On a 'Mysterious' Case of a Quadratic Hamiltonian
We show that one of the five cases of a quadratic Hamiltonian, which were recently selected by Sokolov and Wolf who used the Kovalevskaya-Lyapunov test, fails to pass the Painlevé test for integrability.
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irk-123456789-1461032019-02-08T01:23:03Z On a 'Mysterious' Case of a Quadratic Hamiltonian Sakovich, S. We show that one of the five cases of a quadratic Hamiltonian, which were recently selected by Sokolov and Wolf who used the Kovalevskaya-Lyapunov test, fails to pass the Painlevé test for integrability. 2006 Article On a 'Mysterious' Case of a Quadratic Hamiltonian / S. Sakovich // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 10 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 70H07; 37J30; 34M35 http://dspace.nbuv.gov.ua/handle/123456789/146103 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We show that one of the five cases of a quadratic Hamiltonian, which were recently selected by Sokolov and Wolf who used the Kovalevskaya-Lyapunov test, fails to pass the Painlevé test for integrability. |
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Sakovich, S. On a 'Mysterious' Case of a Quadratic Hamiltonian Symmetry, Integrability and Geometry: Methods and Applications |
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Sakovich, S. |
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Sakovich, S. |
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On a 'Mysterious' Case of a Quadratic Hamiltonian |
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On a 'Mysterious' Case of a Quadratic Hamiltonian |
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On a 'Mysterious' Case of a Quadratic Hamiltonian |
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On a 'Mysterious' Case of a Quadratic Hamiltonian |
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On a 'Mysterious' Case of a Quadratic Hamiltonian |
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on a 'mysterious' case of a quadratic hamiltonian |
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Інститут математики НАН України |
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2006 |
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On a 'Mysterious' Case of a Quadratic Hamiltonian / S. Sakovich // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 10 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT sakovichs onamysteriouscaseofaquadratichamiltonian |
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2025-07-10T23:10:33Z |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 064, 4 pages
On a ‘Mysterious’ Case of a Quadratic Hamiltonian
Sergei SAKOVICH
Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus
E-mail: saks@tut.by
Received June 02, 2006, in final form July 18, 2006; Published online July 28, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper064/
Abstract. We show that one of the five cases of a quadratic Hamiltonian, which were
recently selected by Sokolov and Wolf who used the Kovalevskaya–Lyapunov test, fails to
pass the Painlevé test for integrability.
Key words: Hamiltonian system; nonintegrability; singularity analysis
2000 Mathematics Subject Classification: 70H07; 37J30; 34M35
1 Introduction
Recently, Sokolov and Wolf [1] applied the Kovalevskaya–Lyapunov test for integrability to
a class of quadratic Hamiltonians and selected in this way five cases, out of which three cases were
previously known to be integrable and one case turned out to be a new integrable Hamiltonian
on so(3, 1) with an additional sixth-degree polynomial integral. Integrability of the fifth case
remained unknown, and the authors of [1] wrote the following:
Case (e) is a mysterious one. We have verified that the Hamiltonian has no additional
polynomial integrals of degrees less than or equal to 8. On the other hand, on all
Kowalewski solutions all Kowalewski exponents are integers. It would be interesting
to verify whether the equations of motion in case (e) satisfy the standard Painlevé
test.
In the present short note, we apply the Painlevé test for integrability to this ‘mysterious’ case (e)
and show that it must be nonintegrable due to some movable logarithmic branching of solutions.
We use the Ablowitz–Ramani–Segur algorithm of singularity analysis of ODEs [2] (see also the
review [3]).
2 The studied quadratic Hamiltonian
Sokolov and Wolf [1] considered the following family of Poisson brackets:
{Mi,Mj} = εijkMk, {Mi, γj} = εijkγk, {γi, γj} = κεijkMk, (1)
where Mi and γi are components of three-dimensional vectors M and Γ, εijk is the totally
skew-symmetric tensor, and κ is a parameter. The cases κ > 0 and κ < 0 correspond to
the so(4) and so(3, 1) Lie algebras, whereas the e(3) Lie algebra case κ = 0 was not studied
in [1]. Since the brackets (1) have two Casimir functions, J1 = (M ,Γ) and J2 = κ|M |2 +
|Γ|2 (with standard notations for the vector dot product and module), only one additional
integral functionally independent of the Hamiltonian and the Casimir functions is required for
the Liouville integrability of the equations of motion.
