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Stability of relative equilibria. II: Application to nonlinear elasticity. (English) Zbl 0738.70011

One of the authors, J. E. Marsden, is a pioneer in the investigations of Hamiltonian systems with symmetries given by Lie groups \(G\) acting on symplectic manifolds [see e.g. J. E. Marsden and A. Weinstein, Rep. Math. Phys. 5, 121–130 (1974; Zbl 0327.58005)]. These techniques were applied by Marsden and other authors to several fields of mathematical physics, see e.g. the applications to three different models of plasma physics [J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid, and R. G. Spencer, Modern developments in analytical mechanics, Vol. I: Geometrical dynamics, Proc. IUTAM-ISIMM Symp., Torino/Italy 1982, 289–340 (1983; Zbl 0577.58013)]. In the two present papers the main goal is a new, explicit, and readily understood analysis of the stability of the relative equilibria of Hamiltonian systems with symmetry on a cotangent bundle \(T^*Q\); one calls \(z_ e\) is a relative equilibrium if the trajectory of the Hamilton’s equations through \(z_ e\) coincides with the orbit through \(z_ e\)of the one-dimensional subgroup \(\exp[\varepsilon \xi_ e]\) generated by some vector \(\xi_ e\) belonging to the algebra \({\mathcal G}\) of the Lie group of symmetry \(G\).
The approach presented in Part I and Part II constitutes an improvement over former techniques, in the sense that it involves only the configuration space \(Q\), and not the full phase space \(T^*Q\); the authors restrict their attention to systems for which the symplectic action \(\Psi_ g\) of a Lie group \(G\) on \(T^*Q\), \(\Psi_ g: T^*Q\to T^*Q\), \(g\in G\), is the cotangent lift of an action on the configuration space \(Q\). The authors consider the papers by V. Arnol’d [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)] and S. Smale [Invent. Math. 10, 305–331 (1970; Zbl 0202.23201)] as the starting points of the study of the stability of relative equilibria.
This earlier analysis of stability, later called the “energy-Casimir method”, is here improved to the so-called “reduced energy-momentum method”; the key idea is a representation of the Hamiltonian function in terms of the momentum mapping, by using the crucial notion of “locked inertia” tensor. This allows the authors to by-pass certain intrinsic difficulties of the former theory; for example, the method automatically enforces the constraint of constant momentum mapping without resort to Lagrange multipliers; furthermore, the second variation of the augmented kinetic energy can be precisely estimated, leading to sharp stability conditions for relative equilibria. An application to homogeneous elasticity (i.e. affine deformations) is explicitly worked out.
In part II, the authors illustrate in detail the formulation and application of their method in the specific context of classical three- dimensional elasticity. However, in contrast with the finite-dimensional case, here the stability results are only formal (note that “formal stability” is a well defined notion in these papers); but this fact is due to well known difficulties in the general theory of elasticity, and, in general, in the treatment of Lyapunov stability for infinite-dimensional dynamical systems.
Reviewer: Franco Cardin

MSC:

70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
74B20 Nonlinear elasticity
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