1 Introduction

The governing equations for an incompressible inviscid fluid were formulated by Euler in 1757  [11] and have since been cardinal in the ever growing field of hydrodynamics. On a general orientable Riemannian manifold M, Euler’s equations are

$$\begin{aligned} \dot{{\mathbf {v}}} + \nabla _{{\mathbf {v}}}{\mathbf {v}} = -\nabla p, \qquad {\text {div}}{\mathbf {v}} = 0, \end{aligned}$$
(1)

where \({\mathbf {v}}\) is a vector field on M describing the motion of infinitesimal fluid particles, p is the pressure function, and \(\nabla _{\mathbf {v}}\) denotes the co-variant derivative along \({\mathbf {v}}\). Arnold [2] discovered a rich and beautiful geometry underlying these equations; they can be interpreted as the geodesic equation for a right-invariant Riemannian metric on the infinite-dimensional Lie group of volume preserving diffeomorphism of M. For an entry-level introduction to Arnold’s discovery see the lecture notes by Modin [32]. More detailed expositions are given by Arnold and Khesin [3] and by Khesin and Wendt [18].

101 years after Euler’s influential paper on incompressible fluids, Helmholtz [14] showed that the 2D Euler equations exhibit special solutions consisting of a finite number of point-vortices. These are non-smooth solutions where \({\text {curl}}{\mathbf {v}}\), called vorticity, is a finite sum of Dirac distributions

$$\begin{aligned} \omega :={\text {curl}}{\mathbf {v}} = \sum _{i=1}^N \Gamma _i \delta _{{\mathbf {r}}_i}. \end{aligned}$$

Here, \(\Gamma _i \in \mathbb {R}\backslash \{0\}\) is the strength of vortex i and \({\mathbf {r}}_i \in M\) is its position (M can be any two-dimensional orientable Riemannian manifold). If \(G:M\times M \rightarrow \mathbb {R}\) denotes the Green’s function on M for the Laplacian, i.e., for \({\mathbf {r}}\in M\) we have

$$\begin{aligned} \Delta G(\cdot ,{\mathbf {r}}) = \delta _{{\mathbf {r}}}, \end{aligned}$$

then the equations of motion for N point-vortices are given by

$$\begin{aligned} \dot{{\mathbf {r}}}_i = \sum _{i\ne j} \Gamma _j \nabla ^\bot _{\mathbf{r}_i} G({\mathbf {r}}_i,{\mathbf {r}}_j). \end{aligned}$$

Here, \(\nabla ^\bot _{{\mathbf {r}}_i}\) is the skew gradient with respect to \({\mathbf {r}}_i\), i.e., the operation which first takes the gradient with respect to \({\mathbf {r}}_i\) and then rotates this vector by \(\pi /2\) in positive orientation. The point-vortex equations stated like this are rather abstract; in section 2 below we present these equations explicitly for different manifolds M.

Kirchhoff [21] was the first to point out that the point-vortex system has a Hamiltonian structure; in the planar case \({\mathbf {r}}_i = (x_i,y_i)\) the canonical coordinates are given by pairs of Euclidean coordinates \(x_i\) and \(y_i\) scaled by the strengths \(\Gamma _i\). Thus, point-vortex dynamics fits within the realm of Hamiltonian mechanics, a central observation for understanding its phase space geometry.

The classical way of thinking is that point-vortices constitute a formal ansatz for solutions of the Euler equations. By following Arnold’s approach, however, Marsden and Weinstein [28] showed that point-vortex solutions automatically appear from a systematic study of the symmetries of the system.

The interest in point-vortex dynamics has been growing steadily since the work of Helmholtz. Historically, it is fair to say that the field has emerged in two branches. One branch, originating from Onsager [38], is to study a large number \(N\gg 1\) of point-vortices via the tools provided by statistical mechanics for Hamiltonian dynamics. Onsager’s work laid out a statistical theory of hydrodynamics for predicting the formation of coherent structures in 2D turbulence. Since then, mathematicians and physicists have followed up on his approach, which has led to many deep and beautiful results; see [25] and references therein. However, the question concerning the long-time behaviour of classical solutions to the 2D Euler equations is still unanswered; numerical simulations suggest a more complicated generic behaviour than predicted by theories based on statistical mechanics (see Sect. 7 below).

The other branch is to study a few number of point-vortices, and ask whether the dynamics is integrable. Early contributors are Gröbli [12] and Poincaré [39], who explicitly (Gröbli) and implicitly (Poincaré) proved integrability of the planar \(N=3\) case. Since then, many results were obtained, on various domains, on integrability, equilibrium solutions, and relative equilibrium solutions; for an overview see the survey by Aref [1] and the monograph by Newton [37].

The purpose of this paper is to advocate symplectic reduction theory—an underused tool in dynamical systems—as a universal framework for proving integrability results of point-vortex dynamics (and variations thereof) on two-dimensional manifolds. That symplectic reduction theory can be used to obtain integrability results was already stressed in the paper by Marsden and Weinstein [28]. Indeed, concerning the planar case they say: “For \(N=3\) one can check that the motion is (completely) integrable in the sense that the (non-abelian) reduced phase spaces are points”. They then go on to say: “However one can also see that the dynamics of 3 point-vortices is (completely) integrable by exhibiting 3 independent integrals in involution...” In the literature since, mostly the second method—to find enough first integrals in involution—has been used. Kimura [20], in particular, used this method to prove integrability of the \(N=3\) case on any surface of constant curvature. However, in many ways, the symplectic reduction approach is more fundamental, as it starts from only the Hamiltonian structure and the symmetries.

Concerning \(N=4\), Ziglin [46] gave the earliest results on non-integrability. A few years later Marsden and Weinstein [28] said: “The motion of four vortices is generally believed to be chaotic.” Today, we know that \(N=4\) is a border-liner: depending on the geometry of the domain, and on other conditions such as total circulation or momentum, the \(N=4\) case may be integrable. For example, Eckhardt [10] showed that the planar case yields integrable dynamics when restricted to the subset of solutions with vanishing total circulation and momentum. More recently, Sakajo [42] showed that the \(N=4\) case on the sphere with vanishing total linear momentum is integrable. The proofs of these results are based on explicit calculations in specific coordinates. A main point of our paper is to show that the same results quickly fall out from the symplectic reduction framework essentially without any calculations.

We remark that there is also another approach to integrability for point-vortex dynamics, based on noncommutative integrability where the integrals of motion form a Lie algebra [5, 6]. It is possible that the symplectic reduction approach considered here and the noncommutative integrability approach are two viewpoints of the same theory. However, such a connection is not investigated here.

Our motivation for questions of integrability of point-vortex dynamics originates from recent numerical results for 2D Euler equations indicating that integrability of point-vortex dynamics, rather than prevailing statistical mechanics based theories, is central for predicting the long-time behaviour of solutions [34].

The paper is organized as follows. In Sect. 2, we describe, in more detail, the point-vortex equations and their Hamiltonian structures on the sphere, the plane, the hyperbolic plane, and the flat torus. Integrability results are given in Sect. 3. A brief review of symplectic reduction theory is given in Sect. 4. In Sect. 5, we prove the integrability results using symplectic reduction. A summary of the known non-integrability results are given in Sect. 6. Thereafter, Sect. 7 contains an outline of the connection between long-time behaviour of 2D Euler equations and point-vortex integrability. Finally, in Appendix A, we provide a gallery of point-vortex solutions—a sort of ‘visual summary’ of the paper.

