jointworkwith VladimirDragovi´candBorislavGaji´c...

Singular motion of a symmetric Manakov top

Božidar Jovanović

Mathematical Institute SANU

joint work withVladimir Dragović and Borislav GajićXIX GEOMETRICAL SEMINAR

Dedicated to the memory of Professor Mileva PrvanovićZlatibor, Serbia, August 28-September 4, 2016

Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 1 / 28

Symmetric Euler top

Euler equations na so(n)

Ṁij =n∑

k=1

bi − bj(bk + bi)(bk + bj)

MikMkj

M ∈ so(n) - angular momentum.

Ω ∈ so(n) - angular velocity.

Mij = (bi + bj)Ωij, B = diag(b1, . . . , bn) - mass tensor.

Kinetic energy: H = 12∑

i

Symmetric Euler top

Mishchenko integrals

Mishchenko [1970]: The following polynomials

Jk =k∑

p=1

tr(Bp−1MBk−pΩ), k ≥ 0,

are independent integrals of the Euler equations. In particular, theEuler top on so(4) is completely integrable.

Proof — by calculating the Hamiltonian vector fields at the point

M0 =∑

i

Ei ∧ Ei+1,


Symmetric Euler top

Manakov integrals

Manakov [1976]: Euler equations imply LA pair

L̇(λ) = [L(λ),N(λ)], L(λ) = M + λA, N(λ) = Ω + λB, A = B2,

and vice verse for bi 6= bj. Manakov integrals:

L = {tr(M + λA)k | k = 1, 2, . . . , n, λ ∈ R},

Theorem 1. [Mishchenko and Fomenko, 1978] Assume ai 6= aj. We have

ρso(n) =12

(dim so(n) + rank so(n)) =12

(n(n− 1)

2+[n2

])independent polynomials in L. L is a complete commutative set onso(n).


Symmetric Euler top

Symmetric rigid body

If bi = bj then Ṁij = 0, i.e., Mij is the first integrals. Suppose

a1 = · · · = an1 = α1, . . . , an+1−nr = · · · = an = αr,n1 + n2 + · · ·+ nr = n, n1 ≤ n2 ≤ n3 · · · ≤ nr.

Let so(n)A = {X ∈ so(n) | [X,A] = 0} = so(n1)× · · · × so(nr).

S - linar functions on so(n)A - Noether integrals. {L,S} = 0, tj.,Manakov integrals are AdSO(n)A–invariant, whereSO(n)A ∼= SO(n1)× · · · × SO(nr) is subgroup of SO(n) with the Liealgebra so(n)A.

Theorem 2. [Bolsinov, 1995], [Dragovic, Gajic, B.J, 2009] L+ S is acomplete set on so(n). Symmetric Euler top is completely integrable ina non-commutative sense.


Geodesics on SO(n)/SO(n1) × · · · × SO(nr)

Algebra of SO(n)A–invariant functions

Orthogonal decomposition

so(n) = so(n)A ⊕ v ∼= so(n1)⊕ so(n2)⊕ · · · ⊕ so(nr)⊕ v,

Consider the restriction of the Lie–Poisson bracket

{f, g}v(M) = −〈M, [∇vf(M),∇vg(M)]〉, f, g ∈ F .

F : AdSO(n)A–invariant functions on v.∼= SO(n)–invariant functions on T∗(SO(n)/SO(n)A).∼= functions on singular Poisson variety v/SO(n)A.

Theorem 3. [Dragovic, Gajic, B.J, 2009, 2014], [Mykytyuk 2014].Lv is a complete commutative subset of F : we have

ρv = dim v−12dimOSO(n)(M) (= ρso(n) − dim so(n)A for M regular),

for a generic M ∈ v, independent polynomials among restrictions Lv.Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 6 / 28

Geodesics on SO(n)/SO(n1) × · · · × SO(nr)

Geodesic flows on homogeneous spaces G/H

Mishchenko and Fomenko [1981] stated the conjecture thatnoncommutative integrability implies the usual Liouville one by meansof integrals that belong to the same functional class as noncommutativeintegrals. The conjecture is solved in C∞–smooth case forinfinite-dimensional algebras of integrals, as well as in polynomial andanalytic cases for finite-dimensional algebras of integrals [Bolsinov, BJ,2003], [Sadetov, 2004].

