We examine existence and stability of relative equilibria of the $n$-vortex problem specialized to the case where $N$ vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued, and its linear stability is determined by properties of the potential. For $N\geq 3$ there are at least three distinct families of critical points associated to the limiting problem. Assuming nondegeneracy, one of these families continues to a linearly stable class of relative equilibria with small and large circulation of the same sign. This class becomes unstable as the small circulation passes through zero and changes sign. Another family of critical points which is always nondegenerate continues to a configuration with small vortices arranged in an $N$-gon about the strong central vortex. This class of relative equilibria is linearly unstable regardless of the sign of the small circulation when $N\geq 4$. Numerical results suggest that the third family of critical points of the limiting problem also continues to a linearly unstable class of solutions of the full problem independent of the sign of the small circulation. Thus there is evidence that linearly stable relative equilibria exist when the large and small circulation strengths are of the same sign, but that no such solutions exist when they have opposite signs. The results of this paper are in contrast to those of the analogous celestial mechanics problem, for which the $N$-gon is the only relative equilibrium for $N$ sufficiently large, and is linearly stable if and only if $N\geq 7$.
We analyze existence, stability, and symmetry of point vortex relative equilibria with one dominant vortex and N vortices with infinitesimal circulation. The dimension of the problem can be reduced by taking an infinitesimal circulation limit, resulting in the so-called (1+N)-vortex problem. In this work, we first generalize the reduction to allow for circulations of varying signs and weights. We then prove that symmetric configurations require equality of two circulation parameters in the (1+3)-vortex problem and show that there are stable asymmetric relative equilibria. In a number of exampl…
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.
This article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that, there exist infinitely many normalised relative periodic orbits on a dense subset of all energy levels, which are neither fixed points nor relative equilibria.
Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative…
We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number.
We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The $S_3$ invariance obtained from the equality of vorticities is used to reduce the defining equations and organize the solutions into two distinct groups based on the ordering of the vortices along the line. The number and type of solutions are given, along with a discussion of the bifurcations that occur. The linear stability of all solutions is investigated rigorously and stable solutions are found to exist for cases where the vorticities have mixed signs. We emp…
This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hypersurface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appea…
Helmholtz's equations provide the motion of a system of N vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we are interested in relative equilibria formed of concentric regular polygons of vortices. We show that in the case of one regular polygon (and a possible vortex at the center) with more than three vertices (two if there is a vortex at the center), a relative equilibrium requires equal vorticities (on …
We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vo…
We consider the $N$-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the discrete reduction of the system by twisted subgroups of the full symmetry group that rotates and permutes the vortices. Our approach formalises and extends ideas outlined previously by Tokieda (C. R. Acad. Sci., Paris I 333 (2001)) and Soulière and Tokieda (J. Fluid Mech. 460 (2002)) and allows us to prove the existence of several 1-parameter families of per…
In this paper we derive the equations of motion for two-layer point vortex motion on the upper half plane. We study the invariants using symmetry, including the Hamiltonian and show that the two vortex problem is integrable. We characterize all two vortex motions for the cases where the vortex strengths are both equal, $Γ_{1}=Γ_{2}=1$ and when they are opposite $Γ_{1}=-Γ_{2}=1$. We also prove that there are no equilibria for the two vortex problem when $Γ_{1}=-Γ_{2}=1$.\ We show that there is only one relative equilibrium configuration when $Γ_{1}=Γ_{2}=1$ and the vortices are in different lay…
We prove that if for the curved $n$-body problem the masses are given, the minimum distance between the point masses of a specific type of relative equilibrium solution to that problem has a universal lower bound that is not equal to zero. We furthermore prove that the set of all such relative equilibria is compact. This class of relative equilibria includes all relative equilibria of the curved $n$-body problem in $\mathbb{S}^{2}$, $\mathbb{H}^{2}$ and a significant subset of the relative equilibria for $\mathbb{S}^{3}$ and $\mathbb{H}^{3}$.
This article is concerned with the study of existence and properties of stationary solutions for the dynamics of $N$ point vortices in an idealised fluid constrained to a bounded two--dimen\-sional domain $Ω$, which is governed by a Hamiltonian system \[ \left\{\begin{aligned} Γ_i\frac{d x_i}{d t} &=\frac{\partial H_Ω}{\partial y_i}(z_1,\dots,z_N)\\ Γ_i\frac{d y_i}{d t} &=-\frac{\partial H_Ω}{\partial x_i}(z_1,\dots,z_N) \end{aligned} \hspace{2cm}\text{where}\ z_i=(x_i,y_i),\ i=1,\dots,N, \right. \] where $H_Ω(z):=\sum_{j=1}^NΓ_j^2h(z_j)+\sum_{i,j=1, i\not=j}^NΓ_iΓ_jG(z_i,z_j)$ is the so--call…
In this paper we find the families of relative equilibria for the three body problem in the plane, when the interaction between the bodies is given by a quasi-homogeneous potential, which is the sum of two homogeneous functions. The number of the relative equilibria depends of the values of the masses and of the size of the system, measured by the moment of inertia.
