Archive for Rational Mechanics and Analysis

, Volume 115, Issue 1, pp 61–100 | Cite as

Stability of relative equilibria. Part II: Application to nonlinear elasticity

  • J. C. Simo
  • T. A. Posbergh
  • J. E. Marsden

Glossary: Summary of notation employed for elasticity

Q = Emb+(ℬ, ℝ3)

Configuration Space, with elements denoted byϕQ


State Space; points in the state space correspond to configurations and velocities and are denoted by\((\varphi ,\dot \varphi )\)

P =T*Q

Phase Space; points inP correspond to configurations and momenta and are denoted by z = (ϕ, p)

(δϕ, δp)

Configuration-momentum variations inTϕQ ×Tϕ*P


Special orthogonal group; orthogonal 3 × 3 matrices with determinant 1


Lie algebra of SO(3); 3 × 3 skew symmetric matrices


Infinitesimal generator;ηQ =η × ϕ

〈·, ·〉g

Riemannian metric; for elasticity the inner product\(\left\langle {\delta \varphi _1 ,\delta \varphi _2 } \right\rangle _g = \int\limits_B {\rho _{ref} \delta \varphi _1 \cdot \delta \varphi _2 dV} \).

Locked inertia tensor; defined as


First elasticity tensor; defined as\(A(\varphi ) = \left. {\frac{{\partial ^2 W}}{{\partial F\partial F}}} \right|_{F = D\varphi } \)


Angular momentum map;J(ϕ, p)· n = < 〈p,ηQ(ϕ)〉

K:P → ℝ

Kinetic energy

V:Q → ℝ

Potential energy

H:P → ℝ

Hamiltonian function;H=K + V

Hξ:P × ℝ3→ ℝ

Energy-momentum functional (Routhian)