2 Point-Vortex Equations and Their Conservation Laws

In this section, we give more detailed presentations of point-vortex dynamics in four specific cases: the sphere \(\mathbb {S}^2\), the plane \(\mathbb {R}^2\), the hyperbolic plane \(\mathbb {H}^2\), and the flat torus \(\mathbb {T}^2=\mathbb {R}^2/\mathbb {Z}^2\). From the point of view of symplectic reduction, these cases correspond to different structures of the symmetry group: compact non-Abelian, semi-direct product, non-compact but semi-simple, and Abelian. These structures illustrate well the different scenarios that can occur in symplectic reduction, as we shall see in Sect. 4 below.

2.1 The Sphere

Consider the sphere \(\mathbb {S}^2\) embedded in Euclidean 3-space as the subset of unitary vectors. The standard area form equips \(\mathbb {S}^2\) with a symplectic structure, denoted \(\Omega _{\mathbb {S}^2}\). The corresponding Poisson bracket is

$$\begin{aligned} \{F,G \}({\mathbf {r}}) = (\nabla F \times \nabla G)\cdot {\mathbf {r}} . \end{aligned}$$

We may think of \(\nabla F\) and \(\nabla G\) as the intrinsic Riemannian gradients on \(\mathbb {S}^2\), but equally well as Euclidean gradients for extensions of F and G to a neighbourhood of \(\mathbb {S}^2\) in \(\mathbb {R}^3\); the parts of \(\nabla F\) and \(\nabla G\) not orthogonal to \(\mathbb {S}^2\) are cancelled out in the triple product.

The phase space of N point-vortices on \(\mathbb {S}^2\) is given by

$$\begin{aligned} P^N_{\mathbb {S}^2} = \{ ({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in (\mathbb {S}^2)^N \mid {\mathbf {r}}_i \ne {\mathbf {r}}_j \text { for } i\ne j \}, \end{aligned}$$

equipped with the direct product symplectic structure

$$\begin{aligned} \Omega _N({\mathbf {u}},{\mathbf {w}}) = \sum _{i=1}^N \Gamma _i \Omega _{\mathbb {S}^2}({\mathbf {u}}_i,{\mathbf {w}}_i), \end{aligned}$$

where as before, \(\Gamma _i\) should be interpreted as the vortex strengths and \({\mathbf {u}}\) and \({\mathbf {w}}\) are tangent vectors of \(P^N_{\mathbb {S}^2}\). The equations of motion for a Hamiltonian \(H = H({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)\) are then given by

$$\begin{aligned} \dot{{\mathbf {r}}}_i = \underbrace{\frac{1}{\Gamma _i}{\mathbf {r}}_i \times \nabla _{{\mathbf {r}}_i}H}_{X_H} , \end{aligned}$$
(2)

where \(X_H\) is called the Hamiltonian vector field. Notice that, for any choice of Hamiltonian, the right hand side is tangent to \(\mathbb {S}^2\) so the dynamics evolves on \((\mathbb {S}^2)^N\). Since the Green’s function on \(\mathbb {S}^2\) is given by \(G({\mathbf {r}},{\mathbf {r}}') = -\frac{1}{4\pi }\log (1-{\mathbf {r}}\cdot {\mathbf {r}}')\), the specific Hamiltonian corresponding to point-vortex dynamics is

$$\begin{aligned} H({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = -\dfrac{1}{4\pi }\sum _{i\ne j}\Gamma _i\Gamma _j\log (1-\mathbf{r}_i\cdot {\mathbf {r}}_j), \end{aligned}$$
(3)

which leads to the point-vortex equations on the sphere

figure a

Let us now turn to the symmetries for the system (4). Clearly, the Hamiltonian (3) is invariant under the diagonal action of \(\mathrm {SO}(3)\) on \(P^N_{\mathbb {S}^2}\). That is, if \(R\in \mathrm {SO}(3)\) then

$$\begin{aligned} H(R{\mathbf {r}}_1,\ldots ,R{\mathbf {r}}_N) = H(\mathbf{r}_1,\ldots ,{\mathbf {r}}_N). \end{aligned}$$

Furthermore, this action preserves the symplectic form \(\Omega _{N}\) (since it is isometric and, therefore, preserves the area form on each sphere). By the Hamiltonian version of Nöther’s theorem (cf. [27, Thm. 11.4.1]) the symmetry, together with the fact that the action is symplectic, leads to a conservation law stated in terms of the momentum map associated with the action of \(\mathrm {SO}(3)\) on \(P^N_{\mathbb {S}^2}\). Recall that the momentum map \({\mathbf {J}}:P^N_{\mathbb {S}^2}\rightarrow \mathfrak {so}(3)^*\) is defined by the condition that, for any \(\varvec{\xi }\in \mathfrak {so}(3)\), the Hamiltonian vector field

$$\begin{aligned} X_{\langle {\mathbf {J}}(\cdot ),\varvec{\xi }\rangle }(\mathbf{r}_1,\ldots ,{\mathbf {r}}_N) = \frac{1}{\Gamma _i}{\mathbf {r}}_i \times \nabla _{{\mathbf {r}}_i} \langle {\mathbf {J}}(\mathbf{r}_1,\ldots ,{\mathbf {r}}_N),\varvec{\xi }\rangle \end{aligned}$$
(5)

corresponds to the infinitesimal action of \(\xi \) on \(P^N_{\mathbb {S}^2}\). If we identify \(\mathfrak {so}(3)\) with \(\mathbb {R}^3\) via the usual hat map (cf. [27, Eq. 9.2.7]), then the infinitesimal action is given by

$$\begin{aligned} \varvec{\xi }\cdot ({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = (\varvec{\xi }\times {\mathbf {r}}_1,\ldots ,\varvec{\xi }\times \mathbf{r}_N), \end{aligned}$$
(6)

i.e., infinitesimal rotation of each \({\mathbf {r}}_i\) about the axis \(\varvec{\xi }\). From (2) it follows that if

$$\begin{aligned} \langle {\mathbf {J}}({\mathbf {r}}_1,\ldots ,\mathbf{r}_N),\varvec{\xi }\rangle = \sum _{i=1}^N\Gamma _i {\mathbf {r}}_i\cdot \varvec{\xi }\end{aligned}$$

then the right hand side of (5) equal that of (6). Thus, identifying \(\mathfrak {so}(3)^*\) with \(\mathbb {R}^3\) via the Euclidean inner product, we get

$$\begin{aligned} {\mathbf {J}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i . \end{aligned}$$
(7)

From Nöther’s theorem, it follows that the three components of \({\mathbf {J}}\), called total linear momentum, are conserved for the point-vortex flow (4), or, more generally, for any \(\mathrm {SO}(3)\)-invariant Hamiltonian flow on \(P^N_{\mathbb {S}^2}\).

We now come to a property of momentum maps that is central in the symplectic reduction framework, namely equivariance. A momentum map is called equivariant if it commutes with the symplectic action of the underlying symmetry group. Thus, the momentum map \({\mathbf {J}}:P^N_{\mathbb {S}^2}\rightarrow \mathfrak {so}(3)^*\) is equivariant if

$$\begin{aligned} {{\,\mathrm{Ad}\,}}_R^*{\mathbf {J}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = \mathbf{J}(R{\mathbf {r}}_1,\ldots ,R{\mathbf {r}}_N) \quad \forall \;R\in \mathrm {SO}(3), \end{aligned}$$

where \({{\,\mathrm{Ad}\,}}_R^*\) is the coadjoint action, here defined as matrix-vector multiplication. It is easy to check that this condition is fulfilled, so the momentum map \({\mathbf {J}}\) on \(P^N_{\mathbb {S}^2}\) is indeed equivariant. A general result states that if the symmetry group is semi-simple, then an equivariant momentum map always exists. In the next section, however, we encounter a symplectic action that does not have an equivariant momentum map, which has consequences for the symplectic reduction.

