Classical Mechanics and Point Vortices

Classical Mechanics and Point Vortices

Abstract: The point vortex model for inviscid fluids is aHamiltonian system. In this lecture I’ll explain how ideas fromclassical mechanics can be used to establish the existence ofcertain classes of physically interesting solutions of these equations.

The part of the lecture that represents my own work is jointresearch with A. Barry and G.R. Hall.

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References

• A. Barry, G.R. Hall, and C.E. Wayne, Relative Equilibria of the(1+N)-Vortex Problem, J. Nonlinear Sci., 22, 63-83 (2012).

• P. Newton, The N-Vortex Problem: AnalyticalTechniques, Vol. 145, Springer, (2001).

• K.M. Khanin, Quasi-periodic motions of vortex systems,Physica D: Nonlinear Phenomena, 4, 261-269, (1982).

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Recall:

• N-vortices with locations (xi , yi )Ni=1, and “strength” Γi.• The dynamics of these vortices is given by the system of

ODE’s

xi = − 1

2π

∑j 6=i

Γj(yi − yj)

(xi − xj)2 + (yi − yj)2

yi =1

2π

∑j 6=i

Γj(xi − xj)

(xi − xj)2 + (yi − yj)2

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Hamiltonian Form

Let qi = (xi , yi ) and define

H(q1, . . . , qN) = −∑i<j

ΓiΓj log |qi − qj |

Then the point vortex equations can be written:

Γi xi =∂H

∂yi

Γi yi = −∂H

∂xi(1)

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Conserved quantities

Conserved quantities:

• “Energy” (i.e. the Hamiltonian)

• “Momentum”

P =∑i

Γixi , Q =∑i

Γiyi

• “Angular Momentum”

I =∑i

Γi |qi |2

Remark: The fact that the vortex strength Γi can be both positiveand negative, unlike the mass of a point particle means that onemust be careful about drawing immediate conclusions fromanalogy with classical mechanics.

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Small numbers of vortices

• The presence of conserved quantities allows one to reduce theeffective number of degrees of freedom.

• Thus, systems of N = 2 or N = 3 vortices are alwaysintegrable Hamiltonian systems.

• For N = 4 or more vortices, the systems are not integrable.

• However, there may still be initial conditions that lead toregular behavior (e.g. KAM theory).

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Vortex Crystals

One configuration that is observed surprisingly often in nature is astate in which a system of N vortices rotates with a constantvelocity, but maintain a fixed relationship to one another in arotating frame.

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Vortex Crystals

More interesting examples occur in a host of experimental settings:Hurricanes:

Figure: Hurricane Isabel (2003); from J.P. Kossin and W.H. Schubert.Mesovortices in hurricane Isabel. Bulletin of the American MeteorologicalSociety, 85:151153, 2004.

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(More) Vortex Crystals

Electron columns:

Figure: Stable vortex configurations in electron columns as described inD. Durkin and J. Fajans. Experiments on two-dimensional vortexpatterns. Physics of Fluids, 12:289,2000.

Remark: Note that these are stable (at least experimentally) butnot symmetric.

Remark: Note that they also involve one vortex that is significantlystronger than the rest.

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Electron columns:




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Electron columns:




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Vortex crystals (theory)

One explicit vortex crystal solution is to take N vortices of equalstrength at the vertices of a regular polygon.

By adjusting the speed of rotation, one can show that this is avortex crystal.

One can also add an additional vortex at the center of thisconfiguration (which doesn’t need to be of the same strength.

This is similar to the configurations observed in the hurricane, butnot in the electron columns.

In celestial mechanics, similar solutions of the N-body problemhave been investigated where they are known as relative equilibria.

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Motivated by the celestial mechanics problem, we began to searchfor relative equilibria/vortex crystals consisting of 1 “large” vortex,(whose vorticity we normalize to Γ0 = 1) and N “small” vorticesall of which have vorticity Γj = ε.

Note that ε can be either positive or negative. We will show thatone can construct vortex crystals with ε of either sign, but thestability of the resulting solutions may differ, depending on the signof ε.

Our method of constructing these solutions will be to identifycertain limiting configurations as ε→ 0 which are“non-degenerate” and prove that for ε small there are nearbyrelative equilibria for the full point vortex equations.