Let G/H be a homogeneous space of a compact connected Lie group Gendowed with a normal metric ds20 induced from a bi-invariantRiemannian metric on G. The geodesic flow of ds20 is integrable in thenoncommutative sense by means of analytic integrals polynomial inmomenta F + G [Bolsinov, BJ, 2001, 2004].

F - the algebra of G-invariant functions on T∗G/H, G - the algebra ofNoether integrals with respect to the Hamiltonian G-action on T∗G/H.

The Mishchenko–Fomenko conjecture reduces to the construction of acomplete commutative set L of F .

If the required set L exists, we say that (G,H) is an integrable pair [?].Apart of (SO(n), SO(n1)× · · · × SO(nr)), there are several knownclasses of integrable pairs, but the general problem is still unsolved. Inparticular, if G/H is a symmetric space, the algebra of G–invariantfunctions is already commutative and (G,H) is a trivial example of anintegrable pair.


Proof of the completeness

Regular case: so(n)A = so(n− r), 2r ≥ n

By expanding Manakov integrals in λ, we get integrals in the form

pkl = tr(M2kAl + (other permutations with 2k M and l A)),

k = 1, 2, . . . , [n/2], l = 0, 1, . . . , n− 2k.

Note that the total number of polynomials pkl is:

(n− 1) + (n− 3) + · · ·+ (n + 1− 2[n/2]) = ρso(n).

Instead of A = diag(a1, a2, . . . , ar−1, ar, ar+1, . . . , ar+1), we can takeA = diag(ar+1, a1, ar+1, a2, ar+1, . . . , ar) and consider the vector spaces

Wk = span{E1 ∧ E2k,E2 ∧ E2k+1, . . . }, k = 1, 2, . . . , [n/2]Vk = span{E1 ∧ E2k+1,E2 ∧ E2k+2, . . . },vk = Vk ∩ v, k = 1, 2, . . . , [n/2].


Proof of the completeness

Consider the gradients ∇pkl |v ∼ Pkl

Pkl = M2k−10 A

l + (other permutations with 2k-1 M0 and l A)

at the pointM0 =

∑i

Ei ∧ Ei+1 ∈ v.

Then

Pkl ∈W1 ⊕W2 ⊕ · · · ⊕Wk ⊂ v,Qkl = [M0,P

kl ] ∈ v1 ⊕ v2 ⊕ · · · ⊕ vk,

Uk = span{Qkl } ⊂ v1 ⊕ v2 ⊕ · · · ⊕ vk. (1)

Lemma 1. prvk(Uk) = vk.

Therefore, the relation (1) becomes an equality, and we obtain therequired number of integrals.Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 9 / 28

The Euler top with SO(n − r)–symmetry

Noncompleteness of the integrals L+ S on vLemma 2. [Dragovic, Gajic, BJ, 2009] L+ S is a complete set on so(n)at the points of v iff so(n)A is a commutative subalgebra.

From Theorem 3 we know that the system induces completelyintegrable system on v/SO(n)A.

Assume SO(n)A = SO(n− r). Let:

vi = (Mi,r+1, . . . ,Mi,n)T ∈ Rn−r, i = 1, . . . , r

The following polynomials are basic AdSO(n−r)–invariants on v:

xij = Mij, 1 ≤ i, j ≤ r,ykl = (vk, vl), 1 ≤ k, l ≤ r,

and, for n = 2r, z = det(v1, . . . , vr), which is functionally dependentfrom the other invariants.Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 10 / 28


Reduced system

Lemma 3. The Hamiltonian flow of the reduced rigid body metric

H =12

∑1≤i


The reduced bracket

{xij, xjk}v = −xik,{xij, xkl}v = 0 for {i, j} ∩ {k, l} = ∅,{xij, yjj}v = −2yij,{xij, yji}v = yjj − yii,{xij, yjk}v = −yik,{xij, ykl}v = 0 for {i, j} ∩ {k, l} = ∅, (3){yii, yjj}v = 4xijyij,{yii, yij}v = 2xijyii,{yij, yjk}v = xijyjk + yijxjk + xikyjj,{yii, yjk}v = 2xijyik + 2xikyij,{yij, ykl}v = xilyjk + xjlyik + xikyjl + xjkyil for {i, j} ∩ {k, l} = ∅.