2.2 The Plane

Consider now the Euclidean plane \(\mathbb {R}^2\), with standard coordinates \({\mathbf {r}} = (x,y)\). The canonical symplectic structure is \(\Omega _{\mathbb {R}^2} \equiv dx\wedge d y\), which gives the Poisson bracket

$$\begin{aligned} \{F,G \}({\mathbf {r}}) = \nabla F \cdot \nabla ^\perp G, \end{aligned}$$

where \(\nabla \) is the standard gradient operator and \(\nabla ^\perp \) is the skew-gradient: \(\nabla ^\perp G = (\partial _y G,-\partial _x G)\). The phase space of N point-vortices on \(\mathbb {R}^2\) is given by

$$\begin{aligned} P^N_{\mathbb {R}^2} = \{ ({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in (\mathbb {R}^2)^N \mid {\mathbf {r}}_i \ne {\mathbf {r}}_j \text { for } i\ne j \}, \end{aligned}$$

equipped with the scaled direct product symplectic structure

$$\begin{aligned} \Omega _N({\mathbf {u}},{\mathbf {w}}) = \sum _{i=1}^N \Gamma _i \Omega _{\mathbb {R}^2}({\mathbf {u}}_i,{\mathbf {w}}_i). \end{aligned}$$

The equations of motion for a Hamiltonian \(H = H(\mathbf{r}_1,\ldots ,{\mathbf {r}}_N)\) are the set of scaled canonical Hamiltonian equations

$$\begin{aligned} \dot{{\mathbf {r}}}_i = \frac{1}{\Gamma _i} \nabla _{{\mathbf {r}}_i}^\perp H. \end{aligned}$$
(8)

Since the Green’s function for the Laplacian on \(\mathbb {R}^2\) is \(G({\mathbf {r}},{\mathbf {r}}') = -\frac{1}{4\pi }\log (|\mathbf{r}-{\mathbf {r}}' |^2)\), the Hamiltonian for point-vortex dynamics is

$$\begin{aligned} H({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = -\dfrac{1}{4\pi }\sum _{i\ne j}\Gamma _i\Gamma _j \log (|\mathbf{r}_i-{\mathbf {r}}_j|^2). \end{aligned}$$
(9)

From (8), it follows that the corresponding point-vortex equations on the plane are

figure b

Concerning the symmetries of the equations (10), it is clear that the Hamiltonian (9) is invariant to the diagonal action of the Euclidean group \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\). We recall that the semidirect group product is defined as:

$$\begin{aligned} (R_1,{\mathbf {u}}_1)\cdot (R_2,{\mathbf {u}}_2) = (R_1R_2,{\mathbf {u}}_1 + R_1{\mathbf {u}}_2) \end{aligned}$$

for \(R_1,R_2\in \mathrm {SO}(2)\) and \({\mathbf {u}}_1,\mathbf{u}_2\in \mathbb {R}^2\) and the action of \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) on \(\mathbb {R}^2\) is:

$$\begin{aligned} (R,{\mathbf {u}})\cdot {\mathbf {r}} = R{\mathbf {r}} + {\mathbf {u}}, \end{aligned}$$

for \(R\in \mathrm {SO}(2)\) and \({\mathbf {u}},{\mathbf {r}}\in \mathbb {R}^2\). The action of the Euclidean group is clearly symplectic, being area-preserving. Therefore, we get the following momentum map \(\mathbf{J }:P^N_{\mathbb {R}^2}\rightarrow (\mathfrak {so}(2)\ltimes \mathbb {R}^2)^*\):

$$\begin{aligned} {\mathbf {J}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = \left( \dfrac{1}{2}\sum _{i=1}^N \Gamma _i |{\mathbf {r}}_i|^2, \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i\right) , \end{aligned}$$
(11)

whose first component is the total angular momentum, and the second component contains the total linear momenta. However, the momentum map (11) is not, in general, equivariant; the condition for equivariance is that the total circulation vanishes, i.e., \(\sum _{i}\Gamma _i = 0\). In fact, the coadjoint action of \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) on \((\mathfrak {so}(2)\ltimes \mathbb {R}^2)^*\) is:

$$\begin{aligned} (R,{\mathbf {u}})\cdot (\xi ,{\mathbf {w}}) = (\xi + \mathbf{u}^\top (R{\mathbf {w}}),R{\mathbf {w}}) \end{aligned}$$

for \((R,{\mathbf {u}})\in \mathrm {SO}(2)\ltimes \mathbb {R}^2\) and \((\xi ,{\mathbf {w}})\in (\mathfrak {so}(2)\ltimes \mathbb {R}^2)^*\) (see [26, ch. 4.2]). Therefore,

$$\begin{aligned} {{\,\mathrm{Ad}\,}}^*_{(R,{\mathbf {u}})} ({\mathbf {J}}({\mathbf {r}}_1,\ldots ,\mathbf{r}_N))=\left( \dfrac{1}{2}\sum _{i=1}^N \Gamma _i |{\mathbf {r}}_i|^2 + {\mathbf {u}}^\top \left( \sum _{i=1}^N \Gamma _i R{\mathbf {r}}_i\right) , \sum _{i=1}^N \Gamma _i R{\mathbf {r}}_i\right) \end{aligned}$$

and

$$\begin{aligned} {\mathbf {J}}((R,{\mathbf {u}})\cdot {\mathbf {r}}_1,\ldots ,(R,\mathbf{u}) \cdot {\mathbf {r}}_N)=&\left( \dfrac{1}{2}\sum _{i=1}^N \Gamma _i |{\mathbf {r}}_i|^2 + {\mathbf {u}}^\top \left( \sum _{i=1}^N \Gamma _i R{\mathbf {r}}_i\right) \right. \\&\left. + \dfrac{1}{2}\left( \sum _{i=1}^N \Gamma _i\right) |{\mathbf {u}}|^2,\sum _{i=1}^N \Gamma _i R{\mathbf {r}}_i + \left( \sum _{i=1}^N \Gamma _i\right) {\mathbf {u}}\right) \end{aligned}$$

Hence, to satisfy the equivariance condition:

$$\begin{aligned} {{\,\mathrm{Ad}\,}}^*_{(R,{\mathbf {u}})} ({\mathbf {J}}({\mathbf {r}}_1,\ldots ,\mathbf{r}_N)) = {\mathbf {J}}((R,{\mathbf {u}})\cdot \mathbf{r}_1,\ldots ,(R,{\mathbf {u}})\cdot {\mathbf {r}}_N) \qquad \forall \;(R,{\mathbf {u}})\in \mathrm {SO}(2)\ltimes \mathbb {R}^2. \end{aligned}$$

we must have that the total circulation is zero, i.e., \(\sum _{i}\Gamma _i = 0\).