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(N + 1)/(1 + N) vortex problem

Definition: Let εk be a sequence of real numbers such thatεk → 0 as k →∞ and let qk

0 . . . , qkN be a sequence of

configurations which are relative equilibria of the (N + 1)-vortexproblem for each k with corresponding circulations Γk

0 = 1 ,Γkj = εk , j = 1, . . .N . A relative equilibrium of the (1 + N)-vortex

problem is a configuration q0 . . . qN such that qkj → qj as

k →∞. for each j = 0, . . . ,N.

In keeping with the terminology of celestial mechanics, the N + 1vortex problem will refer to the full point vortex equations whilethe 1 + N vortex problem will refer to the limiting problem (inwhich the “small” vortices have vortex strength “zero”.)

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(N + 1)/(1 + N) vortex problem

Definition: Let εk be a sequence of real numbers such thatεk → 0 as k →∞ and let qk

0 . . . , qkN be a sequence of

configurations which are relative equilibria of the (N + 1)-vortexproblem for each k with corresponding circulations Γk

0 = 1 ,Γkj = εk , j = 1, . . .N . A relative equilibrium of the (1 + N)-vortex

problem is a configuration q0 . . . qN such that qkj → qj as

k →∞. for each j = 0, . . . ,N.

In keeping with the terminology of celestial mechanics, the N + 1vortex problem will refer to the full point vortex equations whilethe 1 + N vortex problem will refer to the limiting problem (inwhich the “small” vortices have vortex strength “zero”.)

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An (easy) fact

Suppose we have a sequence of relative equalibria qk0 . . . , q

kN of the

N + 1 body problem in a frame rotating with angular velocity ω = 1.

• If we have a relative equilibrium of the N + 1 body problem withangular frequency ω 6= 1, we can change the angular frequency byrescaling the locations of the particles.

If this family of equalibria converges, as εk → 0 to a relative equilibriumof the (1 + N) vortex problem and if the vortices in the family remainbounded away from each other, and in some bounded set, then

• |qk0 | = O(εk), and

• |qkj |2 − 1 = O(εk).

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An (easy) fact


kN of the




• |qk0 | = O(εk), and

• |qkj |2 − 1 = O(εk).

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An (easy) fact


kN of the




• |qk0 | = O(εk), and

• |qkj |2 − 1 = O(εk).

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An (easy) fact

This follows immediately from the form of the equations for relativeequilibria:

0 = −iωξ0 + ε

N∑i=1

1

ξ∗0 − ξ∗i

0 = −iωξj +1

ξ∗j − ξ∗0+ ε

N∑i=1;i 6=j

1

ξ∗j − ξ∗i

We’re using the representation of the point vortex equation in terms ofcomplex variables here.

So, the relative equilibria of the (1 + N) vortex problem lie on the unitcircle.

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An (easy) fact

This follows immediately from the form of the equations for relativeequilibria:

0 = −iωξ0 + ε

N∑i=1

1

ξ∗0 − ξ∗i

0 = −iωξj +1

ξ∗j − ξ∗0+ ε

N∑i=1;i 6=j

1

ξ∗j − ξ∗i

We’re using the representation of the point vortex equation in terms ofcomplex variables here.

So, the relative equilibria of the (1 + N) vortex problem lie on the unitcircle.

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Location of the relative equilibrium

The key simplification that occurs when studying the (1 + N)vortex problem is that it’s relative equilibria are critical points of apotential function defined on the unit circle.

TheoremLet q = (q0, . . . , qN) be a relative equilibrium of the (1 + N)vortex problem, and let (r , θ) be the representation of (q1, . . . , 1N)in polar coordinates. Then θ is a critical point of

V (θ) = −∑i<j

(cos(θi − θj) +

1

2log(2− 2 cos(θi − θj))

).

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The proof of this proposition depends on the fact that

qεj · qεj = 0 .

Taking the dot product of the differential equations for the vortexmotion, rewriting qj in polar coordinates, and inserting theasymptotics for qεj derived above, one finds that in the limit ε→ 0,the solutions of the equations of motion converge to critical pointsof V .

• Critical points of V are (relatively) easy to find.

• Every family of relative equilbria of the N + 1 vortex problemconverge to a critical point of V .

• Does every critical point of V correspond to a family ofrelative equilbria of the N + 1 vortex problem?

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qεj · qεj = 0 .





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qεj · qεj = 0 .





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qεj · qεj = 0 .





