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


Neural Network Complex System Nonlinear Dynamics Electromagnetism Relative Equilibrium 
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.

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 ×T ϕ * 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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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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