To construct a momentum maps that is always equivariant, we shall explore a less obvious symmetry of equations (10). Consider the group \(G\subset \mathrm {GL}(2N,\mathbb {R})\) generated by the infinitesimal generators \(\xi ,\eta \) defined by

$$\begin{aligned} \xi = \left[ \begin{matrix} 0 &{} \Gamma _1 &{} 0 &{} \Gamma _2 &{} \ldots &{} 0 &{} \Gamma _N \\ -\Gamma _1 &{} 0 &{} -\Gamma _2 &{} 0 &{} \ldots &{} -\Gamma _N &{} 0\\ 0 &{} \Gamma _1 &{} 0 &{} \Gamma _2 &{} \ldots &{} 0 &{} \Gamma _N \\ -\Gamma _1 &{} 0 &{} -\Gamma _2 &{} 0 &{} \ldots &{} -\Gamma _N &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} \Gamma _1 &{} 0 &{} \Gamma _2 &{} \ldots &{} 0 &{} \Gamma _N \\ -\Gamma _1 &{} 0 &{} -\Gamma _2 &{} 0 &{} \ldots &{} -\Gamma _N &{} 0\\ \end{matrix} \right] \eta = \left[ \begin{matrix} 0 &{} 1 &{} 0 &{} 0 &{} \ldots &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 0 &{} \ldots &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} \ldots &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 &{} \ldots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \ldots &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} \ldots &{} -1 &{} 0\\ \end{matrix} \right] . \end{aligned}$$

We notice that \(\eta \) is the infinitesimal generator of \(\mathrm {SO}(2)\) and \(\xi \) is the infinitesimal generator of a 1-dimensional group K. The symplectic form \(\Omega _N\) on \(\mathbb {R}^{2N}\) has the following representation in matrix form:

$$\begin{aligned} \Omega _N = \left[ \begin{matrix} 0 &{} \Gamma _1 &{} 0 &{} 0 &{} \ldots &{} 0 &{} 0 \\ -\Gamma _1 &{} 0 &{} 0 &{} 0 &{} \ldots &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \Gamma _2 &{} \ldots &{} 0 &{} 0 \\ 0 &{} 0 &{} -\Gamma _2 &{} 0 &{} \ldots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \ldots &{} 0 &{} \Gamma _N \\ 0 &{} 0 &{} 0 &{} 0 &{} \ldots &{} -\Gamma _N &{} 0\\ \end{matrix} \right] . \end{aligned}$$

It is straightforward to check that \([\xi ,\eta ]=0\), \(\Omega _N\xi + \xi ^\top \Omega _N=0\) and \(\Omega _N\eta + \eta ^\top \Omega _N=0\). Therefore, \(G=\mathrm {SO}(2)\times K\) is Abelian and the action of G on \(\mathbb {R}^{2N}\) is symplectic. The action of G has momentum map \({\mathbf {L}}:P^N_{\mathbb {R}^2}\rightarrow \mathfrak {so}(2)^*\oplus \mathfrak {k}^*\):

$$\begin{aligned} {\mathbf {L}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = \dfrac{1}{2}\left( \sum _{i=1}^N \Gamma _i |\mathbf{r}_i|^2,\Big |\sum _{i=1}^N \Gamma _i {\mathbf {r}}_i\Big |^2\right) . \end{aligned}$$

It is straightforward to check that \({\mathbf {L}}\) is an equivariant momentum map. Indeed, on one hand, SO(2) acts as a diagonal isometry of \(P^N_{\mathbb {R}^2}\), and therefore, it preserves the Euclidean norms. On the other hand, for the action of K on \({\mathbf {L}}\), we get:

$$\begin{aligned} \dfrac{d}{dt}\bigg |_{t=0}&{\mathbf {L}}(\exp (t\xi )({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N))\\&=\left( \left( \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i\right) ^\perp \cdot \left( \sum _{i=1}^N \Gamma _i \mathbf{r}_i\right) ,\left( \sum _{i=1}^N \Gamma _i\right) \left( \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i\right) ^\perp \cdot \left( \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i\right) \right) =0. \end{aligned}$$

Hence, since \(SO(2)\times K\) is Abelian, the momentum map \(\mathbf{L}\) is equivariant.

Remark 1

The geometric interpretation of the one-parameter group K acting on \(({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)\) is the following. First, if the total linear momentum \({\mathbf {L}}\) vanishes, then K acts as the identity. Second, denote by \(\Gamma =\sum _i \Gamma _i\) the total circulation. If \({\mathbf {L}}\ne 0\), then K rotates synchronously the point vortices \(({\mathbf {r}}_1,\ldots ,\mathbf{r}_N)\) in circles of radius \(r = \frac{\left| {\mathbf {L}} \right| }{\Gamma }\) and with centres \({\mathbf {C}}_i\) which satisfy \({\mathbf {C}}_i + \frac{{\mathbf {L}}}{\Gamma }= {\mathbf {r}}_i\). If the total circulation vanishes, i.e., \(\Gamma =0\), then the radius becomes infinite and the motion is along straight lines.

2.3 The Hyperbolic Plane

Consider the hyperbolic plane model \(\mathbb {H}^2=\lbrace (x,y,z)\in \mathbb {R}^3|z^2-x^2-y^2=1,z>0\rbrace \), with Riemannian structure induced by the bilinear form on \(\mathbb {R}^3\):

$$\begin{aligned} {\mathbf {a}}\cdot _L {\mathbf {b}} = {\mathbf {a}}\cdot (L{\mathbf {b}}), \end{aligned}$$

for any \({\mathbf {a}},{\mathbf {b}}\in \mathbb {R}^3\) and \(L=\text{ diag }(-1,-1,1)\). As on the sphere, the volume form on \(\mathbb {H}^2\) defines a symplectic structure \(\Omega _{\mathbb {H}^2}\). Defining \({\mathbf {a}}\times _L {\mathbf {b}} :=L ({\mathbf {a}}\times {\mathbf {b}})\), the corresponding symplectic form is given by

$$\begin{aligned} \Omega _{\mathbb {H}^2}({\mathbf {c}})({\mathbf {a}},{\mathbf {b}}) = \mathbf{c}\cdot _L({\mathbf {a}}\times _L {\mathbf {b}}). \end{aligned}$$

The phase space of N point-vortices on \(\mathbb {H}^2\) is given by

$$\begin{aligned} P^N_{\mathbb {H}^2} = \{ ({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in (\mathbb {H}^2)^N \mid {\mathbf {r}}_i \ne {\mathbf {r}}_j \text { for } i\ne j \}, \end{aligned}$$

equipped with the so-called direct product symplectic structure

$$\begin{aligned} \Omega _N({\mathbf {u}},{\mathbf {w}}) = \sum _{i=1}^N \Gamma _i \Omega _{\mathbb {H}^2}({\mathbf {u}}_i,{\mathbf {w}}_i), \end{aligned}$$

with \(\Gamma _i\) as before and \({\mathbf {u}}\), \({\mathbf {w}}\) tangent vectors of \(P^N_{\mathbb {H}^2}\). The equations of motion for a Hamiltonian \(H = H({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)\) are

$$\begin{aligned} \dot{{\mathbf {r}}}_i = \frac{1}{\Gamma _i}{\mathbf {r}}_i\times _L \nabla _{{\mathbf {r}}_i}H . \end{aligned}$$
(12)

For point-vortices, the Hamiltonian is

$$\begin{aligned} H = -\frac{1}{4\pi }\sum _{i\ne j}\Gamma _i\Gamma _j \log \left( \dfrac{{\mathbf {r}}_i\cdot _L {\mathbf {r}}_j + 1}{\mathbf{r}_i\cdot _L {\mathbf {r}}_j - 1}\right) . \end{aligned}$$

which gives the point-vortex equations on the hyperbolic plane [15, 16, 36]

figure c

These equations constitute a Lie–Poisson system on \((\mathfrak {sl}(2,\mathbb {R})^*)^N\simeq (\mathbb {R}^3,\times _L)^N\). Equations (13) constrain the vortices to the hyperboloid \(x^2+y^2-z^2=-1\). Furthermore, the \(\mathrm {SL}(2,\mathbb {R})\) symmetry of (13) gives, analogously to \(\mathbb {S}^2\), the equivariant momentum map

$$\begin{aligned} {\mathbf {J}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i. \end{aligned}$$

For further background on point-vortices on the hyperbolic plane, see [15, 16, 36]. For hydrodynamics on the hyperbolic plane, see [17].

