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

DOI: 10.1007/BF01881679

Cite this article as:
Simo, J.C., Posbergh, T.A. & Marsden, J.E. Arch. Rational Mech. Anal. (1991) 115: 61. doi:10.1007/BF01881679

### 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 ×Tϕ*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

## 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