152 research outputs found
Relative equilibria and conserved quantities in symmetric Hamiltonian systems
No full textThese are lectures given at a summer school in 1998. They begin with recalling the basic structure of Hamiltonian systems, and then proceed to discuss the effects of symmetry and the geometry of conservation laws
Symmetric Hamiltonian bifurcations
No full textThe purpose of these notes is to give a brief survey of bifurcation theory of Hamiltonian systems with symmetry; they are a slightly extended version of the 5 lectures given by JM on Hamiltonian Systems with Symmetry at the Peyresq Summer School. Attention is focussed on bifurcations near equilibrium solutions and relative equilibria. [Taken from introduction
Integrability and dynamics of the n-dimensional symmetric Veselova top
No full textWe consider the the n-dimensional generalisation of the nonholonomic Veselova problem.
We derive the reduced equations of motion in terms of the mass tensor of the body and determine some general properties of the dynamics. In particular we give a closed formula for the invariant measure, we indicate the existence of steady rotation solutions, and obtain some results on their stability.
We then focus our attention on bodies whose mass tensor has a specific type of symmetry. We show that the phase space is foliated by invariant tori that carry quasi-periodic dynamics in the natural time variable. Our results
enlarge the known cases of integrability of the multi-dimensional Veselova top. Moreover, they show that in some previously known instances of integrability, the flow is quasi-periodic without the need of a time reparametrisation
Geometric mechanics and symmetry: the Peyresq lectures
No full textThe lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems
Gauge momenta as Casimir functions of nonholonomic systems
No full textWe consider nonholonomic systems with symmetry possessing a certain type of first integrals that are linear in the velocities. We develop a systematic method for modifying the standard nonholonomic almost Poisson structure that describes the dynamics so that these integrals become Casimir functions after reduction. This explains a number of recent results on Hamiltonization of nonholonomic systems, and has consequences for the study of relative equilibria in such systems
Transformations of Feynman path integrals and generalized densities of Feynman pseudomeasures
No full textApplications of transformations of Feynman path integrals and Feynman pseudomeasures to explain arising quantum anomalies are considered. A contradiction in the literature is also explained
A note on the geometry of linear Hamiltonian systems of signature 0 in R4
No full textIt is shown that a linear Hamiltonian system of signature zero on R4 is elliptic, hyperbolic or mixed according to the number of Lagrangian planes in the null-cone H-1 (0), or equivalently the number of invariant Lagrangian planes. A weaker extension to higher dimensions is described. © 2007 Elsevier B.V. All rights reserved
Persistence and stability of relative equilibria
No full textWe consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact
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