Theorem 4. The reduced system (2) is completely integrable. Thedimension of the invariant tori is

∆r = r2 −(r + 1)r

2

and a complete set of independent commuting SO(n− r)–invariantintegrals on v is given by

p10|v, p11|v, . . . , p1r−2|v, p1r−1|v,p20|v, p21|v, . . . , p2r−2|v,...pr−10 |v, p

r−11 |v,

pr0|v.



The reconstruction problem

Lemma 4. Consider the r–dimensional subspaces

wk = span{E1 ∧ Ek,E2 ∧ Ek, . . . ,Er ∧ Ek} ⊂ v, k = r + 1, . . . , n

The Euler equations on wk

Ṁik =r∑

j=1

bi − br+1(bj + bi)(bj + br+1)

xijMjk, 1 ≤ i ≤ r < k ≤ n,

have the integrals

Fk =r∑

i=1

M2ikb2r+1 − b2i

=r∑

i=1

M2ikar+1 − ai

, k = r + 1, . . . , n.



Lemma 5. The r–plane π in Rn−r spanned by the vectors v1, . . . , vr isinvariant under the Euler equations restricted to v. Equivalently, thePlucker coordinates of π

Si1...ir = (i1 . . . ir) minor of [v1, . . . , vr], 1 ≤ i1 < i2 < · · · < ir ≤ n−r,

are integrals of the motion.

Among the Plücker coordinates, there are (n− 2r)r + 1 independentones. The sum of their squares is an SO(n− r)–invariant function, aCasimir function of the reduced bracket (3). Geometrically, by meansof the integrals (15) we reduce the problem to the case n = 2r.



SO(n− 2)–symmetry

In the case r = 2, we have ∆2 = 1, and a generic trajectory onv/SO(n− r) is periodic. The integrals (14), (15) can be written in theform

Fk = (b23 − b22)M21k + (b23 − b21)M22k = (a3 − a2)M21k + (a3 − a1)M22k,Sij = M1iM2j −M1jM2i, k = 3, . . . , n, 3 ≤ i < j ≤ n.

Together with the independent Manakov first integrals p10|v, p20|v, p11|v,they imply the following statement

Theorem 5. In the case of SO(n− 2)–symmetry, the generic trajectorieson v are periodic. The trajectories can be expressed by means of ellipticfunctions.



We can consider v/SO(n− 2) in the coordinates (x1, x2, x3, x4),x1, x4 ∈ R, x2, x3 ≥ 0, where

x1 = x12 = M12, x4 = y12 = M13M23 + · · ·+ M1nM2n,x2 = y11 = M213 + · · ·+ M21n, x3 = y22 = M223 + · · ·+ M22n.

Note that in the case n = 4, there is an additional basicSO(2)–invariant polynomial x5 = M13M24 −M14M23 and the relationx2x3 − x24 = x25. The reduced bracket gets the form:

{x1, x2}v = 2x4, {x1, x3}v = −2x4,{x1, x4}v = x3 − x2, {x2, x3}v = 4x1x4,{x2, x4}v = 2x1x2, {x3, x4}v = −2x1x2,

having two independent Casimir functions, given by AdSO(n)–invariants

p10 = −2x21 − 2x2 − 2x3,p20 = 2x

41 + 4x

21(x2 + x3) + 2x

22 + 2x

23 + 4x

24.



We can write down the above invariants in terms of the quadraticCasimir functions:

I1 = x21 + x2 + x3,I2 = x2x3 − x24 (= x25 for n = 4).

The reduced flow

ẋ1 = (b22 − b21)x4, ẋ2 = 2(b21 − b23)x1x4,ẋ3 = 2(b23 − b22)x1x4, ẋ4 = (b23 − b22)x1x2 + (b21 − b23)x1x3.