2.4 The Flat Torus

The easiest way to work with the flat torus \(\mathbb {R}^2/\mathbb {Z}^2\) is to extend everything to \(\mathbb {R}^2\) and then assure that all operations and equations are \(2\pi \)-periodic in both directions. Thus, with definitions as in subsection 2.2, the Hamiltonian for point-vortex dynamics on the flat torus, expanded to a Hamiltonian on the plane, is given by

$$\begin{aligned} H({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = -\dfrac{1}{4\pi }\sum _{i\ne j}\Gamma _i\Gamma _jh(x_i-x_j,y_i-y_j), \end{aligned}$$
(14)

where \(h(x,y) = -\dfrac{x^2}{2\pi }+\sum _{m=-\infty }^{+\infty }\log \left( \dfrac{\cosh (x-2\pi m)-\cos (y)}{\cosh (2\pi m)}\right) \). The corresponding equations of motion (cf. [45]) are

figure d

These equations are (equivalent to) the point-vortex equation on a flat torus, as explained in the following remark.

Remark 2

The equations (15) can be derived in two ways. The first, proposed in [45], starts from a \(2\pi \)-periodic distribution of vortices in the point-vortex equation on \(\mathbb {R}^2\) and sums up the infinite number of contributions given by the different vortices, obtaining in the limit a well defined vector field for the right hand side in (15). As shown in [45], this vector field is \(2\pi \)-periodic. Therefore, it can be seen as a vector field on \(\mathbb {T}^2\). The second way consist of explicitly calculate the Green’s function of the Laplacian on a flat two-torus. The calculations given in [24] confirm the equivalence of the two approaches, and the description of the Green’s function in terms of the First Jacobi Theta function guarantees the smoothness of the vector field on \(P^N_{\mathbb {R}^2}\). We stress that this equivalence is crucial in the analysis of the equations (15). In fact, on one hand, we have that the two-torus has non-trivial first de Rham cohomology, whereas on the other hand for \(\mathbb {R}^2\) the cohomology is trivial, and so for any infinitesimal symmetry, we get a conservation law. In particular, we get two more first integrals as stated below.

The Hamiltonian (14) is invariant with respect to the diagonal action of \(\mathbb {R}^2\). Since this action is symplectic, we get the (linear) momentum map

$$\begin{aligned} {\mathbf {J}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i. \end{aligned}$$
(16)

As noticed in remark 2, it is convenient to consider the point-vortex equations in \(\mathbb {R}^2\) according to the Hamiltonian (14). In this setting, the \(\mathbb {R}^2\) action on \(\mathbb {R}^2\) is free and proper, being the map R in Definition 5 invertible. However, the momentum map (16) is in general not equivariant. In fact, since \(\mathbb {R}^2\) is an Abelian group, the \({{\,\mathrm{Ad}\,}}\) operator is the identity. Hence, the equivariance equation \({{\,\mathrm{Ad}\,}}^*_{g^{-1}}\circ \mu =\mu \circ \rho _g\) implies that \(\mu =\mu \circ \rho _g\). We notice that this is true only if the circulation is zero, i.e., \(\Gamma _1+\dots +\Gamma _N=0\). Moreover, since \(\mathbb {R}^2\) is neither compact nor semisimple, Theorem [27, Thm 11.5.2] cannot be applied to get a modified equivariant momentum map. We notice that considering the point-vortex equation directly on the flat two-torus and the action of \(U(1)^2\) instead of \(\mathbb {R}^2\) would have in principle guarantee the equivariance of the momentum map, but the non-trivial first group of De Rham cohomology of the two-torus would have prevent the existence of the momentum map itself.

3 Integrability Results

In this section, we state results on integrability for the point-vortex equations presented in the previous section. As we shall see in the sections below, all of these results, which are valid not just for the point-vortex equations but for all invariant Hamiltonian systems on the respective phase spaces, follow directly from the symplectic reduction framework. Most of the results are known since before; the point here is to demonstrate how the results naturally fall out from symplectic reduction in a purely geometric way. In particular, symplectic reduction gives a clear geometric understanding for why the vanishing momentum and circulation cases are structurally different from the non-vanishing cases.

Before proceeding to the specific manifolds \(\mathbb {S}^2\), \(\mathbb {R}^2\), \(\mathbb {H}^2\), and \(\mathbb {T}^2\), we recall the following general definition:

Definition 1

Let P be a phase space manifold for a dynamical system and assume that P is acted upon by a Lie group G. A solution \(t\mapsto {\mathbf {r}}(t) \in P\) is called a G-relative equilibrium if there exists an \({\mathbf {r}}_0\in P\) and \(\varvec{\xi }\in \mathfrak {g}\) such that \({\mathbf {r}}(t) = \exp (t\varvec{\xi })\cdot {\mathbf {r}}_0\).

3.1 The Sphere

Integrability of the point-vortex equations (4) for \(N=3\) is given by Sakajo [41]. The proof is based on specific coordinates for which one finds three first integrals in involution. Eight years later Sakajo [42] also proved integrability of the vanishing momentum \(N=4\) case, essentially by reducing it to the \(N=3\) case. See also the work by Borisov et al. [7].

Theorem 1

(\(\mathbb {S}^2\) with non-vanishing momentum) Consider Hamilton’s equations (2) on \(P_{\mathbb {S}^2}^N\) for an \(\mathrm {SO}(3)\) invariant Hamiltonian, restricted to the part of phase space with non-vanishing total linear momentum

$$\begin{aligned} \{({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in P^N_{\mathbb {S}^2}\mid \mathbf{J}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \ne {\mathbf {0}} \}. \end{aligned}$$

  • If \(N=2\) all solution are relative equilibria for the \(\mathrm {SO}(2)\)-action, the isotropy group of the momentum \({\mathbf {J}}\).

  • If \(N=3\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (4) on \(P_{\mathbb {S}^2}^N\).

Theorem 2

(\(\mathbb {S}^2\) with vanishing momentum)] Consider Hamilton’s equations (2) on \(P_{\mathbb {S}^2}^N\) for an \(\mathrm {SO}(3)\) invariant Hamiltonian, restricted to the part of phase space with vanishing total linear momentum

$$\begin{aligned} \{({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in P^N_{\mathbb {S}^2}\mid \mathbf{J}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = {\mathbf {0}} \}. \end{aligned}$$
(17)

  • If \(N=2\) all solutions are equilibria of antipodal points.

  • If \(N=3\) all solution are relative equilibria for the \(\mathrm {SO}(3)\)-action.

  • If \(N=4\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (4) on \(P_{\mathbb {S}^2}^N\).

3.2 The Plane

Integrability of point-vortices in the plane were the first to be studied. Early results for \(N=3\) were given by Gröbli [12] and by Poincaré [39]. The \(N=4\) result was given by Eckhardt [10].

Theorem 3

(\(\mathbb {R}^2\) with non-vanishing circulation or momentum) Consider Hamilton’s equations (8) on \(P_{\mathbb {R}^2}^N\) for an \(\mathrm {SO}(2)\times K\) invariant Hamiltonian, restricted to the case of non-vanishing circulation

$$\begin{aligned} \begin{array}{ll} \sum _{i=1}^N\Gamma _i \ne 0\\ \end{array} \end{aligned}$$

or restricted to the part of phase space with non-vanishing total linear momentum

$$\begin{aligned} \begin{array}{ll} \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i \ne 0. \end{array} \end{aligned}$$

  • If \(N=2\) and \(\sum _i \Gamma _i \ne 0\) (non-vanishing circulation), then the solutions are \(\mathrm {SO}(2)\times K\) relative equilibria.

  • If \(N=2\) and \(\sum _i \Gamma _i = 0\) (vanishing circulation), then the solutions are \(\mathbb {R}^2\)-relative equilibria (travelling vortex dipoles).

  • If \(N=3\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (10) on \(P_{\mathbb {R}^2}^N\).

