E. &: J.H. Maddocks. Nonlinear dynamical theory of the elastica. Proc. Roy. Soc. Edinburgh 99A (1984) 1–23.
Coleman, B.D. & J-M. Xu. On the Interaction of Solitary Waves of Flexure in Elastic Rods. Acta Mechanica 110 (1995) 173–182.
Dichmann, D.J. Hamiltonian Dynamics of an Elastica and Stability of Solitary Waves. Ph. D. thesis, University of Maryland (1994).
Dichmann, D.J., J.H. Maddocks & R.L. Pego. Hamiltonian Dynamics of an Elastica. Arch. Rational Mech. Anal, forthcoming.
Dichmann, D.J., J.H. Maddocks & J-M. Xu. Three-Dimensional Hamiltonian Dynamics of an Elastica and the Stability of Solitary Waves. In preparation.
Dirac, P.A.M. On generalized Hamiltonian dynamics. Can. J. Math. 2 (1950) 129–148.
Doedel, E. AUTO 86 User Manual. (Caltech, Dept. Applied Mathematics, Pasadena)
Domokos, G. A group-theoretic approach to the geometry of elastic rings. J. Nonlinear Science 5 (1995) 453.
Domokos, G. & R. Paffenroth. PCR - A Visualization Tool for Multi-Point Boundary Value Problems. Technical Note BN-1167. (University of Maryland, Institute for Physical Science and Technology, College Park, 1994).
Falk, R.S. & J-M. Xu. Convergence of a Second-Order Scheme for the Nonlinear Dynamical Equations of Elastic Rods. SIAM J. Numerical Analysis 32 (1995).
Goldstein, H. Classical Mechanics, Second Edition (Addison-Wesley, Reading, MA, 1981).
Ilyukhin, A.A. Spatial problems of non-linear theory of elastic rods, (Naukova Dumka, Kiev, 1979). [in Russian]
James, R.D. The equilibrium and post-buckling behaviour of an elastic curve governed by a non-convex energy. J. Elasticity 11 (1981) 239–269.
Jülicher, F. Supercoiling transitions of closed DNA. Phys. Rev. E 49 (1994) 2429–2435.
Kirchhoff, G.R. Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. Gesammelte Abhandlungen (Leipzig, 1882).
Klapper, I. & M. Tabor. Dynamics of Twisted Elastic Rods. Preprint.
Landau, L.D. & E.M. Lifshitz. Theory of Elasticity (Pergamon Press, New York, 1970).
Langer, J. & D. Singer. Lagrangian Aspects of the Kirchhoff Elastic Rod. SIAM Review forthcoming
LeBret, M. Catastrophic Variation of Twist and Writhing of Circular DNAs with Constraint? Biopolymers 18 (1979) 1709–1725.
Li, Y. & J.H. Maddocks. On the Computation of Equilibria of Elastic Rods, Part I: Integrals, Symmetry and a Hamiltonian Formulation. Submitted to J. Comp. Physics
Li, Y. &: J.H. Maddocks. On the Computation of Equilibria of Elastic Rods, part II: Effects of Self-Contact. In preparation.
Li, Y., J.H. Maddocks & D.J. Dichmann. On the Computation of Equilibria of Elastic Rods, part III: Effects of Shear. In preparation.
Love, A.E.H. A Treatise on the Mathematical Theory of Elasticity, (Dover, New York, 1944).
Maddocks, J.H. & D.J. Dichmann. Conservation Laws in the Dynamics of Rods. J. Elasticity 34 (1994) 83–96.
Olver, P.J. Applications of Lie Groups to Differential Equations, (Springer-Verlag, New York, 1986).
Shi, Y., A. Borovik & J.E. Hearst. Elastic Rod Model Incorporating Shear and Extension, Generalized Schrödinger Equations, and Novel Closed-Form Solutions for Supercoiled DNA. J. Chem. Physics 103 3166 (1995)
Shi, Y. & J.E. Hearst. The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling. J. Chem. Phys. 101 (1994) 5186–5200.
Schlick, T. Modeling Superhelical DNA: Recent Analytical and Dynamical Approaches. Current Opinions in Structural Biology, ed. B. Honig. 5 (1995).
Schlick, T., W.K. Olson, T. Westcott & J.P. Greenberg. On Higher Buckling Transitions in Supercoiled DNA. Biopolymers 34 (1994) 565–597.
Shuster, M.D. A Survey of Attitude Representations. J. Astronautical Sciences 41 (1994) 439–518.
Simo, J.C., J.E. Marsden &: P.S. Krishnaprasad. The Hamiltonian Structure of Nonlinear Elasticity: The Material and Convective Representations of Solids, Rods and Plates. Arch. Rational Mech. Anal. 104 (1988) 125–183.
Simo, J.C. & L. Vu-Quoc. A Three-Dimensional Finite-Strain Rod Model. Part II: Computational Aspects. Comput. Meths. Appl. Mech. Engrg. 58 (1986) 79–116.
Starotsin, E.L. Three-Dimensional Conformations of Looped DNA in an Elastome-chanical Approximation. Proc. 2nd International Conference on Nonlinear Mechanics. Ed. Chien Wei-zang. (Peking University Press, Peking, 1993).
Tsuru, H. Equilibrium Shapes and Vibrations of Thin Elastic Rods. J. Phys. Soc. Japan 56 (1987) 2309–2324.
Wadati, M. & H. Tsuru. Elastic Model of Looped DNA. Physica D 21 (1986) 213–226.
Yang, Y., I. Tobias & W.K. Olson. Finite Element Analysis of DNA Supercoiling. J. Chem. Phys. 98 (1993) 1673–1686.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Dichmann, D.J., Li, Y., Maddocks, J.H. (1996). Hamiltonian Formulations and Symmetries in Rod Mechanics. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds) Mathematical Approaches to Biomolecular Structure and Dynamics. The IMA Volumes in Mathematics and its Applications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4066-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4066-2_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94838-6
Online ISBN: 978-1-4612-4066-2
eBook Packages: Springer Book Archive