Here we supposed (b1 + b2)(b2 + b3)(b3 + b1) = 1.The equations transform to

ẋ21 = P(x1; I1, I2,F),

where the polynomial P(x; I1, I2,F) is given by

P = (b21−b23)(b23−b22)(I1−x2)2+F(b21−2b23+b22)(I1−x2)−F2−(b22−b21)2I2.

and F is linear first integral F = (b22 − b23)x2 + (b21 − b23)x3.Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 18 / 28


Spectral curve

The spectral curve Γ : P(λ, µ) = det(M + λA− µI) = 0, has theform

Γ : P(λ, µ) = (λa1−µ)(λa2−µ)(λa3−µ)2+P20µ2+P11µλ+P02λ2+P00 = 0.

The coefficients Pij are first integrals and they are given explicitly:

P00 = (M13M24 −M14M23)2 = x25 = I2,P20 = M212 + M

213 + M

214 + M

223 + M

224 = x

21 + x2 + x3 = I1,

P11 = −2a3M212 − (a3 + a1)(M224 + M223)− (a3 + a2)(M214 + M213)= −2a3x21 − (a2 + a3)x2 − (a1 + a3)x3,

P02 = a3(a3M212 + a2(M214 + M

213) + a1(M

224 + M

223))

= a3(a3x21 + a2x2 + a1x3).

There is a relation among these first integrals: a23P20 + a3P11 + P02 = 0.



The curve Γ is regular in the affine part. It is a 4–fold coveringπ : Γ→ P1(λ). The intersection of the curve with the line at infinity isπ−1(∞) = P1 + P2 + P3, where P3 is a singular point.

P3 is not an ordinary double point. The multiplicity of P3 is 2, thenumber of local brunches is 2 (with a common tangent), and itsδ–invariant is equal to 2. Therefore, the normalization Γ̃ of Γ is anelliptic curve:

genus(Γ̃) =3 · 32− 2 = 1.

In the Manakov nonsymmetric case (a3 6= a4) the spectral curve isnonsingular and of genus 3.

In the symmetric case a3 = a4, with the additional condition M34 6= 0,the spectral curve is singular. But, opposite to the case studied above,the point P3 is an ordinary double point, and thegenus of thenormalization of the spectral curve is 2.



SO(n− 3)–symmetry

For r = 3 there are 6 independent Manakov integrals{p10|v, p20|v, p30|v, p11|v, p21|v, p12|v} and the dimension of the invariant torion v/SO(n− 3) is ∆3 = 3.

We can reduce the problem to the case n = 6 when dim v = 12.

Then apart from the Manakov first integrals, there are two independentquadratic first integrals among F4,F5,F6. As a result, the invariantmanifolds are 4-dimensional.



Reconstruction

For b1, b2, b3 < b4, in the new coordinates

ω1 = −x23

b2 + b3

√(b4 − b2)(b4 − b3)(b4 + b2)(b4 + b3)

, γ1k =M1k√b24 − b21

,

ω2 =x13

b3 + b1

√(b4 − b1)(b4 − b3)(b4 + b1)(b4 + b3)

, γ2k =M2k√b24 − b22

,

ω3 = −x12

b1 + b2

√(b4 − b1)(b4 − b2)(b4 + b1)(b4 + b2)

, γ3k =M3k√b24 − b23

,

we get the reconstruction equations in the form of the Poisson equations

γ̇k = γk × ω(t), k = 4, . . . , n,

where γk = (γ1k, γ2k, γ3k), ω = (ω1, ω2, ω3).



Integrability

In general, the Poisson equation are not necessarily solvable byquadratures, e.g., see [Fedorov, Maciejewski, Przybylska, 2009]. Tosolve by quadratures, one needs one particular solution [Kozlov, 2014].

It can be shown that the commuting Hamiltonian vector fields of theManakov integrals p11, p

21, p

12 are tangent toMc. Since we have an

invariant volume form onMc, from the Euler-Jacobi-Lie theorem[Kozlov, 2013], we get:

Theorem 6. v is almost everywhere foliated on 4 dimensional manifoldsMc. The motion onMc is solvable by quadratures.



Spectral curve

For simplicity, we denotex1 = x32, x2 = x13, x3 = x21, y1 = y11, y2 = y22, y3 = y33.The polynomial P(µ, λ), determining the spectral curveΓ : det(M + λA− µI) = P(µ, λ) = 0, reads

P(µ, λ) =6∏

i=1

(λai − µ) +2∑

k=0

∑i+j=2k

Pijµiλj.

