Hamiltonian Systems & Symplectic Geometry: Abstracts
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Papers are all in pdf format (unless otherwise mentioned)
Dynamics of poles with variable strengths and optical analogies
(with Tadashi Tokieda) (7 pp.)
Dynamics of point vortices is generalized to complex, variable strengths (poles), and several
exact solutions with optical analogues, notably Snell's law and the law of reflection, are given.
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paper in pdf (120k)
Symplectic group actions and covering spaces
(with Juan-Pablo Ortega)
J. Diff Geom. and Appl. 27 (2009), 589-604.
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 covering.
At the same time we give a classification of all Hamiltonian coverings of a given symplectic
group action. The main properties of the lifting of a group action to a cover are studied.
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paper in pdf (172k)
A note on the geometry of linear Hamiltonian systems
of signature 0 in R4
J. Diff Geom. and Appl. 25 (2007), 344–350
It is shown that a linear Hamiltonian system of signature zero on R4 is elliptic or hyperbolic
according to the number of Lagrangian planes in the null-cone H-1(0), or equivalently the number
of invariant Lagrangian planes. An extension to higher dimensions is described.
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paper in pdf (60k)
Stability of relative equilibria of point vortices on the sphere
(with Frédéric Laurent-Polz and Mark Roberts).
improved version in preparation
We describe the linear and nonlinear stabilities of certain configurations of
point vortices on the sphere forming relative equilibria. These configurations
consist of up to two rings, with and without polar vortices. Such configurations
have dihedral symmetry, and the symmetry is used to block diagonalize the relevant
matrices, enabling their eigenvalues to be calculated.
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paper in pdf (430k)
Bifurcation and forced symmetry breaking in Hamiltonian systems
(with Féthi Grabsi and Juan-Pablo Ortega).
C.R. Acad. Sci. Paris Ser I 338 (2004), 565--570.
We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian
system on a symplectic manifold. In particular we study the persistence of an
initial relative equilibrium subjected to this forced symmetry breaking. We
see that, under certain non-degeneracy conditions, an estimate can be made on
the number of relative equilibria bifurcating from a given one.
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The relation between local and global dual pairs
(with Juan-Pablo Ortega and Tudor Ratiu).
Math Research Letters 11
(2004), 355-363.
We clarify the relationship between the local and global definitions of dual
pairs in Poisson geometry. It turns out that these are not equivalent. For the
passage from local to global one needs a connected fibre hypothesis (this is
well known), while the converse requires a dimension condition (which appears
not to be known). We also provide examples illustrating the necessity of the
extra conditions.
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Relative periodic orbits in symmetric Lagrangian systems
(with Chris McCord, Mark Roberts and Luca Sbano).
Proceedings of Equadiff, 2003.
We announce two topological results that may be used to estimate the number
of relative periodic orbits of different homotopy classes that are possessed
by a symmetric Lagrangian system. The results are illustrated by applications
to systems on tori an to strong force N-centre problems.
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Vortex dynamics on cylinders
(with Anik Soulière and Tadashi Tokieda)
SIAM J. on Applied Dynamical
Systems 2 (2003), 417-430.
Point vortices on a cylinder (periodic strip) are studied geometrically, using
local integrals of motion. 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.
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Openness of momentum maps and persistence of extremal relative equilibria
(with Tadashi Tokieda)
Topology 42 (2003), 833-844.
We prove that for every proper Hamiltonian action of a Lie group G in
finite dimensions the momentum map is locally G-open relative to its
image (i.e.\ images of G-invariant open sets are open). As an application
we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries,
extremal relative equilibria persist for every perturbation of the value of
the momentum map, provided the isotropy subgroup of this value is compact. We
also demonstrate how this persistence result applies to an example of ellipsoidal
figures of rotating fluid, and provide an example with plane point vortices
which shows how the compactness assumption is related to persistence.
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Relative equilibria of point vortices on
the sphere.
(with Chjan Lim and Mark Roberts)
Physica D
148 (2001), 97-135.
We prove the existence of many different symmetry types of relative equilibria
for systems of identical point vortices on a non-rotating sphere. The proofs
use the rotational symmetry group SO(3) and the resulting conservation laws,
the time-reversing reflectional symmetries in O(3), and the finite symmetry
group of permutations of identical vortices. Results include both global existence
theorems and local results on bifurcations from equilibria. A more detailed
study is made of relative equilibria which consist of two parallel rings with
$n$ vortices in each rotating about a common axis. The paper ends with discussions
of the bifurcation diagrams for systems of 3, 4, 5 and 6 identical vortices.
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A note on semisymplectic actions of Lie groups
(with Mark
Roberts)
Comptes Rendues de l'Acad. des Sciences 330
(2000), 1079-1084.
A semisymplectic action of a Lie groups on a symplectic manifold is one where
each element of the group acts either symplectically or antisymplectically.
We find conditions that ensure a semisymplectic action descends to an action
on the symplectic reduced spaces. We consider a few examples, and in particular
apply these ideas to reduction of $N$-body systems with Galilean invariance.
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Perturbing a symmetric resonance: the magnetic spherical
pendulum.
SPT98 -- Symmetry and Perturbation Theory II.