mailto:saks@tut.by
http://www.emis.de/journals/SIGMA/2006/Paper064/
2 S. Sakovich
The special class of Hamiltonians studied by Sokolov and Wolf [1] has the following form:
H = c1(a, b)|M |2 + c2(a,M)(b,M) + (b,M × Γ), (2)
where c1 and c2 are parameters, the constant vectors a and b are taken to be a = (a1, 0, a3)
and b = (0, 0, 1), a2
1 + a2
3 = −κ, and × denotes the vector skew product. The Hamiltonians (2),
referred to as ‘vectorial’ Hamiltonians in [1], belong to a wider class of quadratic Hamiltonians
which have numerous applications (two-spin interactions, motion of a three-dimensional rigid
body in a constant-curvature space or in an ideal fluid, motion of a body with ellipsoidal cavities
filled with an ideal fluid, etc).
Sokolov and Wolf [1] applied the Kovalevskaya–Lyapunov test for integrability to the class
of Hamiltonians (2) and selected in this way the following five cases: (a) c1 is arbitrary, c2 = 0;
(b) c1 = 1, c2 = −2; (c) c1 = 1, c2 = −1; (d) c1 = 1, c2 = −1
2 ; (e) c1 = 1, c2 = 1. It was pointed
out in [1] that the case (a) obviously possesses the linear additional integral I = (b,M), whereas
the cases (b) and (c) correspond to two recently discovered integrable Hamiltonian systems with
polynomial additional integrals of degrees three [4] and four [5], respectively. It is remarkable
that the case (d) turned out to be a new integrable quadratic Hamiltonian discovered by the
Kovalevskaya–Lyapunov test, with an additional sixth-degree polynomial integral [1]. However,
in the case (e), which also passed the Kovalevskaya–Lyapunov test well, Sokolov and Wolf [1]
failed to find an additional integral.
In the next section, we will show that the equations of motion in this ‘mysterious’ case (e)
fail to pass the Painlevé test and must be nonintegrable in the Liouville sense. It is worthwhile
to remember, however, that the fact of not passing the Painlevé test does not necessarily imply
nonintegrability of a tested system, and the way is essential how the test is failed. If the Painlevé
test is failed at its first step, when the leading exponents are determined, or at its second
step, when the positions of resonances are determined, it is sometimes possible to improve the
behavior of solutions by a transformation of variables so that the transformed system passes the
Painlevé test well. In particular, rational leading exponents and rational positions of resonances
are allowed by the so-called weak Painlevé property which may [6] – as well as may not [7] –
correspond to integrability. However, if the Painlevé test is failed at its third step, when the
compatibility of recursion relations is checked at the resonances, one only have to introduce some
logarithmic terms into the expansions of solutions, and this logarithmic branching of solutions is
generally believed to be a clear symptom of nonintegrability [8]. We will see that, in the case (e)
of the Hamiltonian (2), the Painlevé test is failed in this last – hopeless – way.
3 Demystifying the ‘mysterious’ case
We are going to show that the system of six ODEs
ṁ1 = m1m2 + 2am2m3 −m1g3 + m3g1,
ṁ2 = −m2
1 − 2am1m3 + m2
3 −m2g3 + m3g2,
ṁ3 = −m2m3,
ġ1 =
(
1 + a2
)
m1m3 + m1g2 − 2am2g3 + 4am3g2 + g1g3,
ġ2 =
(
1 + a2
)
m2m3 −m1g1 + 2am1g3 − 4am3g1 + m3g3 + g2g3,
ġ3 = −
(
1 + a2
) (
m2
1 + m2
2
)
− 2am1g2 + 2am2g1 −m3g2 − g2
1 − g2
2, (3)
where the dot denotes d
dt and a is a parameter, must be nonintegrable unless a = 0. The
system (3) represents the equations of motion in the case (e) of the Hamiltonian (2). The
notations mi, gi and a used in (3) correspond to Mi, γi and a3 used in (2), respectively, whereas
On a ‘Mysterious’ Case of a Quadratic Hamiltonian 3
for the nonzero parameter a1 of (2) we have set a1 = 1 by rescaling Mi and γi (note that the
case (e) with a1 = 0 falls under the case (a) which possesses a linear additional integral).
