James Montaldi

The University of Manchester · Dept of Mathematics

This is the first of two companion papers, in which we investigate vortex motion on non-orientable two dimensional surfaces. We establish the `Hamiltonian' approach to point vortex motion on non-orientable surfaces through describing the phase space, the Hamiltonian and the local equations of motion. This paper is primarily focused on the dynamics...

We first provide a classification of the pure rotational motion of 2 particles on a sphere interacting via a repelling potential. This is achieved by providing a simple geometric equivalence between repelling particles and attracting particles, and relying on previous work on the similar classification for attracting particles. The second theme of...

We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles interacting via a cotangent potential, we sho...

Suitable for advanced undergraduates, postgraduates and researchers, this self-contained textbook provides an introduction to the mathematics lying at the foundations of bifurcation theory. The theory is built up gradually, beginning with the well-developed approach to singularity theory through right-equivalence. The text proceeds with contact equ...

We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles, interacting via a cotangent potential, we sh...

We first provide a classification of the pure rotational motion of 2 particles on a sphere interacting via a repelling potential. This is achieved by providing a simple geometric equivalence between repelling particles and attracting particles, and relying on previous work on the similar classification for attracting particles. The second theme of...

We 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...

Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed in a way that the perturbation preserves only some symmetries of the original system. We give a geometric approach to study this phenomenon in the setting of equivariant Hamiltonian systems. A lower bound for the number of orbits of equilibria and...

This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space $\CP ^2$ interacting via a simple Hamiltonian function. The system has symmetry group SU(3). The first paper describes all possible momentum p...

This is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the point vortex systems familiar in 2-D. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 4-space. For 2 copies, the momentum polytope is simply a line segmen...

We perform the reduction of the two-body problem in the two dimensional spaces of constant non-zero curvature and we use the reduced equations of motion to classify all relative equilibria (RE) of the problem and to study their stability. In the negative curvature case, the nonlinear stability of the stable RE is established by using the reduced Ha...

We 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...

We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection between the sets of adjoint and coadjoint orbits of such groups. In addition, we show that the corresponding orbi...

The LS-category of a topological space is a numerical homotopy invariant, introduced originally in a course on the global calculus of variations by Lyusternik and Schnirelmann, to estimate the number of critical points of a smooth function. When the topological space is a smooth manifold equipped with a proper action of a Lie group, we give a local...

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 call a Lebesgue-Feynman measure (LFM) any generalized measure (distribution in the sense of Sobolev and Schwartz) on a locally convex topological vector space E which is translation invariant. In the present paper, we investigate transformations of the LFM generated by transformations of the domain and also discuss the connections of these trans...

Applications of transformations of Feynman path integrals and Feynman pseudomeasures to explain arising quantum anomalies are considered. A contradiction in the literature is also explained.

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...

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...

In this note we describe algorithms for obtaining formulae for transformations of measures on infinite dimensional topological vector spaces or manifolds, generated by transformations of the domains of the measures and by transformations of the range.

We investigate the dynamical system of point vortices on the hyperboloid. This system has non-compact symmetry $SL(2, R)$ and a coadjoint equivariant momentum map. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equil...

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n 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...

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...

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Bir...

We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $\d_{X_M} = \d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd inv...

We present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with non-trivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied...

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Bir...

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten...

We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the rel...

We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu and by Lerman and Singer. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subspace of...

In earlier work [D.S. Broomhead, J.P. Huke, M.R. Muldoon, and J. Stark, Iterated function system models of digital channels, Proc. R. Soc. Lond. A 460 (2004), pp. 3123--3142], aimed at developing an approach to signal processing that can be applied as well to nonlinear systems as linear ones, we produced mathematical models of digital communication...

The dynamics of point vortices is generalized in two ways: first by making the strengths complex, which allows for sources and sinks in superposition with the usual vortices, second by making them functions of position. These generalizations lead to a rich dynamical system, which is nonlinear and yet has conservation laws coming from a Hamiltonian-...

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...

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 and to strong force N -centre problems.

In the visualization of the topology of second rank symmetric tensor fields in the plane one can extract some key points (degenerate points), and curves (separatrices) that characterize the qualitative behaviour of the whole tensor field. This can provide a global structure of the whole tensor field, and effectively reduce the complexity of the origin...

It 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.

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 bifurcatio...

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1). As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ fo...

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 nondegeneracy conditions, an estimate can be made on the number of bifurcating relative eq...

In this note 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 fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also...

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...

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...

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 vortic...

Singularity theory is a broad subject with vague boundaries. It draws on many other areas of mathematics, and in turn has contributed to many areas both within and outside mathematics, in particular differential and algebraic geometry, knot theory, differential equations, bifurcation theory, Hamiltonian mechanics, optics, robotics and computer visi...