Theorem 4

(\(\mathbb {R}^2\) with vanishing circulation and momentum) Consider Hamilton’s equations (8) on \(P_{\mathbb {R}^2}^N\) for an \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) invariant Hamiltonian, restricted to case of vanishing circulation and total linear momentum:

$$\begin{aligned} \sum _{i=1}^N\Gamma _i = 0, \quad \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i = 0. \end{aligned}$$

  • If \(N=2\), the point-vortex equation (10) is not defined.

  • If \(N=3\) all solutions are \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) relative equilibria.

  • If \(N=4\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (10) on \(P_{\mathbb {R}^2}^N\).

3.3 The Hyperbolic Plane

Integrability results on the hyperbolic plane reflect the results on the sphere, as from a symplectic reduction point of view, the two settings are almost the same, with an equivariant momentum map for a semi-simple three-dimensional symmetry group.

Theorem 5

(\(\mathbb {H}^2\) with non-vanishing momentum) Consider Hamilton’s equations (12) on \(P_{\mathbb {S}^2}^N\) for an \(\mathrm {SL}(2)\) invariant Hamiltonian, restricted to the part of phase space with non-vanishing total linear momentum

$$\begin{aligned} \{({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in P^N_{\mathbb {H}^2}\mid \mathbf{J}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \ne {\mathbf {0}} \}. \end{aligned}$$

  • If \(N=2\) all solution are relative equilibria for the action of the isotropy subgroup of the momentum \({\mathbf {J}}\).

  • If \(N=3\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (13) on \(P_{\mathbb {H}^2}^N\).

Theorem 6

(\(\mathbb {H}^2\) with vanishing momentum) Consider the point-vortex equation (13) restricted to the part of phase space with vanishing total linear momentum

$$\begin{aligned} \{({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in P^N_{\mathbb {H}^2}\mid \mathbf{J}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) = {\mathbf {0}} \}. \end{aligned}$$

  • If \(N=2\) the point-vortex equation (13) is not defined.

  • If \(N=3\) all solution are relative equilibria for the \(\mathrm {SL}(2)\)-action.

  • If \(N=4\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (13) on \(P_{\mathbb {H}^2}^N\).

3.4 The Flat Torus

Integrability results for point-vortex dynamics on the flat torus were given by Stremler and Aref [43], in the case of zero total circulation, and Kilin and Artemova [19]. As they do, we take here the view-point that the phase space is embedded as periodic solutions in \(P_{\mathbb {R}^2}^N\).

Theorem 7

(\(\mathbb {T}^2\) with non-vanishing circulation) Consider Hamilton’s equations (8) on \(P_{\mathbb {R}^2}^N\) for an \(\mathbb {R}^2\) invariant Hamiltonian, restricted to the case of non-vanishing circulation:

$$\begin{aligned} \sum _{i=1}^N\Gamma _i \ne 0. \end{aligned}$$

  • If \(N=2\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (15) on \(P_{\mathbb {T}^2}^N\).

Theorem 8

(\(\mathbb {T}^2\) with vanishing circulation) Consider Hamilton’s equations (8) on \(P_{\mathbb {R}^2}^N\) for an \(\mathbb {R}^2\) invariant Hamiltonian, restricted to the case of vanishing circulation

$$\begin{aligned} \sum _{i=1}^N\Gamma _i = 0. \end{aligned}$$

  • If \(N=2\) all solutions are \(\mathbb {R}^2\)-relative equilibria.

  • If \(N=3\) the system is completely integrable (but solutions are typically not relative equilibria).

In particular, these results are valid for the point-vortex equations (15) on \(P_{\mathbb {T}^2}^N\).

4 Symplectic Reduction Theory

To prove the theorems stated in Sect. 3, we need to recall some definitions and notations. We will denote in the following the smooth action of a Lie group G on a manifold M with:

$$\begin{aligned}&\rho : G\times M\rightarrow M\\&\rho (g,p)=:\rho _g (p)=:g\cdot p, \end{aligned}$$

for each \(g\in G,p\in M\).

Definition 2

(Symplectic action) Let G be a Lie group acting on a smooth symplectic manifold \((M,\omega )\). Then the action of G is said to be symplectic if \(\rho ^*_g\omega = \omega \), for every \(g\in G\).

Definition 3

(Momentum map) Let G be a Lie group and \(\mathfrak {g}\) its Lie algebra. Assume that G acts symplectically on a smooth symplectic manifold \((M,\omega )\) and that \(\xi _p:=d_e\rho _{\exp (\xi )}(p)\), for every \(p\in M\), is its inifinitesimal action. Then a map \(\mu :M\rightarrow \mathfrak {g}^*\), fulfilling

$$\begin{aligned} d\langle \mu (p),\xi \rangle =\iota _{\xi _p}\omega , \end{aligned}$$

is called a momentum map.

When G is compact, and a momentum map exists, it can always be chosen to satisfy the equivariance identity:

$$\begin{aligned} \langle \mu (p),[\xi ,\eta ]\rangle =\lbrace \langle \mu (p),\xi \rangle ,\langle \mu (p),\eta \rangle \rbrace , \end{aligned}$$

for \(\xi ,\eta \in \mathfrak {g}\) and \(p\in M\), where the bracket \(\lbrace \cdot ,\cdot \rbrace \) is the Poisson bracket on M. In this case the Lie group action will be said to be Hamiltonian. A crucial fact on momentum maps is that they are conserved quantities for Hamiltonian systems with symmetries.

Theorem 9

[27] Let M be a smooth symplectic manifold and let G be a Lie group with Hamiltonian action on it and momentum map \(\mu :M\rightarrow \mathfrak {g}^*\). Let \(H:M\rightarrow \mathbb {R}\) be a smooth function such that, for each \(g\in G,x\in M\), \(H(g\cdot x)=H(x)\). Then, if \(\phi ^H_t\) is the flow of the Hamiltonian vector field \(X_H\), we have that \((\phi ^H_t)^*\mu =\mu \), for each \(t\ge 0\).

Let us now continue defining some properties of the group actions.

Definition 4

(Free action) Let G be a Lie group acting on a smooth manifold M. Then the action of G is said to be free if, for each \(x\in M\), \(g\cdot x=x\) implies \(g=e\).

Definition 5

(Proper action) Let G be a Lie group acting on a smooth manifold M. Then the action of G is said to be proper if the map \(R :G\times M\rightarrow M\times M\), \(R(g,p)=(\rho _g(p),p)\) is a proper map of topological spaces.

Definition 6

(Isotropy subgroup) Let G be a Lie group acting on a smooth manifold M. Then the set:

$$\begin{aligned} G_p = \lbrace g\in G \mid g\cdot p = p\rbrace \end{aligned}$$

is a Lie subgroup of G called isotropy subgroup.

Let us now recall the symplectic reduction theorem:

Theorem 10

(Symplectic reduction theorem, cf. [26]) Let G be a Lie group with Hamiltonian free and proper action on a smooth symplectic manifold \((M,\omega )\) and let \(\mu :M\rightarrow \mathfrak {g}^*\) be an equivariant momentum map with respect to this action. Then, for each \(\phi \in \mathfrak {g}^*\), the quotient:

$$\begin{aligned} J_\phi :=\mu ^{-1}(\phi )/G_\phi \end{aligned}$$

is a symplectic manifold of dimension \(d:=\dim (M)-2\dim (G_\phi )\), with symplectic form \(\omega _\phi \) uniquely characterized by

$$\begin{aligned}\pi ^*\omega _\phi =\iota ^*\omega ,\end{aligned}$$

where \(\pi :\mu ^{-1}(\phi )\rightarrow \mu ^{-1}(\phi )/G_\phi \) is the projection and \(\iota : \mu ^{-1}(\phi )\rightarrow M\) is the inclusion.

