125 research outputs found
Existence of symmetric central configurations
Central configurations have been of great interest over many years, with the
earliest examples due to Euler and Lagrange. There are numerous results in the
literature demonstrating the existence of central configurations with specific
symmetry properties, using slightly different techniques in each. The aim here
is to describe a uniform approach by adapting to the symmetric case the
well-known variational argument showing the existence of central
configurations. The principal conclusion is that there is a central
configuration for every possible symmetry type, and for any symmetric choice of
masses. Finally the same argument is applied to the class of balanced
configurations introduced by Albouy and Chenciner.Comment: 14 pages, to appear in Cel Mech and Dyn Ast
Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry
For Hamiltonian systems with spherical symmetry there is a marked difference
between zero and non-zero momentum values, and amongst all relative equilibria
with zero momentum there is a marked difference between those of zero and those
of non-zero angular velocity. We use techniques from singularity theory to
study the family of relative equilibria that arise as a symmetric Hamiltonian
which has a group orbit of equilibria with zero momentum is perturbed so that
the zero-momentum relative equilibrium are no longer equilibria. We also
analyze the stability of these perturbed relative equilibria, and consider an
application to satellites controlled by means of rotors.Comment: 24 pp; to appear in J. Geometric Mechanic
Periodic orbits in Hamiltonian systems with involutory symmetries
We study the existence of families of periodic solutions in a neighbourhood
of a symmetric equilibrium point in two classes of Hamiltonian systems with
involutory symmetries. In both classes, involutions reverse the sign of the
Hamiltonian function. In the first class we study a Hamiltonian system with a
reversing involution R acting symplectically. We first recover a result of
Buzzi and Lamb showing that the equilibrium point is contained in a three
dimensional conical subspace which consists of a two parameter family of
periodic solutions with symmetry R and there may or may not exist two families
of non-symmetric periodic solutions, depending on the coefficients of the
Hamiltonian. In the second problem we study an equivariant Hamiltonian system
with a symmetry S that acts anti-symplectically. Generically, there is no
S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we
prove the existence of at least 2 and at most 12 families of non-symmetric
periodic solutions. We conclude with a brief study of systems with both forms
of symmetry, showing they have very similar structure to the system with
symmetry R
Hamiltonian Relative Equilibria with Continuous Isotropy
In symmetric Hamiltonian systems, relative equilibria usually arise in
continuous families. The geometry of these families in the setting of free
actions of the symmetry group is well-understood. Here we consider the question
for non-free actions. Some results are already known in this direction, and we
use the so called bundle equations to provide a systematic treatment of this
question which both consolidates the known results, extending the scope of the
results to deal with non-compact symmetry groups, as well as producing new
results. Specifically we address questions about the stability, persistence and
bifurcations of these relative equilibria
Symplectic Group Actions and Covering Spaces
For symplectic group actions which are not Hamiltonian there are two ways to
define reduction. Firstly using the cylinder-valued momentum map and secondly
lifting the action to any Hamiltonian cover (such as the universal cover), and
then performing symplectic reduction in the usual way. We show that provided
the action is free and proper, and the Hamiltonian holonomy associated to the
action is closed, the natural projection from the latter to the former is a
symplectic cover. At the same time we give a classification of all Hamiltonian
covers of a given symplectic group action. The main properties of the lifting
of a group action to a cover are studied.Comment: 19 page
Vortex dynamics on a cylinder
Point vortices on a cylinder (periodic strip) are studied geometrically. The
Hamiltonian formalism is developed, a non-existence theorem for relative
equilibria is proved, equilibria are classified when all vorticities have the
same sign, and several results on relative periodic orbits are established,
including as corollaries classical results on vortex streets and leapfrogging.Comment: LaTeX2e, 13 pages, 5 figure
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