We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0 . Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectivel...

The authors analyse the existence of nonlinear normal modes of a (nonlinear) Hamiltonian system, i.e. periodic solutions that approximate periodic solutions of the system linearised around an elliptic (and semisimple) equilibrium point. In particular they consider systems with symmetry, including time-reversible symmetry which involves an antisympl...

A nonlinear normal mode of a Hamiltonian system is a periodic solution near equilibrium with period close to that of a periodic trajectory of the linearised vector field. This paper is a sequel to the preceding paper in this issue, in which the authors developed methods for proving the existence of nonlinear normal modes of a Hamiltonian system tha...

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 also persist when shifting to nearby values of the energy-momentum...

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 also persist when shifting to nearby values of the energy-momentum...

Singularity theory encompasses many different aspects of geometry and topology, and an overview of these is represented here by papers given at the International Singularity Conference held in 1991 at Lille. The conference attracted researchers from a wide variety of subject areas, including differential and algebraic geometry, topology, and mathem...

This article grew out of a series of lectures I gave at the ICMSC in July 1992, preceding the conference. I would particularly like to thank Maria Ruas for inviting me to give the lectures, for organizing a wonderful conference, and finally for encouraging me to write up the lectures for publication in these proceedings. I would also like to thank...

We show that (in the nice dimensions) if a stable map is restricted to a generic submanifold of the source manifold, then the resulting map will also be stable. At the same time, the analogous result for versal unfoldings is proved. A few examples of applications to the extrinsic geometry of submanifolds of Euclidean space are discussed.

sense the zeros and poles of the form respectively. These modules are finite dimensional vector spaces, and we prove in $1 that the difference in dimension is preserved under deformation of both the form and the curve. In the case that r = dg for some holomorphic function g, this enables us to find the number of critical points of a small generic d...

We consider the effects of symmetry on the dynamics of a nonlinear hamiltonian system invariant under the action of a compact Lie group Gamma , 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, asser...

Two hundred years ago Meusnier established that for any surface in R3 the set of osculating circles at a point and with a given tangent direction form a sphere [14]. (An osculating circle is one with at least 3-point contact with the surface.)

In this paper 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 result we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examin...

Consider a linear action of the group C on X = C n+1 . We study the fundamental algebraic properties of the sheaves of invariant and basic dierential forms for such an action, and use these to dene an algebraic notion of multiplicity for critical points of functions which are invariant under the C -action. We also prove a theorem relating the cohom...

We show how the path formulation of bifurcation theory can be made to work, and that it is (essentially) equivalent to the usual parametrized contact equivalence of Golubitsky and Schaeffer. Introduction In their original paper on imperfect bifurcation theory [GS79], Golubitsky and Schaeffer consider the so-called path formulation of bifurcation th...

We 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 e...

this paper is to study this geometry. For brevity, we consider only the cases of symmetry

This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family: [Formula: see text] where z is a complex dynamic (phase) variable, [Formula: see text] its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter...

. 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 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 sys...

A recent result of J. Damon's [4] relates the A e -versal unfoldings of a map-germ f with the KD(G) -versal unfoldings of an associated map germ which induces f from a stable map G. We extend this result to the case where the source is a complete intersection with an isolated singularity. In a similar vein, we also relate the bifurcation theoretic...

How many cusps does a swallowtail have, After it becomes a stable map, And how many swallowtails does a butterfly have, After it . . . (with apologies to B. Dylan) Introduction Consider the map F : C 2 ! C 2 (x; y) 7! (x; y 4 + xy); (which is a section of the swallowtail singularity) and its perturbation F " (x; y) = (x; y 4 + xy + "y 2 ): The sing...

We consider the projection to configuration space of invariant tori in a time reversible Hamiltonian system at a point of zero momentum. At such points the projection has rank zero 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.

Abstract – I. Melbourne [9] has recently announced,an existence theorem for generic bifurcation of rotating waves with maximal,isotropy for vector fields which are equivariant under an absolutely irreducible action of a compact Lie group. Melbourne’s proof relies on recent results of Field [4, 5]. Rotating waves can also be interpreted as equilibri...

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 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 i...

This consists of lecture notes from 6 courses held at 2 summer schools in Peyresq, France in 2000 and 2001. The notes were written up by the lecturers together with some participants.

Dynamics of point vortices is generalized to complex, variable strengths (poles), and several exact solutions with optical analogues, notably Snell's law, are given.

Given an action of a Lie group $G$ on a manifold $M$, and a cover $N$ of $M$ (such as the universal cover $\tilde M$) the natural question arises of whether the action lifts to a cover of $M$. In these notes, we address this question and determine whether the $G$-action itself lifts or whether it is necessary to pass to a cover of $G$ (there is alw...