5 Proofs by Symplectic Reduction

In this section we prove the results stated in Sect. 3. All of our proofs are based on the symplectic reduction Theorem 10, which makes them very streamlined. Essentially, the proofs consist in checking that the action is free and proper, that the momentum map is equivariant, and then counting the dimensions of the reduced phase space.

5.1 The Sphere

For the sphere, the \(\mathrm {SO}(3)\) action on \(P_{\mathbb {S}^2}^N\) is free for any \(N\ge 3\) and for \(N=2\) if the points are not antipodal. It is also proper since \(\mathrm {SO}(3)\) is compact. Furthermore, the momentum map (7) is equivariant, as one can directly check.

Proof of Theorem 1

For \(N=2\), the \(\mathrm {SO}(3)\) action is free, unless the two vortices are antipodal, which from (4) implies that solutions are equilibria. Hence, Theorem 10 tells us that systems evolves on a zero-dimensional manifold, since \(G_\phi \simeq \mathrm {SO}(2)\). Thus, the reconstructed motion is an \(\mathrm {SO}(3)\) relative equilibrium, i.e., a steady rotation of the initial positions.

For \(N=3\), the \(\mathrm {SO}(3)\) action is always free and Theorem 10 tells us that the system evolves on a two-dimensional manifold. Any Hamiltonian system on a two-dimensional manifold is integrable, so the reconstructed system is also integrable, as follows, for example, from the standard Floquet theory (cf. [33, Proof of thm 3.1]). \(\square \)

Proof of Theorem 2

For \(N=2\), the vanishing momentum condition (17) implies that \({\mathbf {r}}_1,{\mathbf {r}}_2\) are antipodal points with equal strength. From (4) it follows directly that solutions are equilibria.

For \(N=3\), the \(\mathrm {SO}(3)\) action is free and Theorem 10 tells us that the reduced system evolves on a zero-dimensional manifold, so solutions are relative equilibria.

For \(N=4\), the \(\mathrm {SO}(3)\) action is free and by Theorem 10 the system evolves on a two-dimensional reduced manifold, so the dynamics is integrable. \(\square \)

5.2 The Plane

The action of \(\mathrm {SO}(2)\) on \(P_{\mathbb {R}^2}^N\) is free for \(N=1\) unless \({\mathbf {r}}_1 ={\mathbf {0}}\). The action is always free for \(N\ge 2\). The action of K is free and proper only under certain conditions, as stated in the following lemmas.

Lemma 1

If \(\sum \Gamma _i{\mathbf {r}}_i\ne 0\), the action \(\rho :K\times P_{\mathbb {R}^2}^N\rightarrow P_{\mathbb {R}^2}^N\) is free.

Proof

Since K is one-dimensional, the action of K is non-free if and only if K has fixed points. Let \(\xi \) be the infinitesimal generator of K, as defined in Sect. 2.2. Then, the action \(\rho \) of K is free if and only if the kernel of \(\xi \) is trivial. It is straightforward to check that the kernel of \(\xi \) is given by:

$$\begin{aligned} \text{ ker }(\xi )=\lbrace ({\mathbf {r}}_1,\dots ,{\mathbf {r}}_N)\in P_{\mathbb {R}^2}^N | \sum \Gamma _i{\mathbf {r}}_i = 0 \rbrace . \end{aligned}$$

Hence, under the non-vanishing linear momentum condition the action of K is free at any time. \(\square \)

Since the linear momentum is a first integral of the Eq. (9), it follows from Lemma 1 that the K action is free provided that the linear momentum is non-vanishing for the initial condition.

Lemma 2

If \(\sum \Gamma _i\ne 0\), the action \(\rho :K\times P_{\mathbb {R}^2}^N\rightarrow P_{\mathbb {R}^2}^N\) is proper.

Proof

Let \(\xi \) be the infinitesimal generator of K, as defined in Sect. 2.2. Then, it is straightforward to check that \(\xi \) has rank 2, for any \(N\ge 1\) and its non-zero eigenvalues are purely imaginary and equal to \(\pm i\sum \Gamma _i\ne 0\). Hence, the group K is bounded and closed in the operator norm topology. Therefore, it is compact and so its action is proper. \(\square \)

For \(N\ge 2\), the action of \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) on \(P_{\mathbb {R}^2}^N\) is free and proper. Indeed, if \(R\in SO(2)\) is not the identity, \(v\in \mathbb {R}^2\) and \((p,q)\in P_{\mathbb {R}^2}^2\), then \(Rp+v=p\) and \(Rq+v=q\) imply \(p=q\), since R does not have real eigenvectors, and so the action is free. Moreover, SO(2) is compact and the map R in Definition 5 for \(\mathbb {R}^2\)-action has continuous inverse, therefore, the semidirect product action is a composition of proper maps and so it is proper.

Proof of Theorem 3

We divide the proof in two cases.

Case 1: \(\sum _i\Gamma _i\ne 0\). Without loss of generality due to the translational invariance of equations (9) and to the non-vanishing total circulation, we can assume that the linear momentum is non-zero. Then, Lemmas 1-2 ensure the \(\mathrm {SO}(2)\times K\) action is free and proper, for any \(N\ge 2\).

For \(N=2\), Theorem 10 tells us that the reduced Hamiltonian system has dimension 0, and therefore the motion is a \(\mathrm {SO}(2)\times K\)-relative equilibrium.

For \(N=3\), Theorem 10 with respect to the \(\mathrm {SO}(2)\times K\) action tells us that the reduced Hamiltonian system has dimension 2, which implies that it is integrable.

Case 2: \(\sum _i\Gamma _i = 0\) If the circulation is zero, the linear momentum map due to the action of \(\mathbb {R}^2\) is equivariant. Moreover, the \(\mathbb {R}^2\) action is always free and proper for \(N\ge 1\).

For \(N=2\), Theorem 10 tells us that the reduced Hamiltonian system has dimension 0 when \(N=2\). Therefore the motion can only be up to translations; this explains the vortex dipole solutions.

For \(N=3\), Theorem 10 tells us that the reduced Hamiltonian system has dimension 2, which implies that it is integrable. \(\square \)

Proof of Theorem 4

For \(N=2\), the zero-circulation and zero-momentum conditions imply that \({\mathbf {r}}_1={\mathbf {r}}_2\), so the point-vortex equation 10 is not defined.

For \(N=3\), the \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) the momentum map is equivariant, so Theorem 10 tells us that the reduced systems evolves on a zero-dimensional manifold. Therefore, the motion of the point-vortices is a \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) relative equilibrium.

For \(N=4\), the \(\mathrm {SO}(2)\ltimes \mathbb {R}^2\) the momentum map is equivariant, so Theorem 10 tells us that the reduced systems evolves on a on a two-dimensional manifold. Integrability then follows. \(\square \)

5.3 The Hyperbolic Plane

To get a free action for \(N\ge 2\), we need to restrict to \(\mathrm {PSL}(2)\cong \mathrm {SL}(2)/\lbrace \pm 1\rbrace \). Furthermore, it is known that the action is proper [36].

Proof of Theorem 5

For \(N=2\), Theorem 10 tells us that systems evolves on a zero-dimensional manifold, since \(G_\phi \) is a one-dimensional Lie group (see [36]). Thus, solutions are relative equilibria with respect to Möbius transformations.

For \(N=3\), Theorem 10 tells us that systems evolves on a two-dimensional manifold, so the dynamics is integrable. \(\square \)

Proof of Theorem 6

For \(N=2\), the zero momentum condition (17) implies that \({\mathbf {r}}_1,{\mathbf {r}}_2\) lie on common line through the origin. Hence, \({\mathbf {r}}_1={\mathbf {r}}_2\) and so the equations (13) are not defined.

