Linearized Dynamics of Symmetric Lagrangian Systems
The structure induced by the Lagrangian or Hamiltonian character of a mechanical system and its symmetries can make explicit analyses of the stability and bifurcation behavior tractable even for moderately large systems. The variational characterization of relative equilibria, i.e. steady motions generated by elements of the symmetry group, greatly simplifies many of the necessary calculations. Local minima modulo symmetries of an appropriate energy-momentum functional correspond to nonlinearly orbitally stable motions; if the second variation of the energy-momentum functional is positive (negative) definite on an appropriate subspace, the relative equilibrium is said to be formally stable. For finite dimensional systems, formal stability typically implies nonlinear orbital stability.
KeywordsTangent Space Internal Variation Relative Equilibrium Nonlinear Stability Reference Configuration
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