References
R. Abraham & J. Marsden [1978] Foundations of Mechanics, Second Edition, Addison Wesley.
R. Abraham, J. Marsden & T. Ratiu [1983] Manifolds, Tensor Analysis, and Applications, Addison-Wesley, Second Edition, Springer-Verlag, New York, 1988.
S. S. Antman [1972] The Theory of Rods, Handbuch der Physik, Vol. VIa/2, Springer, Berlin.
S. S. Antman [1974] Kirchhoff's Problem for Nonlinearly Elastic Rods, Quart. J. of Appl. Math. 32, 221–240.
S. S. Antman & K. B. Jordan [1975] Qualitative Aspects of the Spatial Deformation of Non-linearly Elastic Rods, Proc. Roy. Soc. Edinburgh Sect. A 73 (5), 85–105.
S. S. Antman [1976] Ordinary Differential Equations of Nonlinear Elasticity I: Foundations of the Theories of Non-linearly Elastic Rods and Shells, Arch. Rational Mech. Anal. 61 (4), 307–351.
S. S. Antman [1978] Buckled States of Nonlinearly Elastic Plates, Arch. Rational Mech. Anal. 67 (2), 111–149.
S. Antman & C. S. Kenney [1981] Large Buckled States of Nonlinearly Elastic Rods Under Torsion, Thrust, and Gravity, Arch. Rational Mech. Anal. 76 (4), 289–338.
S. S. Antman [1984] Large Lateral Buckling of Nonlinearly Elastic Beams, Arch. Rational Mech. Anal. 84 (4), 293–305.
V. Arnold [1966] Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, Grenoble 16, 319–361.
A. Chorin, T.J. R. Hughes, M. F. McCracken & J. E. Marsden [1978], Product formulas and Numerical Algorithms, Comm. Pure. Appl. Math. 31, 205–256.
Y. Choquet-Bruhar, C. Dewitt-Morette & M. Dillard-Bleick [1984] Analysis, Manifolds, and Physics, North Holland, second edition.
P. Ciarlet [1988] Mathematical Elasticity. Volume 1: Three Dimensional Elasticity, Studies in Mathematics and its Applications, North-Holland, Amsterdam
H. Cohen & C. N. DeSilva [1966] Nonlinear Theory of Elastic Directed Surfaces, J. Math. Phys. 7, No. 6, 960–966.
D. Ebin & J. E. Marsden [1970] Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, Ann. Math. 92, 102–163.
J. L. Ericksen & C. Truesdell [1958] Exact Theory of Stress and Strain in Rods and Shells, Arch. Rational Mech. Anal. 1, 295–333.
H. Goldstein [1980] Classical Mechanics, Second Edition, Addison-Wesley.
M. Golubitsky & I. Stewart [1987] Generic Bifurcation of Hamiltonian Systems with Symmetry, Physica 24 D, 391–405.
A. E. Green, P. M. Naghdi & W. L. Wainwright [1965] A General Theory of a Cosserat Surface, Arch. Rational Mech. Anal. 20, 287–308.
A. E. Green & W. Zerna [1968] Theoretical Elasticity, Oxford U. Press.
J. Guckenheimer & P. Holmes [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York.
D. D. Holm & B. A. Kuperschmidt [1983a] Poisson Brackets and Clebsch Representations for Magnetohydrodynamics, Multifluid Plasmas, and Elasticity, Physica 6 D, 347–363.
D. D. Holm, J. E. Marsden & T. Ratiu [1986] The Hamiltonian Structure of Continuum Mechanics in the Material, Inverse Material, Spatial, and Convective Representations, Séminaire de Mathématiques Supérieures, 100, Les Presses de l'Université de Montréal.
D. D. Holm, J. E. Marsden, T. S. Ratiu & A. Weinstein [1985] Nonlinear Stability of Fluid and Plasma Equilibria, Phys. Rep. 123, 1–116.
P. Holmes & J. Marsden [1983] Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups, Indiana Univ. Math. J. 32, 273–310.
T. J. R. Hughes & J. Winget [1980] Finite Rotation Effects in Numerical Integration of Rate Constitutive Equations Arising in Large-Deformation Analysis, Int. J. Num. Meth. Engng. 15, 1862–1867.
T. J. R. Hughes, W. K. Liu & P. Caughy [1978] Transient Finite Element Formulations that Preserve Energy, J. Applied Mechanics 45, 366–370.
Z. R. Iwinski & L. A. Turski [1976] Canonical Theories of Systems Interacting Electromagnetically, Lett. in Appl. and Eng. Sci. 4, 179–191.
P. S. Krishnaprasad [1985] Lie Poisson Structures, Dual Spin Spacecraft and Asymptotic Stability, Nonlinear Analysis, Theory, Methods, and Appl. 9, 1011–1035.
P. S. Krishnaprasad & J. E. Marsden [1987] Hamiltonian Structure and Stability for Rigid Bodies with Flexible Attachments, Arch. Rational Mech. Anal. 98, 71–93.
P. S. Krishnasaprad, J. E. Marsden, T. Posbergh & J. C. Simo [1988] Nonlinear Stability of Coupled Rigid Body, Rod and Plate Structures (in preparation).
D. Lewis, J. E. Marsden, R. Montgomery & T. Ratiu [1986] The Hamiltonian Structure for Dynamic Free Boundary Problems, Physica 18 D, 391–404.