A. Degasperis and G. Gaeta eds., World Scientific (1999).
The periodic orbits of the spherical pendulum are well-known. There are two
types of modes: the planar oscillations and the circular modes (discovered by
Huygens). The stable equilibrium is resonant, in that it has a pair of double
eigenvalues. Adding a (constant vertical) magnetic field breaks the reflexional
symmetry and so breaks the resonance. Indeed, one of the Huygens modes will
rotate faster than the other. Moreover the planar modes will not persist as
such, although the corresponding family of periodic orbits will continue to
exist, albeit slightly perturbed. The object of this note is to describe the
fate of the families of periodic orbits as the magnetic field is increased from
0. We assume that the magnetic perturbation preserves the rotational symmetry.
The method is also applied to the analogous symmetry-breaking problem with square
symmetry rather than the full rotation group. We use singularity theoretic and
normal form techniques.
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Relative equilibria of molecules.
(with Mark Roberts).
J. Nonlinear Science 9 (1999), 53-88.
We apply and extend results from the paper "Persistence and Stability of Relative
Equilibria" (below) to prove the existence of relative equilibria of molecules
for small angular momentum. In this case, relative equilibria are pure rotational
motions. For a generic molecule, there are 6 rotational modes, as for the rigid
body. If the molecule is symmetric (e.g. methane), we show that there are (many)
more, classifying them by their symmetry. We also consider their stability.
For example, for methane, which has tetrahedral symmetry, there are 26 rotational
modes, of which 6 are Lyapounov stable, 8 are linearly stable and 12 are unstable,
corresponding to the three distinct symmetry types.
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Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques.
Comptes Rendues de l'Acad. des Sciences 324 (1997), 553-558.
It was known to Poincaré that a non-degenerate
periodic orbit in a Hamiltonian system persists to nearby
energy-levels. In this note, we consider the analogous
problem for relative periodic orbits in symmetric Hamiltonian
systems. We show that non-degenerate relative periodic orbits
persist to nearby values of the energy-momentum map, under the
hypothesis that the group of symmetries acts freely. Get
paper (131K)
Persistence and stability of relative equilibria.
Nonlinearity 10 (1997), 449-466.
We consider relative equilibria in symmetric Hamiltonian systems, and their
persistence or otherwise as the momentum is varied. The symmetry group in question
is assumed to be compact. In particular, we extend a result about persistence
of relative equilibria for values of the momentum map that are regular for the
coadjoint action, to arbitrary values, provided that either the action on the
phase space is locally free, or that the relative equilibrium is at a local
extremum of the reduced Hamiltonian. We also consider the Lyapounov stability
of such extremal relative equilibria.
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Caustics in time reversible Hamiltonian systems.
In
Singularity Theory and its Applications, Warwick 1989, Part 2
(ed. M. Roberts & I. Stewart), Lecture Notes in Maths
1463, Springer, 1991.
We consider the projection to configuration space of invariant tori in time-reversible Hamiltonian systems at a point of zero momentum. At such points the projection has rank 0 and the resulting caustic has a corner. We use caustic equivalence of Lagrangian mappings to find a normal form for such a corner in 3 degrees of freedom. This extends previous work of Delos for the analogous problem in 2 degrees of freedom. We also use the Lefschetz fixed point theorem to show that such a torus in n degrees of freedom has 2n of these corners.
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Existence of nonlinear normal modes in symmetric Hamiltonian systems
Stability of nonlinear normal modes in symmetric Hamiltonian systems
(with Mark Roberts and Ian Stewart),
Nonlinearity 3 (1990), 695-730 and 731-772.
A nonlinear normal mode in nonlinear Hamiltonian system
near an equilibrium point is a family of periodic trajectories
approximating the periodic trajectories (normal modes) of the
linear system. In the first of these papers, we employ a
combination of traditional methods such as Birkhoff normal
form with more recent Singularity Theory methods to prove
existence of such nonlinear normal modes in symmetric
Hamiltonian systems. In the second paper, we show how to
compute the spectral stability of these nonlinear normal
modes. Both papers illustrate the methods with a number of
examples.
Get papers: Existence or Stability (both from Journal)
Periodic Solutions near Equilibria of Symmetric Hamiltonian
Systems
(with Mark Roberts and Ian Stewart),
Proc. Roy. Soc. London, 325 (1988), 237-293.
We consider the effects of symmetry on the dynamics of
nonlinear Hamiltonian systems under the action of a compact
Lie group, in the vicinity of an isolated equilibrium: in
particular the local existence and stability of periodic
trajectories. The main existence result, an equivariant
version of the Weinstein-Moser theorem, asserts the existence
of periodic trajectories with certain prescribed symmetries,
independently of the precise nonlinearities. We then describe
the constraints put on the Floquet operator of these periodic
trajectories by the action of the Lie group. This description
has three ingredients: an analysis of the linear symplectic
maps that commute with a symplectic representation, a study of
the momentum mapping and its relation to Floquet multipliers,
and Krein theory. We find that for some symmetry groups,
which we call cyclospectral, all eigenvalues of the
Floquet operator are forced by the symmetry to lie on the unit
circle; that is the periodic trajectory is spectrally stable.
Similar results for equilibria are described briefly. The
results are applied to a number of simple examples, such as
SO(2), O(2), cyclic and dihedral groups, and also to the
irreducible symplectic representations of O(3) on spaces of
complex spherical harmonics, modelling oscillations of a
liquid drop.
Non-linear normal modes of symmetric Hamiltonian systems
(with Mark Roberts and Ian Stewart). In
The physics of structure formation (ed. G. Dangelmayr and W.Güttinger), Springer, 1987.
This is a short exposition of the
main results of the paper listed above
James Montaldi
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