We consider the $n$--body problem defined on surfaces of constant negative curvature. For the case of $n$--equal masses we prove that the hyperbolic relative equilibria with a regular polygonal shape do not exist. In particular the Lagrangian (three equal distances) hyperbolic relative equilibria do not exist. We also show the existence of a new class of hyperbolic collinear relative equilibria for the five body problem on surfaces of constant negative curvature.
We consider the $n$ body problem defined on surfaces of constant positive curvature. For the 5 and 7 body problem in a collinear symmetric configuration we obtain initial positions which lead to relative equilibria. We give explicitly the values of masses in terms of the initial positions. For positions for which relative equilibria exist, there are infinitely many values of the masses that generate such solutions. For the 5 and 7 body problem, the set of parameters (masses and positions) leading to relative equilibria has positive Lebesgue measure.
We prove that if for the curved $n$-body problem in $\mathbb{S}^{k-1}$, $k\geq 3$, the masses are given, the minimum distance between the point masses of a specific type of relative equilibrium solution that is a generalisation of positive elliptic relative equilibria and positive elliptic-elliptic relative equilibria has a universal lower bound that is not equal to zero.
We prove that if for relative equilibrium solutions of a generalisation of the $n$-body problem of celestial mechanics the masses and rotation are given, then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero. We furthermore prove that the set of such relative equilibria is compact.
We prove that all non-degenerate relative equilibria of the planar Newtonian $n$--body problem can be continued to spaces of constant curvature $κ$, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces.
We give a method to determine relative periodic orbits in point vortex systems: it consists mainly into perform a symplectic reduction on a fixed point submanifold in order to obtain a two-dimensional reduced phase space. The method is applied to point vortices systems on a sphere and on the plane, but works for other surfaces with isotropy (cylinder, ellipsoid, ...). The method permits also to determine some relative equilibria and heteroclinic cycles connecting these relative equilibria.
We prove that for some potentials (including the Newtonian one, and the potential of Helmholtz vortices in the plane) relative equilibria consisting of two homothetic regular polygons of arbitrary size can only occur if the masses at each polygon are equal. The same result is true for many regular polygons as long as the ratio between the radii of the polygons are sufficient large. Moreover, under these hypotheses, the relative equilibrium always exist.
We study a classical model for the atom that considers the movement of $n$ charged particles of charge $-1$ (electrons) interacting with a fixed nucleus of charge $μ>0$. We show that two global branches of spatial relative equilibria bifurcate from the $n$-polygonal relative equilibrium for each critical values $μ=s_{k}$ for $k\in \lbrack 2,...,n/2]$. In these solutions, the $n$ charges form $n/h$-groups of regular $h$-polygons in space, where $h$ is the greatest common divisor of $k$ and $n$. Furthermore, each spatial relative equilibrium has a global branch of relative periodic solutions for…
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, …
We prove that if the masses of the $n$-body problem of celestial mechanics are given, the number of classes of relative equilibria that solve the $n$-body problem is finite.
We prove that if for relative equilibrium solutions of a generalisation of quasi-homogeneous $n$-body problems the masses and rotation are given, then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero. We furthermore prove that the set of such relative equilibria is compact and prove related results for $n$-body problems in spaces of constant Gaussian curvature.
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 relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.
Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central mass, central circulation or amplitude of the oscillation). The symmetries of the problem are used in order to find the irreducible representations, the linearization and, with the help of a degree theory, the symmetries of the bifurcated solutions.
We revisit the dynamics of a permanent-magnetic rigid body submitted to a spatially-uniform steadily-rotating magnetic field in Stokes flow. We propose an analytical parameterisation of the full set of equilibria depending on two key experimental parameters, and show how it brings further understanding that helps to optimise magnetisation and operating parameters. The system is often bistable when it reaches its optimal swimming velocity. A handling strategy is proposed that guarantees that the correct equilibrium is reached.