For \(N=3\), the \(\mathrm {PSL}(2)\) action is free and Theorem 10 tells us that systems evolves on a zero-dimensional manifold. Thus, solutions are relative equilibria with respect to Möbius transformations.

For \(N=4\), the \(\mathrm {PSL}(2)\) action is free and Theorem 10 tells us that systems evolves on a two-dimensional manifold, so the dynamics is integrable. \(\square \)

5.4 The Flat Torus

The action of \(\mathbb {R}^2\) on \(\mathbb {T}^2\) is not free, but as we have seen in Sect. 2, we understand the point-vortex equations on \(\mathbb {T}^2\) as a special case of the point-vortex dynamics on \(\mathbb {R}^2\). Therefore, the action of \(\mathbb {R}^2\) on itself via translations is free and proper. We can also consider the action of the group K, as defined in Sect. 2.2. Since its momentum map only depends on the conservation of the \(\mathbb {R}^2\) momentum map, K is a symmetry also of any translational invariant Hamiltonian on the torus.

Proof of Theorem 7

For \(N=2\), without loss of generality due to the translational invariance of equations (14) and to the non-vanishing total circulation, we can assume that the linear momentum is non-zero. Then, Lemmas 1-2 ensure the K action is free and proper. Therefore, Theorem 10 tells us that systems evolves on a two-dimensional manifold, so the dynamics is integrable. \(\square \)

Proof of Theorem 8

For \(N=2\), the \(\mathbb {R}^2\) action is free and Theorem 10 tells us that systems evolves on a zero-dimensional manifold. This means that the initial condition can only be transformed via translations, which conserve the Hamiltonian.

For \(N=3\), the \(\mathbb {R}^2\) action is free and Theorem 10 tells us that systems evolves on a two-dimensional manifold. By definition, any Hamiltonian system on a two-dimensional manifold is integrable. \(\square \)

6 Non-integrability Results

In this section, we briefly summarize the results in literature on non-integrability of point-vortex dynamics. Unlike the integrability results, which have been addressed extensively, only the planar case has been fully analysed and completed. Some results concern non-integrability for a restricted model of the point-vortex dynamics, defined as follows:

Definition 7

The restricted problem of N point-vortices consists of a partition of the vortices in two sets A and B, such that the vortices in A have non-zero strength and do not interact with vortices in B, and the vortices in B have vorticity \(0\le \varepsilon \ll 1\) and interact with all the other vortices.

Theorem 11

(\(\mathbb {S}^2\) with non-vanishing momentum, [4]) Consider the point-vortex equations (4) on \(P_{\mathbb {S}^2}^N\), restricted to the part of phase space with non-vanishing total linear momentum

$$\begin{aligned} \{({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \in P^N_{\mathbb {S}^2}\mid \mathbf{J}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N) \ne {\mathbf {0}} \}. \end{aligned}$$

  • If \(N=4\), the restricted model with A of three identical vortices and B of one single vortex with strength \(\varepsilon =0\) is non-integrable.

Theorem 12

(\(\mathbb {R}^2\) with non-vanishing circulation or momentum, [22, 46, 47]) Consider Hamilton’s equations (10) on \(P_{\mathbb {R}^2}^N\), either restricted to the case non-vanishing circulation

$$\begin{aligned} \sum _{i=1}^N\Gamma _i \ne 0 \end{aligned}$$

or restricted to the part of phase space with non-vanishing total momentum

$$\begin{aligned} \sum _{i=1}^N \Gamma _i {\mathbf {r}}_i \ne 0. \end{aligned}$$

  • If \(N=4\), there exist configurations of point-vortices for which the motion is non-integrable.

  • If \(N\ge 5\), there exist configurations of point-vortices for which the Arnold diffusion occurs.

Regarding the flat torus, Kilin and Artemova [19] gave numerical evidence of non-integrability for three point-vortices on \(\mathbb {T}^2\) with non-zero total circulation. Our point-vortex gallery in Appendix A also indicate chaotic behaviour in this case. For hyperbolic space, we have also conducted numerical simulations in Appendix A indicating the same behaviour as for the sphere: chaotic behaviour for four point-vortices with non-vanishing momentum. To the best of our knowledge there are yet no rigorous non-integrability results available for the torus or the hyperbolic space.

7 Outlook: Long-Time Predictions for 2D Euler Equations

The prevailing theories for the long-time behaviour of the Euler equations (1) on a two-dimensional manifold are those given by Miller [31] and by Robert and Sommeria [40], here referred to as MRS. The MRS approach is a generalization of Onsager’s [38] ideas, from discrete to continuous vorticity fields. These theories state that, in the long-time limit, the vorticity field evolves towards a state where the entropy of a course-grain probability distribution of macroscopic states is maximized under the constraint of conservation of energy and Casimirs. Consequently, this leads to a course-grain steady vorticity state, characterized by a functional dependence between vorticity and stream function. For a survey of MRS and the statistical approach to 2D turbulence, see the article by Bouchet and Venaille [8].

However, in a numerical study for 2D Euler equations on the sphere, Dritschel et al. [9] gave results that contradict MRS theory, yielding, for randomly generated initial conditions, a seemingly non-steady vortex blob configuration. The numerical method used in [9] did not conserve Casimir functions, which raised questions of the reliability since MRS theory is based on conservation of Casimirs. As a remedy, we develop in [34] a Casimir preserving numerical method for Euler equations on the sphere that captures all the features of 2D Euler equations: conservation of Casimirs, energy, and the Lie–Poisson structure. Using this method we obtain, again, strong evidence against the MRS predictions, but now with a more reliable, structure preserving method. Furthermore, we found a new mechanism that connects the long-time behaviour with point-vortex integrability results. Here is a brief outline of how it works:

  1. (1)

    As is well known in 2D turbulence, the inverse energy cascade, discovered by Kraichnan [23], forces smaller vortex formations of the same sign to merge into larger ones by vortex stretching, forming positive and negative vortex blobs.

  2. (2)

    As long as the vortex blobs are not too close to each other, so they are not ripped apart and merged, their dynamics are well described by point-vortex dynamics. Theoretical results in this direction are given, for example, in the monograph by Marchioro and Pulvirenti [25].

  3. (3)

    The vortex merging continue, into fewer and larger vortex blobs, until the blob dynamics become integrable, with well separated vortex blob trajectories. Because of quasi-periodicity, the vortex blobs are then ‘stuck’ in this part of phase space and no further merging occurs.

  4. (4)

    A prediction for the final number of vortex blobs N is thus given by integrability results for point-vortex dynamics: for the given fluid configuration (in terms of circulation, energy, momentum, etc.), find the largest N such that the dynamics is integrable for N vortices, but non-integrable for \(N+1\) vortices.

For the Euler equations on the sphere, Casimir preserving numerical simulations with randomly generated initial conditions, presented in [34], perfectly align with this mechanism: if the total linear momentum is zero (or very small) we see the formation of four vortex blobs interacting in a quasi-periodic, non-steady fashion reflecting the results in Theorem 2 above. If the linear momentum is non-zero, we see the formation of three vortex blobs, reflecting Theorem 1. We anticipate that the long-time behaviour on other domains also shall be reflected in the corresponding point-vortex integrability results, at least for the plane and the hyperbolic plane. (As we can see in the proofs above, the flat torus is, from the symplectic reduction viewpoint, more complicated than the other cases; the connection to the corresponding Euler equations is not direct.)

But of course, numerical simulations alone, even if they preserve all the underlying structures, are not enough and must be accompanied with rigorous mathematical analysis, attempting to prove the connection between integrability and the long-time behaviour. We consider the paper at hand the first step in this direction.