Article PDF
Avoid common mistakes on your manuscript.
Abbreviations
- Q = Emb+(ℬ, ℝ3):
-
Configuration Space, with elements denoted byϕ ∈Q
- TQ :
-
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 inT ϕ Q ×T *ϕ P
- SO(3):
-
Special orthogonal group; orthogonal 3 × 3 matrices with determinant 1
- so(3):
-
Lie algebra of SO(3); 3 × 3 skew symmetric matrices
- η Q(ϕ):
-
Infinitesimal generator;η Q =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
- A(ϕ):
-
First elasticity tensor; defined as\(A(\varphi ) = \left. {\frac{{\partial ^2 W}}{{\partial F\partial F}}} \right|_{F = D\varphi } \)
- J:P →so *(3):
-
Angular momentum map;J(ϕ, p)· n = < 〈p,η Q(ϕ)〉
- K:P → ℝ:
-
Kinetic energy
- V:Q → ℝ:
-
Potential energy
- H:P → ℝ:
-
Hamiltonian function;H=K + V
- H ξ:P × ℝ3→ ℝ:
-
Energy-momentum functional (Routhian)
- £d b :
-
Lie derivative ofb in directiona
- ϱ ref B(ϕ):
-
Configuration dependent body force with potentialL: Q → ℝ
References
R. Abraham &J. E. Marsden [1978],Foundations of Mechanics, Second Edition, Addison-Wesley, Reading.
R. Abraham, J. E. Marsden &T. S. Ratiu [1988],Manifolds, Tensor Analysis and Applications, Second Edition, Springer, New York.
V. I. Arnold [1966a], An a priori estimate in the theory of hydrodynamic stability,Izv. Vyssh. Uchebn. Zaved. Matematicka 54, 3–5 (Russian).
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.
V. I. Arnold [1978],Mathematical Methods of Classical Mechanics, Springer, Berlin.
V. I. Arnold [1988],Encyclopedia of Dynamical Systems III, Springer, Berlin.
J. M. Ball [1977], Convexity conditions and existence theorems in nonlinear elasticity,Arch. Rational Mech. Anal. 63, 337–403.
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.
S. Chandrasekhar [1977],Ellipsoidal Figures of Equilibrium, Dover, New York.
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.
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.
P. G. Ciarlet [1978],The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.
P. G. Ciarlet [1988],Mathematical Elasticity, North-Holland, Amsterdam.
K. F. Graff [1975],Wave Motion in Elastic Solids, Ohio State University Press, Dayton.
G. H. Golub &C. F. van Loan [1989],Matrix Computations, Second Edition, The John Hopkins University Press, Baltimore.
D. D. Holm, J. E. Marsden, T. Ratiu &A. Weinstein [1985], Nonlinear stability of fluid and plasma equilibria,Physics Reports 123, 1–116.
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.
H. Goldstein [1981],Classical Mechanics, Addison-Wesley, Reading.
P. S. Krishnaprasad &J. E. Marsden [1987], Hamiltonian structure and stability for rigid bodies with flexible attachements,Arch. Rational Mech. Anal. 98, 71–93.
D. Lewis [1989], Nonlinear stability of a rotating liquid drop,Arch. Rational Mech. Anal. 106, 287–333.
D. Lewis &J. C. Simo [1990], Nonlinear stability of pseudo-rigid bodies,Proc. Royal Society London A 427, 281–319.
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.
J. E. Marsden &A. Weinstein [1974], Reduction of symplectic manifolds with symmetry,Rep. Math. Phys. 5, 121–130.
J. E. Marsden &T. J. R. Hughes [1983],Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs.
J. E. Marsden &T. Ratiu [1986], Reduction of Poisson manifolds,Letters in Math. Physics 11, 161–169.
J. E. Marsden, R. Montgomery &T. Ratiu [1990],Symmetry, Reduction and Geometric Phases in Mechanics, Memoirs of the AMS.
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.
L. A. Pars [1965],A Treatise on Analytical Dynamics, Wiley, New York.
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.
B. Riemann [1860], Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides,Abh. d. Königl. Gesell. der Wiss. zu Göttingen 9, 3–36.
E. J. Routh [1877],A Treatise on the Stability of a Given State of Motion, MacMillan, London.
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.
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.
S. S. Smale [1970a], Topology and mechanics, Part I,Inventions Math. 10, 161–169.
S. S. Smale [1970b], Topology and mechanics, Part II,Inventions Math. 11, 45–64.
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.
Author information
Authors and Affiliations
Additional information
Communicated by P.Holmes
Rights and permissions
About this article
Cite this article
Simo, J.C., Posbergh, T.A. & Marsden, J.E. Stability of relative equilibria. Part II: Application to nonlinear elasticity. Arch. Rational Mech. Anal. 115, 61–100 (1991). https://doi.org/10.1007/BF01881679
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01881679