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Perturbing solutions of the (1 + N)problem

Critical points of V correspond to a family of relative equilbria ofthe N + 1 vortex problem if a non-degeneracy condition holds.

Suppose that we linearize the point vortex equations about afamily of relative equilibria:

• We always have at least four zero eigenvalues.

• Two of these come from the invariance of the center ofvorticity. We can eliminate these by definingq0 = −ε(q1 + · · ·+ qn) and then ignoring the equations for q0.

• The second pair is related to the rotational symmetry and tothe scaling invariance of the problem (i.e. the fact that if wehave one relative equilibrium, we can construct a family ofthem by rescaling qj and simultaneously changing ω.

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Non-degeneracy of relative equilibria

If we rewrite the equations of motion for qj in polar coordinatesand linearize about a family of equilibria of the (N + 1) vortexproblem the matrix in the linearized equations have the form:

M =

(−εA +O(ε2) εVθθ +O(ε2)−2I +O(ε2) εA +O(ε2)

)Here, A is the matrix whose matrix elements are

aij = sin(θj − θi ) , i 6= j

aii =∑j 6=i

sin(θi − θj)

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As remarked above the matrix M always has two zero eigenvaluesdue to symmetries.

We define M to be non-degenerate if it has exactly twoeigenvalues.

TheoremIf θ∗ is a critical point of V with the property that the matrix M,when evaluated at θ∗, is non-degenerate, then the point (1, θ∗) isthe limit of a sequence of relative equilibria of the (N + 1) vortexproblem.

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Proof: The proof is a (relatively) straightforward application of theImplicit Function Theorem.

• The only difficulty is dealing with the zero eigenvalues.

• We eliminated them “by hand”.

• For example, one of the zero eigenvalues comes from therotation invariance - this corresponds to the fact thatv0 = (1, . . . , 1)T is in the null space of Vθθ.

• We eliminate this degeneracy by restricting the variations inthe angles to the orthogonal complement of v0.

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Stability

Can we use the limiting problem to say anything about (dynamical)stability of solutions of relative equilibria of the (N + 1)-body problem?

Relies heavily on method developed in: R. Moeckel; Relative equilibriawith clusters of small masses. J. Dynam. Differential Equations,9(4):507–533, (1997).

Idea: Use the symplectic structure of the matrix M to deduce informationabout the spectrum of M from knowledge of the spectrum of Vθθ.

We’ll focus on linear stability which means showing that all eigenvalues ofthe linearized equation are pure imaginary. In this context we appeal to:

LemmaIf the eigenvectors v of M, satisfy Ω(v , v∗) 6= 0, then the eigenvalues ofM are pure imaginary

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Stability






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Stability






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Stability




We’ll focus on linear stability which means showing that all eigenvalues ofthe linearized equation are pure imaginary.

In this context we appeal to:


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Stability






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Stability of relative equilibria

TheoremSuppose that (ρε, θε) are a family of relative equilibria of the (N + 1)vortex problem that converge to an equilibrium (1, θ∗) of the (1 + N)vortex problem, where θ∗ is a non-degenerate critical point of V . Ifε > 0, (sufficiently small) then (ρε, θε) is linearly stable if and only if θ∗

is a local minimum of V . If ε < 0, (sufficiently small) then (ρε, θε) islinearly stable if and only if θ∗ is a local maximum of V .

First step: Use the facts that:

• Vθθ has exactly one zero eigenvalue, and

• the form of the matrix M, plus

• a straightforward perturbative argument

to conclude that M has two zero eigenvalues (this follows from thesymplectic structure) with all other eigenvalues a distance of at leastO(√ε) away from zero.

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Stability (continued)

In fact, the perturbative argument shows that the eigenvalues are of theform

λ(ε) =√εγ0 +O(ε)

where γ0 is a solution of the equation

det(γ20 + 2Vθθ) = 0 .

Suppose that ε > 0 and θ∗ is not a minimum of Vθθ. Then Vθθ has atleast one negative eigenvalue which implies that there are at least two realsolutions γ0 of this equation and hence at least two real eigenvalues of M

Thus, this relative equilibrium is unstable, and the only possibility forstability is if θ∗ is a minimum of Vθθ.

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λ(ε) =√εγ0 +O(ε)


det(γ20 + 2Vθθ) = 0 .



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λ(ε) =√εγ0 +O(ε)


det(γ20 + 2Vθθ) = 0 .



