Archive for Rational Mechanics and Analysis

, Volume 115, Issue 1, pp 61–100

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

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

## Keywords

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

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 inTϕQ ×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

ηQ(ϕ)

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

A(ϕ)

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

J:Pso*(3)

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)

£db

Lie derivative ofb in directiona

ϱrefB(ϕ)

Configuration dependent body force with potentialL: Q → ℝ

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## 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