The coefficients

P00 = det(v1, v2, v2)2 = det(yij),P20 = 2(x1x2y12 + x3x1y31 + x2x3y23)− (y212 + y223 + y231)

+x21y1 + x22y2 + x

23y3 + y1y2 + y2y3 + y3y1,

P40 = x21 + x22 + x

23 + y1 + y2 + y3,

are the restriction to v of AdSO(6)–invariant polynomials (Casimirfunctions)Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 24 / 28


The other AdSO(3)–invariant integrals:

P11 = −2a4(x21y1 + x22y2 + x23y3) + (a1 + a4)(y223 − y2y3) + (a2 + a4)(y213 − y1y3)+(a3 + a4)(y212 − y1y2)− 4a4(x1x2y12 + x3x1y31 + x2x3y23),

P0,2 = a24(x21y1 + x

22y2 + x

23y3) + a1a4(y2y3 − y223) + a2a4(y3y1 − y213)

+a3a4(y1y2 − y212) + 2a24(x1x3y13 + x2x3y23 + x1x2y12),P31 = −(a1 + a2 + 2a4)y3 − (a1 + a3 + 2a4)y2 − (a2 + a3 + 2a4)y1

−(a1 + 3a4)x21 − (a2 + 3a4)x22 − (a3 + 3a4)x23,P22 = (a2a3 + 2a2a4 + 2a3a4 + a24)y1 + (a1a3 + 2a3a4 + 2a1a4 + a

24)y2

+(2a1a4 + 2a2a4 + a1a2 + a24)y3+3(a24 + a1a4)x

21 + 3(a

24 + a2a4)x

22 + 3(a

24 + a3a4)x

23,

P13 = −a24((3a1 + a4)x21 + (3a2 + a4)x

22 + (3a3 + a4)x

23)

−a4 ((2a2a3 + a2a4 + a3a4)y1 + (2a1a3 + a3a4 + a1a4)y2 + (2a1a2 + a1a4 + a2a4)y3) ,P04 = a24

(a4(a1x21 + a2x

22 + a3x

23) + a2a3y1 + a3a1y2 + a1a2y3

).



The description of Γ is a modification of a description of a spectralcurve in the regular case considered by Haine [1984], where ai 6= aj. Fora generic initial conditions xij, yij, the curve is regular in the affine part.

Γ is a 6–fold covering π : Γ→ P1(λ) and π−1(∞) = P1 + P2 + P3 + P4,where the points P1,P2,P3 are regular, while the point P4 is a singularpoint with multiplicity 3 and δ–invariant equal to 6.

Let Γ̃ be the normalization of Γ. Its genus is

genus(Γ̃) =5 · 42− 6 = 4.

Note that there is an involution σ : (λ, µ) 7→ (−λ,−µ). It can be provedthat the curve Γ̃/σ is an elliptic curve, thus the dimension of the Prymvariety Prym(Γ̃, σ) is 3 and it is equal to the dimension of the invarianttori of the reduced flow.



Dragović, V., Gajić B. and Jovanović, B.: Singular Manakov Flows andGeodesic Flows of Homogeneous Spaces of SO(n), Transfomation Groups,(2009), arXiv:0901.2444

Dragović, V., Gajić B. and Jovanović, B.: Systems of Hess–Appelrot typeand Zhukovskii property , International Journal of Geometric Methods inModern Physics, (2009), arXiv:0912.1875.

Dragović, V., Gajić B. and Jovanović, B.: On the completeness of theManakov integrals, Fundametalnaya i prikldnaya matematika, (2015)arXiv:1504.07221 (Dedicated to Academician Anatoly TimofeevichFomenko on the occasion of his 70th birthday)

Dragović, V., Gajić B. and Jovanović, B.: Note on Free Symmetric RigidBody Motion, Regular and Chaotic Dynamics, (2015). (Dedicated toAcademician Valery Vasil’evich Kozlov on the occasion of his 65thanniversary)

Mykytyuk, I. V.: Integrability of geodesic flows for metrics on suborbitsof the adjoint orbits of compact groups, Transfomation Groups, (2016),arXiv:1402.6526.



Thank you for your attention!


Symmetric Euler topGeodesics on SO(n)/SO(n1)…SO(nr)Proof of the completenessThe Euler top with SO(n-r)–symmetry

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