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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
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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
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DOI: https://doi.org/10.1007/BF01881679