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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
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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
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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,
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 3 / 28
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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).
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 4 / 28
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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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 5 / 28
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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
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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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 7 / 28
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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].
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 8 / 28
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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
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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
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The Euler top with SO(n − r)–symmetry
Reduced system
Lemma 3. The Hamiltonian flow of the reduced rigid body metric
H =12
∑1≤i
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The Euler top with SO(n − r)–symmetry
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} = ∅.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 12 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 13 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 14 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 15 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 16 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 17 / 28
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The Euler top with SO(n − r)–symmetry
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
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 19 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 20 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 21 / 28
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The Euler top with SO(n − r)–symmetry
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).
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 22 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 23 / 28
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The Euler top with SO(n − r)–symmetry
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
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The Euler top with SO(n − r)–symmetry
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
).
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 25 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 26 / 28
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The Euler top with SO(n − r)–symmetry
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.
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 27 / 28
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The Euler top with SO(n − r)–symmetry
Thank you for your attention!
Božidar Jovanović ( MI SANU) XIX GEOMETRICAL SEMINAR 28.8-4.9.2016 28 / 28
Symmetric Euler topGeodesics on SO(n)/SO(n1)…SO(nr)Proof of the completenessThe Euler top with SO(n-r)–symmetry