D. Lewis, J. E. Marsden & T. Ratiu [1986a] Formal Stability of Liquid Drops with Surface Tension, in Perspectives in Nonlinear Dynamics, ed. by M. F. Schlessinger and others, World Scientific, 71–83.
D. Lewis, J. E. Marsden & T. Ratiu [1987] Stability and bifurcation of a rotating planar liquid drop (J. Math. Phys.).
A. Libai & J. G. Simmonds [1983] Nonlinear Elastic Shell Theory. Advances in Applied Mechanics, 23, 271–371, edited by J. Hutchinson and T. Wu.
A. E. H. Love [1944] The Mathematical Theory of Elasticity, 4th edition, Dover, New York.
J. E. Marsden [1982] A Group Theoretical Approach to the Equations of Plasma Physics, Canadian Math. Bull. 25, 129–142.
J. E. Marsden [1988] Lie-Poisson Hamilton Jacobi theory and Lie-Poisson Integrators, Phys. Lett. A. (to appear).
J. E. Marsden & T. Hughes [1983] Mathematical Foundations of Elasticity, Prentice Hall.
J. E. Marsden & T. Ratiu [1986] Reduction of Poisson Manifolds, Letters in Math. Phys. 11, 161–169.
J. E. Marsden, T. Ratiu & A. Weinstein [1984a] Semi-direct Products and Reduction in Mechanics, Trans. Am. Math. Soc. 281, 147–177.
J. E. Marsden, T. Ratiu & A. Weinstein [1984b] Reduction and Hamiltonian Structures on Duals of Semidirect Product Lie Algebras, Cont. Math. AMS 28, 55–100.
J. E. Marsden & A. Weinstein [1974] Reduction of Symplectic Manifolds with Symmetry, Rep. Math. Phys. 5, 121–130.
J. E. Marsden & A. Weinstein [1982] The Hamiltonian Structure of the Maxwell Vlasov Equations, Physica D 4, 394–406.
J. E. Marsden & A. Weinstein [1983] Coadjoint Orbits, Vortices and Clebsch Variables for Incompressible Fluids, Physica 8 D, 305–323.
J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid & R. G. Spencer [1983] Hamiltonian Systems with Symmetry, Coadjoint Orbits and Plasma Physics, Proc. IUTAMISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, June 7–11, 1982, Atti della Academia delle Scienze di Torino 117, 289–340.
R. Montgomery, J. Marsden & T. Ratiu [1984] Gauged Lie-Poisson Structures, Contemporary Mathematics, AMS 28, 101–114.
P. J. Morrison & J. M. Greene [1980] Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics, Phys. Rev. Lett. 45, 790–794.
P. M. Naghdi [1972] The Theory of Plates and Shells, in Handbuch der Physik, Vol. VIa/2, Springer, Berlin.
P. M. Naghdi [1980] Finite Deformations of Elastic Rods and Shells, in Proceedings IUTAM Symposium on Finite Elasticity, Lehigh University, Bethlehem.
R. Reissner [1973] On a one-dimensional, large-displacement, finite-strain beam-theory, Stud. Appl. Math. 52, 87–95.
E. Reissner [1981] On Finite Deformations of Space-Curve Beams, ZAMP 32, 734–744.
J. C. Simo & J. E. Marsden [1984] On the Rotated Stress Tensor and the Material Version of the Doyle-Ericksen formula, Arch. Rational Mech. Analysis 86, 213–231.
J. C. Simo [1985] A Finite Strain Beam Formulation. The Three Dimensional Dynamic Problem. Part I. Comp. Meth. Appl. Mech. Engng. 49, 55–70.
J. C. Simo & L. Vu-Quoc [1986a] A Three-Dimensional Finite Strain Rod Model. Part II: Computational Aspects. Comp. Meth. Appl. Mech. Engng. 58, 79–116.
J. C. Simo & L. Vu-Quoc [1986b] On the Dynamics of Flexible Beams Under Large Overall Motions—The plane Case; Parts I and II, J. Appl. Mech. 54, No. 3.
J. C. Simo & L. Vu-Quoc [1987] On the Dynamics in Space of Rods Urdegoing Large Overall Motions, Comp. Meth. Appl. Mech. Engng. (to appear).
J. C. Simo & D. D. Fox [1988] On a Stress Resultant, Geometrically Exact Shell Model. Part I: Formulation and Optimal Parametrization, Comp. Meth. Appl. Mech. Engng. (to appear).
J. C. Simo [1987] On a Fully Three Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects, Comp. Meth. Appl. Mech. Engng. 60, 137–158.
J. C. Simo & L. Vu-Quoc [1986c] The role of Nonlinear Theories in Transient Dynamic Analysis of Flexible Structures, J. Sound and Vibration, (To appear).
J. C. Simo, D. D. Fox & M. S. Rifai, [1988a] On a Stress Resultant Geometrically Exact Shell Model. Part II: The Linear Theory, Computational Aspects, Comp. Meth. Appl. Mech. Engng. (to appear)
J. C. Simo, D. D. Fox & M. S. Rifai, [1988b] On a Stress Resultant Geometrically Exact Shell Model. Part III: Computational Aspects of the Nonlinear Theory, Comp. Meth. Appl. Mech. Engng. (to appear).
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Simo, J.C., Marsden, J.E. & Krishnaprasad, P.S. The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates. Arch. Rational Mech. Anal. 104, 125–183 (1988). https://doi.org/10.1007/BF00251673
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DOI: https://doi.org/10.1007/BF00251673