127,617 research outputs found
Stability of Zeno Equilibria in Lagrangian Hybrid Systems
This paper presents both necessary and sufficient
conditions for the stability of Zeno equilibria in Lagrangian hybrid systems, i.e., hybrid systems modeling mechanical systems undergoing impacts. These conditions for stability are motivated by the sufficient conditions for Zeno behavior in Lagrangian hybrid systems obtained in [11]âwe show that the same conditions that imply the existence of Zeno behavior near Zeno equilibria imply the stability of the Zeno equilibria. This paper, therefore, not only presents conditions for the stability of Zeno equilibria, but directly relates the stability of Zeno equilibria to the existence of Zeno behavior
Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems
In the presence of noncompact symmetry, the stability of relative equilibria
under momentum-preserving perturbations does not generally imply robust
stability under momentum-changing perturbations. For axisymmetric relative
equilibria of Hamiltonian systems with Euclidean symmetry, we investigate
different mechanisms of stability: stability by energy-momentum confinement,
KAM, and Nekhoroshev stability, and we explain the transitions between these.
We apply our results to the Kirchhoff model for the motion of an axisymmetric
underwater vehicle, and we numerically study dissipation induced instability of
KAM stable relative equilibria for this system.Comment: Minor revisions. Typographical errors correcte
Stability by KAM confinement of certain wild, nongeneric relative equilibria of underwater vehicles with coincident centers of mass and bouyancy
Purely rotational relative equilibria of an ellipsoidal underwater vehicle
occur at nongeneric momentum where the symplectic reduced spaces change
dimension. The stability these relative equilibria under momentum changing
perturbations is not accessible by Lyapunov functions obtained from energy and
momentum. A blow-up construction transforms the stability problem to the
analysis symmetry-breaking perturbations of Hamiltonian relative equilibria. As
such, the stability follows by KAM theory rather than energy-momentum
confinement.Comment: 18 pages, 3 figure
Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry
This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages.
This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles
Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups
We consider the orbital stability of relative equilibria of Hamiltonian
dynamical systems on Banach spaces, in the presence of a multi-dimensional
invariance group for the dynamics. We prove a persistence result for such
relative equilibria, present a generalization of the Vakhitov-Kolokolov slope
condition to this higher dimensional setting, and show how it allows to prove
the local coercivity of the Lyapunov function, which in turn implies orbital
stability. The method is applied to study the orbital stability of relative
equilibria of nonlinear Schr{\"o}dinger and Manakov equations. We provide a
comparison of our approach to the one by Grillakis-Shatah-Strauss
Symmetry breaking for toral actions in simple mechanical systems
For simple mechanical systems, bifurcating branches of relative equilibria
with trivial symmetry from a given set of relative equilibria with toral
symmetry are found. Lyapunov stability conditions along these branches are
given.Comment: 25 page
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