168 research outputs found
Asymptotic Stability, Instability and Stabilization of Relative Equilibria
In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev
Lagrange-Poincare field equations
The Lagrange-Poincare equations of classical mechanics are cast into a field
theoretic context together with their associated constrained variational
principle. An integrability/reconstruction condition is established that
relates solutions of the original problem with those of the reduced problem.
The Kelvin-Noether theorem is formulated in this context. Applications to the
isoperimetric problem, the Skyrme model for meson interaction, metamorphosis
image dynamics, and molecular strands illustrate various aspects of the theory.Comment: Submitted to Journal of Geometry and Physics, 45 pages, 1 figur
Invariant metrics and Hamiltonian Systems
Via a non degenerate symmetric bilinear form we identify the coadjoint
representation with a new representation and so we induce on the orbits a
simplectic form. By considering Hamiltonian systems on the orbits we study some
features of them and finally find commuting functions under the corresponding
Lie-Poisson bracketComment: 16 pages corrected typos, changed contents (Prop. 3.4 and Theorem in
Section 3
Spin chain from membrane and the Neumann-Rosochatius integrable system
We find membrane configurations in AdS_4 x S^7, which correspond to the
continuous limit of the SU(2) integrable spin chain, considered as a limit of
the SU(3) spin chain, arising in N=4 SYM in four dimensions, dual to strings in
AdS_5 x S^5. We also discuss the relationship with the Neumann-Rosochatius
integrable system at the level of Lagrangians, comparing the string and
membrane cases.Comment: LaTeX, 16 pages, no figures; v2: 17 pages, title changed,
explanations and references added; v3: more explanations added; v4: typos
fixed, to appear in Phys. Rev.
Symmetry reduction of Brownian motion and Quantum Calogero-Moser systems
Let be a Riemannian -manifold. This paper is concerned with the
symmetry reduction of Brownian motion in and ramifications thereof in a
Hamiltonian context. Specializing to the case of polar actions we discuss
various versions of the stochastic Hamilton-Jacobi equation associated to the
symmetry reduction of Brownian motion and observe some similarities to the
Schr\"odinger equation of the quantum free particle reduction as described by
Feher and Pusztai. As an application we use this reduction scheme to derive
examples of quantum Calogero-Moser systems from a stochastic setting.Comment: V2 contains some improvements thanks to referees' suggestions; to
appear in Stochastics and Dynamic
On the symmetry breaking phenomenon
We investigate the problem of symmetry breaking in the framework of dynamical
systems with symmetry on a smooth manifold. Two cases will be analyzed: general
and Hamiltonian dynamical systems. We give sufficient conditions for symmetry
breaking in both cases
Algebraic integrability of confluent Neumann system
In this paper we study the Neumann system, which describes the harmonic
oscillator (of arbitrary dimension) constrained to the sphere. In particular we
will consider the confluent case where two eigenvalues of the potential
coincide, which implies that the system has S^{1} symmetry. We will prove
complete algebraic integrability of confluent Neumann system and show that its
flow can be linearized on the generalized Jacobian torus of some singular
algebraic curve. The symplectic reduction of S^{1} action will be described and
we will show that the general Rosochatius system is a symplectic quotient of
the confluent Neumann system, where all the eigenvalues of the potential are
double. This will give a new mechanical interpretation of the Rosochatius
system.Comment: 17 pages, 1 figur
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