arXiv subjects

James Montaldi

Publications and source records attributed to James Montaldi.

At least 19 recordsLinked to original sources

arXiv2025

Piecewise linear circle maps and conjugation to rigid rational rotations

Criteria for piecewise linear circle homeomorphisms to be conjugate to a rigid rotation, $x\to x+\omega~({\rm mod}~1)$, with rational rotation number $\omega$ are given. The consequences of the existence of such maps in families of maps is considered and the results are illustrated using two examples: Herman's classic family of piecewise linear maps with two linear components, and a map derived from geometric optics which has four components. These results show how results for piecewise smooth circle homeomorphisms with irrational rotation numbers have natural correspondences with the case of rational rotation numbers for piecewise linear maps. In natural families of maps the existence of a parameter value at which the map is conjugate to a rigid rotation implies linear scaling of the rotation number in a neighbourhood of the critical parameter value and no mode-locked intervals, in contrast to the behaviour of generic families of circle maps.

math.DS↗

arXiv2023

Equilibria and bifurcations in contact dynamics

We provide a systematic study of equilibria of contact vector fields and the bifurcations that occur generically in 1-parameter families, and express the conclusions in terms of the Hamiltonian functions that generate the vector fields. Equilibria occur at points where the zero-level set of the Hamiltonian function is either singular or is tangent to the contact structure. The eigenvalues at an equilibrium have an interesting structure: there is always one particular real eigenvalue of any equilibrium, related to the contact structure, that we call the principal coefficient, while the other eigenvalues arise in quadruplets, similar to the symplectic case except they are translated by a real number equal to half the principal coefficient. There are two types of codimension 1 equilibria, named Type I, arising where the zero-set of the Hamiltonian is singular, and Type II where it is not, but there is a degeneracy related again to the principal coefficient and the contact of the zero level-set of the Hamiltonian with the contact structure. Both give rise generically to saddle-node bifurcations. Some special features include: (i) for Type II singularities, Hopf bifurcations cannot occur in dimension 3, but they may in dimension 5 or more; (ii) for Type I singularities, a fold-Hopf bifurcation can occur with codimension 1 in any dimension, and (iii) again for Type I, and in dimension at least 5, a fold-multi-Hopf bifurcation (where several pairs of eigenvalues pass through the imaginary axis simultaneously together with one through the origin) may also occur with codimension 1.

math.DS↗

arXiv2022

A Hamiltonian approach for point vortices on non-orientable surfaces

We investigate the motion of point vortices on the Mobius band and Klein bottle. Since these are non-orientable surfaces, the standard Hamiltonian approach does not apply. We therefore begin by establishing a modified Hamiltonian approach which works for arbitrary non-orientable surfaces, through describing the phase space, the Hamiltonian and the local equations of motion. We use a combination of twisted functions and oriented double covers to adapt some of the known notions of vortex dynamics to non-orientable surfaces. For both of the surfaces of interest, we write Hamiltonian-type equations of vortex motion explicitly and follow that by the description of relative equilibria and an investigation of the motion of one and two vortices.

math.DS↗

arXiv2021

Two body problem on a sphere in the presence of a uniform magnetic field

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 show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria.

math-ph↗

arXiv2019

Attracting and repelling 2-body problems on a family of surfaces of constant curvature

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 the paper is to study the 2-body problem on a surface of constant curvature treating the curvature as a parameter, and with particular interest in how families of relative equilibria and their stability behave as the curvature passes through zero and changes sign. We consider two cases: firstly one where the particles are always attracting throughout the family, and secondly where they are attracting for negative curvature and repelling for positive curvature, interpolated by no interaction when the curvature vanishes. Our analysis clarifies the role of curvature in the existence and stability of relative equilibria.

math-ph↗

arXiv2018

Non-Abelian momentum polytopes for products of $CP ^2$

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 segment, which can sit in the positive Weyl chamber in a small number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope, and numerous transition polytopes, all of which are classified here. The type of polytope depends on the weights of the symplectic form on each copy of projective space. In the second paper we use techniques of symplectic reduction to study the possible dynamics of interacting generalized point vortices. The results can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices with double eigenvalues.

math.SG↗

arXiv2018

Generalized point vortex dynamics on $CP ^2$

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 Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices. The different types of polytope depend on the values of the `vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of CP^2. We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points. For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense.

math-ph↗

arXiv2018

Hermitian flag manifolds and orbits of the Euclidean group

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 orbits, although different, are homotopy equivalent. We also provide a geometric description of the adjoint and coadjoint orbits of the Euclidean and orthogonal groups as a special class of flag manifold which we call a Hermitian flag manifold. These manifolds consist of flags endowed with complex structures equipped to the quotient spaces that define the flag.

math.RT↗

arXiv2018

Integrability and dynamics of the n-dimensional symmetric Veselova top

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 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.

math.DS↗

arXiv2017

A localization formula for equivariant Lyusternik-Schnirelmann category

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 localization formula to calculate the equivariant analogue of this category in terms of the minimal orbit-type strata. The formula holds provided that the manifold admits a specific cover. We show that such a cover exists on every symplectic toric manifold. The known result stating that the LS-category of a symplectic toric manifold is equal to the number of fixed points of the torus action follows from our localization formula.

math.GT↗

arXiv2017

Persistence of stationary motion under explicit symmetry breaking perturbation

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 orbits of relative equilibria which persist after a small perturbation is given. This bound is given in terms of the equivariant Lyusternik-Schnirelmann category of the group orbit.

math.DS↗

arXiv2016

Feynman path integrals and Lebesgue-Feynman measures

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 transformations of the LFM with so-called quantum anomalies, improving some recent results of teh authors and co-workers. We revisit the contradiction between the points of view on quantum anomalies presented in the books of Fujikawa and Suzuki on the one hand, and of Cartier and DeWitt-Morette on the other.

math-ph↗

arXiv2016

Gauge momenta as Casimir functions of nonholonomic systems

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 results on Hamiltonization of nonholonomic systems, and has consequences for the study of relative equilibria in such systems.

math.DS↗

arXiv2015

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.

math.DS↗

arXiv2015

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.

math.DS↗

arXiv2014

Point vortices on the hyperbolic plane

We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. 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 equilibria and the stability criteria for a number of cases, focusing on N=2, 3. Contrary to the system on the sphere, relative equilibria with non-compact momentum isotropy subgroup are found, and are used to illustrate the different stability types of relative equilibria.

math.DS↗

arXiv2014

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.

math.DS↗

arXiv2014

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.

math.DS↗