We investigate the generic local structure of relative equilibria in Hamiltonian systems with symmetry $G$ near a completely symmetric equilibrium, where $G$ is compact and connected. Fix a maximal torus $T \subset G$ and identify the equilibrium with the origin within a symplectic representation of $G$. By a previous result, generically, for each $ξ\in \mathfrak t$ such that $V_0:= \ker \mathrm d^2(h-\mathbf J^ξ)(0)$ has linearly independent weights, there is a manifold tangent to $V_0$ that consists of relative equilibria with generators in $\mathfrak t$. Here we determine their isotropy wit…
For the curved n-body problem in S^3, we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium configuration if and only if n is odd and the masses are equal. The equilibrium configuration is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on S^1 embedded in S^2. We then study the stability of the associated relative equilibria on two invariant manifolds, T^*((§^1)^n\D) and T^*((§^2)^n\D). We show that they are Lyapunov stable on S^1, they are Lyapunov stable on S^2 if the absolute value of a…
We consider the linear stabilities of the regular n-gon relative equilibria of the (1+n)-body problem. It is shown that there exist at most two kinds of infinitesimal bodies arranged alternatively at the vertices of a regular n-gon when n is even, and only one set of identical infinitesimal bodies when n is odd. In the case of n even, the regular n-gon relative equilibrium is shown to be linearly stable when n>=14. In each case of n=8,10,12, linear stability can also be preserved if the ratio of two kinds of masses belongs to an open interval. When n is odd, the related conclusion on the linea…
In an earlier paper \cite{yu2021Finiteness}, we showed that there are finitely many stationary configurations (consisting of equilibria, rigidly translating configurations, relative equilibria and collapse configurations) in the planar four-vortex problem. However, we only established finiteness of collapse configurations in the sense of prescribing a collapse constant. In this paper, by developing ideas of Albouy-Kaloshin and Hampton-Moeckel to do an analysis of the singularities, we further show that there really are finitely many collapse configurations in the four-vortex problem. This is a…
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 st…
Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion. Our main result is a sufficient condition to detect spectral (hence linear) instability. Namely, we prove that if the Morse index of an equilibrium point with even nullity is odd, then the associated relative equilibrium is spectrally unstable. The proof is based on some refined formulæ for computing the spectral flow. As a notable …
Given $n$ point masses turning in a plane at a constant speed, this paper deals with the global bifurcation of periodic solutions for the masses, in that plane and in space. As a special case, one has a complete study of n identical masses on a regular polygon and a central mass. The symmetries of the problem are used in order to find the irreducible representations, the linearization, and with the help of the orthogonal degree theory, all the symmetries of the bifurcating branches.
We show existence of relative periodic orbits (a.k.a. relative nonlinear normal modes) near relative equilibria of a symmetric Hamiltonian system under an appropriate assumption on the Hessian of the Hamiltonian. This gives a relative version of the Moser-Weinstein theorem. The paper supersedes an earlier paper (this arxiv math.SG/9906007), which contains a mistake.
We study a classical model of isosceles triatomic "A-B-A" molecules. The atoms, considered mass points, interact mutually via a generic repulsive-attractive binary potential. First we show that the steady states, or relative equilibria (RE), corresponding to rotations about the molecule symmetry axis may be determined qualitatively assuming the knowledge of 1) the shape of the binary interaction potential, 2) the equilibrium diatomic distances (i.e., the equilibrium bond length) of the A-A and A-B molecules, and 3) the distance at which the RE of the diatomic A-A molecule ceases to exist. No a…
The relative equilibria for the spherical, finite density 3 body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical 5 relative equilibria for the point-mass 3-body problem. None of the identified relative equilibria exist or are stable over all values of angular momentum. The stability and bifurcation pathways of these relative equilibria are mapped out as the angular momentum of the system is increased. This is done under the assumption that they have equal and constant densities and that the entire system rotates about …
For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given.
The system of four point vortices in the plane has relative equilibria that behave as composite particles, in the case where three of the vortices have strength $-Γ/3$ and one of the vortices has strength $Γ$. These relative equilibria occur at nongeneric momenta. The reduction of this system, at those momenta, by continuous and then discrete symmetries, classifies the 4-vortex states which have been observed as products of collisions of two such composite particles. In this article I explicitly calculate these reductions, and show they are qualitatively identical one degree of freedom systems…
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…
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 the Lie algebra of the symmetry group, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more easily computed criterion for stability. The result is also extende…
This paper gives an analysis of the movement of n vortices on the sphere. When the vortices have equal circulation, there is a polygonal solution that rotates uniformly around its center. The main result concerns the global existence of relative periodic solutions that emerge from this polygonal relative equilibrium. In addition, it is proved that the families of relative periodic solutions contain dense sets of choreographies.
The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an emphasis on the relation between symmetries of the configurations and symmetries in the masses, or lack thereof. We prove that in the $N=4$, $N=6$, and $N=8$ Newtonian coorbital problems there exist symmetric relative equilibria with asymmetric positive masses. This result can be generalized to other homogeneous potentials, and we conjecture similar results hol…
We consider the 3-body problem in 3-dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist only for equal values of the masses.
In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy-momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between these. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced inst…
We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodríguez-Olmos's criter…
Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability these relative equilibria under momentum changing perturbations is not accessible by Lyapunov functions obtained from energy and momentum. A blow-up construction transforms the stability problem to the analysis symmetry-breaking perturbations of Hamiltonian relative equilibria. As such, the stability follows by KAM theory rather than energy-momentum confinement.
In this paper the non-canonical Hamiltonian dynamics of a gyrostat in the three body problem will be examined. By means of geometric-mechanics methods some relative equilibria of the dynamics of a gyrostat in Newtonian interaction with two rigid bodies will be studied. Taking advantage of the results obtained in previous papers, working on the reduced problem, the bifurcations of these relative equilibria will be studied. The instability of Eulerian relative equilibria if the gyrostat is close to a sphere is proven. Necessary and sufficient conditions will be provided for lineal stability of L…