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Finally, suppose that θ∗ is a local minimum of Vθθ, Then each solutionγ0 of the equation above can be written as

γ0 = iβ0 .

Let λ(ε) = i√εβ0 + o(

√ε) be the corresponding eigenvalue of M and let

v = (vr , vθ) be the corresponding eigenvector.

From the form of M and the equation

Mv = λv

we see that λvr = O(ε) which implies that vr = O(√ε).

A similar consideration of the second component of the eigenvector shows

vr = −√εγ

2vθ +O(ε) .

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γ0 = iβ0 .





Mv = λv



vr = −√εγ

2vθ +O(ε) .

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γ0 = iβ0 .





Mv = λv



vr = −√εγ

2vθ +O(ε) .

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γ0 = iβ0 .





Mv = λv



vr = −√εγ

2vθ +O(ε) .

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The last step

If we now require that v be normalized, we have vθ = 1 +O(ε).

We can now just compute

Ω(v , v∗) = (v , Jv∗) = β√ε+O(ε) 6= 0

Thus, λ is pure imaginary, and since this argument applies to each of thenon-zero eigenvalues of M we have linear stability.

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The last step



Ω(v , v∗) = (v , Jv∗) = β√ε+O(ε) 6= 0


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The last step



Ω(v , v∗) = (v , Jv∗) = β√ε+O(ε) 6= 0


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A surprising special case

Consider the relative equilibrium in which we have N vortices of strengthε arranged at the vertices of a regular polygon around a central vortex ofstrength 1. (Suppose N > 4.)

LemmaFor this choice of θ∗, the matrix Vθθ has at least one positive and onenegative eigenvalue.

Proof: The matrix Vθθ is a circulant matrix in this case and the simpleformula for the eigenvalues of a circulant matrix immediately imply thelemma.

Note that an immediate corollary of this lemma is that the polygonalrelative equilibrium is unstable for ε sufficiently small, a fact alreadyestablished by quite different methods by Cabral and Schmidt.

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The minimizing relative equilibrium

But V (θ) must have a minimum, so we conclude (under mildnon-degeneracy assumptions) that the stable relative equilibrium for the(N + 1) vortex problem with small ε is not symmetric.

This is a strong contrast with the celestial mechanics in which for N > 7the polygonal relative equilibrium is not only stable but is in fact the onlyrelative equilibrium.

In fact, for the point vortex problem we should have at least threefamilies of relative equilibria for each N (again, assuming some mildnon-degeneracy condition.)

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ExtensionsRecently, there has been interest in looking at large numbers of vortices -here, one sometimes sees a sort of swarming behavior:

Figure: From Chen, Kolokolnikov and Zhirov; Collective behavior of largenumber of vortices in the plane: Proc. R. Soc. A, 469, (2013) .

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Extensions

A second physical system in which these ideas have been applied is to theinteraction of vortices in Bose-Einstein condensates:

Figure: From Navarro, et al. Dynamics of a Few Corotating Vortices inBose-Einstein Condensates: PRL, 110, (2013).

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Equations of motion

The authors derive equations of motion for the vortex centers:

xj = −Sjωyj −Ω

2

∑i 6=j

Siyj − yi

r2ij

yj = Sjωsj +Ω

2

∑i 6=j

Sixj − xi

r2ij

These are just the point vortex equations, plus a confining potentialVconf = 1

2Sω(x2j + y2

j ).

Recently analyzed with methods similar to those described today in:Barry and Kevredidis; Few particle vortex clusters...: J. Phys. A, 46,(2013).

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Equations of motion

The authors derive equations of motion for the vortex centers:

xj = −Sjωyj −Ω

2

∑i 6=j

Siyj − yi

r2ij

yj = Sjωsj +Ω

2

∑i 6=j

Sixj − xi

r2ij

These are just the point vortex equations, plus a confining potentialVconf = 1

2Sω(x2j + y2

j ).

Recently analyzed with methods similar to those described today in:Barry and Kevredidis; Few particle vortex clusters...: J. Phys. A, 46,(2013).

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Summary

• The mathematical similarities between the point vortex equationsand the equations of celestial mechanics suggest a large “toolbox”of methods for studying the point-vortex problem.

• There are, however, surprising differences in the phenomenology ofthe two systems.

• There are other physical systems which present very similarequations of motion and should be amenable to similar analysis.

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Summary




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Summary




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Classical Mechanics and Point Vortices