Lie derivative ofb in directiona


Configuration dependent body force with potentialL: Q → ℝ


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  1. R. Abraham &J. E. Marsden [1978],Foundations of Mechanics, Second Edition, Addison-Wesley, Reading.Google Scholar
  2. R. Abraham, J. E. Marsden &T. S. Ratiu [1988],Manifolds, Tensor Analysis and Applications, Second Edition, Springer, New York.Google Scholar
  3. V. I. Arnold [1966a], An a priori estimate in the theory of hydrodynamic stability,Izv. Vyssh. Uchebn. Zaved. Matematicka 54, 3–5 (Russian).Google Scholar
  4. V. I. Arnold [1966b], Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits,Ann. Inst. Fourier, Grenoble 16, 319–361.Google Scholar
  5. V. I. Arnold [1978],Mathematical Methods of Classical Mechanics, Springer, Berlin.Google Scholar
  6. V. I. Arnold [1988],Encyclopedia of Dynamical Systems III, Springer, Berlin.Google Scholar
  7. J. M. Ball [1977], Convexity conditions and existence theorems in nonlinear elasticity,Arch. Rational Mech. Anal. 63, 337–403.Google Scholar
  8. J. M. Ball &J. E. Marsden [1984], Quasiconvexity at the boundary, possitivity of the second variation and elastic stability,Arch. Rational Mech. Anal. 86, 251–277.CrossRefGoogle Scholar
  9. S. Chandrasekhar [1977],Ellipsoidal Figures of Equilibrium, Dover, New York.Google Scholar
  10. D. R. J. Chillingworth, J. E. Marsden &Y. H. Wan [1983], Symmetry and bifurcation in three-dimensional elasticity. Part II,Archive for Rational Mech. Anal. 83, 363–395.CrossRefGoogle Scholar
  11. P. G. Ciarlet &G. Geymonat [1982], Sur les lois de comportement en élasticité nonlinéaire compressible,C. R. Acad. Sci. Paris Sér. II 295, 423–426.Google Scholar
  12. P. G. Ciarlet [1978],The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.Google Scholar
  13. P. G. Ciarlet [1988],Mathematical Elasticity, North-Holland, Amsterdam.Google Scholar
  14. K. F. Graff [1975],Wave Motion in Elastic Solids, Ohio State University Press, Dayton.Google Scholar
  15. G. H. Golub &C. F. van Loan [1989],Matrix Computations, Second Edition, The John Hopkins University Press, Baltimore.Google Scholar
  16. D. D. Holm, J. E. Marsden, T. Ratiu &A. Weinstein [1985], Nonlinear stability of fluid and plasma equilibria,Physics Reports 123, 1–116.CrossRefGoogle Scholar
  17. T. J. R. Hughes, T. Kato &J. E. Marsden [1977], Well-posed quasilinear hyperbolic systems with applications to nonlinear elastodynamics and general relativity,Arch. Rational Mech. Anal. 63, 273–294.CrossRefGoogle Scholar
  18. H. Goldstein [1981],Classical Mechanics, Addison-Wesley, Reading.Google Scholar
  19. P. S. Krishnaprasad &J. E. Marsden [1987], Hamiltonian structure and stability for rigid bodies with flexible attachements,Arch. Rational Mech. Anal. 98, 71–93.Google Scholar
  20. D. Lewis [1989], Nonlinear stability of a rotating liquid drop,Arch. Rational Mech. Anal. 106, 287–333.Google Scholar
  21. D. Lewis &J. C. Simo [1990], Nonlinear stability of pseudo-rigid bodies,Proc. Royal Society London A 427, 281–319.Google Scholar
  22. D. Lewis, J. E. Marsden, T. S. Ratiu &J. C. Simo [1990], Normalizing-connections and the energy-momentum method, inHamiltonian Systems, Transformation Groups and Spectral Methods, J. Harnad & J. E. Marsden, editors. Les Publications CRM, 207–227.Google Scholar
  23. J. E. Marsden &A. Weinstein [1974], Reduction of symplectic manifolds with symmetry,Rep. Math. Phys. 5, 121–130.CrossRefGoogle Scholar
  24. J. E. Marsden &T. J. R. Hughes [1983],Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs.Google Scholar
  25. J. E. Marsden &T. Ratiu [1986], Reduction of Poisson manifolds,Letters in Math. Physics 11, 161–169.CrossRefGoogle Scholar
  26. J. E. Marsden, R. Montgomery &T. Ratiu [1990],Symmetry, Reduction and Geometric Phases in Mechanics, Memoirs of the AMS. Google Scholar
  27. J. E. Marsden, J. C. Simo, D. Lewis &T. A. Posbergh [1989], A block diagonalization theorem in the Energy-Momentum method, inDynamics and Control of Multibody Systems, edited byJ. E. Marsden,et. al,Contemp. Math., American Math. Soc., Providence.Google Scholar
  28. L. A. Pars [1965],A Treatise on Analytical Dynamics, Wiley, New York.Google Scholar
  29. T. A. Posbergh, J. C. Simo &J. E. Marsden [1989], Stability analysis of a rigid body with attached geometrically nonlinear appendage by the energy-momentum method, inDynamics and Control of Multibody Systems, edited byJ. E. Marsden, et al.,Contemp. Math., American Math. Society, Providence.Google Scholar
  30. B. Riemann [1860], Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides,Abh. d. Königl. Gesell. der Wiss. zu Göttingen 9, 3–36.Google Scholar
  31. E. J. Routh [1877],A Treatise on the Stability of a Given State of Motion, MacMillan, London.Google Scholar
  32. J. C. Simo, J. E. Marsden &P. S. Krishnaprasad [1988], The Hamiltonian structure of elasticity, the material and convective representation of solids, rods and plates,Arch. Rational Mech. Anal. 104, 125–183.CrossRefGoogle Scholar
  33. J. C. Simo, T. A. Posbergh &J. E. Marsden [1990], Stability of coupled rigid bodies and geometrically exact rods: Block diagonalization and the energy-momentum method,Physics Reports 193, 280–360.CrossRefGoogle Scholar
  34. S. S. Smale [1970a], Topology and mechanics, Part I,Inventions Math. 10, 161–169.Google Scholar
  35. S. S. Smale [1970b], Topology and mechanics, Part II,Inventions Math. 11, 45–64.CrossRefGoogle Scholar
  36. E. T. Whittaker [1959],A Treatise on the Dynamics of Particles and Rigid Bodies,with an Introduction to the Problem of Three Bodies, Cambridge, 1904; 4th edition, 1937; Dover edition 1959.Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. C. Simo
    • 1
    • 2
    • 3
  • T. A. Posbergh
    • 1
    • 2
    • 3
  • J. E. Marsden
    • 1
    • 2
    • 3
  1. 1.Division of Applied MechanicsStanford UniversityStanford
  2. 2.Department of Aerospace EngineeringUniversity of MinnesotaMinneapolis
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeley

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