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Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the
-body problem: periodic motions where the
bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part, we classify all possible symmetry groups of planar
-body collision-free choreographies. These symmetry groups fall into two infinite families and, if
is odd, three exceptional groups. In the second part, we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset. In particular, we refine the symmetry classification by classifying the connected components of the set of loops with any given symmetry. This leads to the existence of many new choreographies in
-body systems governed by a strong force potential.
The lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems. They succinctly cover an unparalleled range of topics from the basic concepts of symplectic and Poisson geometry, through integrable systems, KAM theory, fluid dynamics, and symmetric bifurcation theory. The lectures are based on summer schools for graduate students and postdocs and provide complementary and contrasting viewpoints of key topics: the authors cut through an overwhelming amount of literature to show young mathematicians how to get to the core of the various subjects and thereby enable them to embark on research careers.
In the summers of 2000 and 2001, we organized two European Summer Schools in Geometric Mechanics. They were both held in the wonderful environment provided by the village-cum-international conference centre at Peyresq in the Alpes de Haute Provence in France, about 100km North of Nice. Each school consisted of 6 short lecture courses, as well as numerous short talks given by participants, of whom there were about 40 at each school. The majority of participants were from Europe with a few coming from West of the Atlantic or East of the Urals, and we were pleased to see a number of participants from the first year returning in the second. Several of the courses and short talks led to collaborations between participants and/or lecturers.
The summer schools were funded principally by the European Commission under the High-Level Scientific Conferences section of the Fifth Framework Programme. Additional funding was very kindly provided by the Fondation Peiresc. The principal aim of the two schools was to provide young scientists with a quick introduction to the geometry and dynamics involved in geometric mechanics and to bring them to a level of understanding where they could begin work on research problems. The schools were also closely linked to the Mechanics and Symmetry in Europe (MASIE) research training network, organized by Mark Roberts, and several of the participants went on to become successful PhD students or postdocs in MASIE.
The 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 five lectures given by JM on Hamiltonian Bifurcations with Symmetry. We focus our attention on bifurcation theory near equilibrium solutions and relative equilibria. The notes are composed of two parts. In the first, we review results on nonlinear normal modes in equivariant Hamiltonian systems, generic movement of eigenvalues in equivariant Hamiltonian matrices, one and two parameter bifurcation of equilibria and the Hamiltonian-Hopf Theorems with symmetry. The second part is about local dynamics near relative equilibria. Particular topics discussed are the existence, stability and persistence of relative equilibria, bifurcations from zero momentum relative equilibria and examples.
We begin with some basic facts on Lie group actions on symplectic manifolds and Hamiltonian systems with symmetry. The reader should refer to Ratiu's lectures for more details and examples.
Semisymplectic actions A Lie group G acts semisymplectically on a symplectic manifold (P, ω) if g*ω = ±ω. In this case the choice of sign determines a homomorphism X : G → Z2 called the temporal character, such that g*ω = X(g)ω. We denote the kernel of X by G+; it consists of those elements acting symplectically, and if G does contain antisymplectic elements then G+ is a subgroup of G of index 2. Some details on semisymplectic actions can be found in [MR00].
We consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector
fields. We give some conditions on the group and representation that imply existence
of equivariant vector fields with such trajectories. Using these results we show very
simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that
imply existence of RHTs, we construct
some new examples of robust relative homoclinic trajectories. We also classify RHTs
of the dihedral and low order symmetric groups by means of their symmetries.