Assuming that a 6= 0 in (3), where all quantities are considered as complex-valued from now
on, and using the expansions
mi = mi,0φ
αi + · · ·+ mi,rφ
r+αi + · · · ,
gi = gi,0φ
βi + · · ·+ gi,rφ
r+βi + · · · ,
i = 1, 2, 3, φ = t− t0, (4)
where mi,j , gi,j , αi, βi and t0 are constants, we find the following singular branches and posi-
tions r of resonances in them (note that we do not consider the possibility of a2 = −1, because
it corresponds to the case of κ = 0, i.e. the e(3) Lie algebra, which was not studied in [1]):
αi = βi = −1, i = 1, 2, 3,
m1,0 =
±a√
−1− a2
, m2,0 = 1, m3,0 =
∓1√
−1− a2
,
g1,0 = −a, g2,0 = ∓
√
−1− a2, g3,0 = 1,
r = −1, 0, 1, 2, 2, 2; (5)
and
αi = βi = −1, i = 1, 2, 3,
m1,0 =
±a√
−1− a2
, m2,0 = 1, m3,0 =
∓1√
−1− a2
,
g1,0 = a, g2,0 = ±
√
−1− a2, g3,0 = −1,
r = −2,−1, 1, 2, 2, 4. (6)
Let us look at the branch (5) first. According to the positions of resonances, this branch must
be a generic one. We see, however, that the position of one resonance is r = 0 there, whereas all
the coefficients m1,0, m2,0, m3,0, g1,0, g2,0 and g3,0 turn out to be fixed. This means that, in the
case of (5), the recursion relations for the coefficients of the expansions (4) have a nontrivial
compatibility condition right in the position r = 0, and we have to modify (4) by introducing
additional logarithmic terms, starting from the terms proportional to φ−1 log φ. Suppose we
do not do this and think that the branch (5) represents not the general solution but a class of
special solutions. Then we find that no compatibility condition appears at the resonance r = 1,
where the coefficient m1,1 remains arbitrary. However, at the triple resonance r = 2, where the
coefficients m1,2, m2,2 and g1,2 remain arbitrary, the nontrivial compatibility condition m1,1 = 0
appears, and we are again forced to introduce logarithmic terms into the expansions of solutions.
The branch (6) gives us the same information: solutions of the system (3) with a 6= 0 possess
movable logarithmic singularities. In this branch, the nontrivial compatibility condition
m2
1,1
[
2a
√
−1− a2
(
2 + a2
)2
m1,2
+
(
2 + a2
)2 (
3 + 2a2
)
m2,2 +
(
4 + 9a2 + 5a4
)
m2
1,1
]
= 0 (7)
appears at the resonance r = 4.
Consequently, the system (3) with a 6= 0 fails to pass the Painlevé test for integrability, and
this, in its turn, explains why Sokolov and Wolf [1] failed to find an additional integral for the
case (e) c1 = 1, c2 = 1 of the quadratic Hamiltonian (2).
The case of (3) with a = 0 is different: it passes the Painlevé test well, as one can easily
verify. However, the fact of integrability of the system (3) with a = 0 is not new: the case (e)
with a3 = 0 of the Hamiltonian (2) falls under the case (c) with a fourth-degree additional
polynomial integral (we are indebted to Prof. T. Wolf who kindly explained this point to us).
4 S. Sakovich
4 Conclusion
We have shown that the ‘mysterious’ case (e) c1 = 1, c2 = 1 of the quadratic Hamiltonian (2),
selected by Sokolov and Wolf [1] who used the Kovalevskaya–Lyapunov test, fails to pass the
Painlevé test for integrability. It is worthwhile to remember that, from the standpoint of the
Painlevé test [2, 3], the Kovalevskaya–Lyapunov test only examines the positions of resonances
for a studied system and is sensitive to the nondominant algebraic branching of solutions, whereas
the Painlevé test also verifies the compatibility of recursion relations at the resonances and can
detect the nondominant logarithmic branching of solutions which is the strongest indication of
nonintegrability. We can add that it happens quite frequently in the singularity analysis practice
that the positions of resonances are good but the recursion relations are incompatible. For
example, in the integrability study of symmetrically coupled higher-order nonlinear Schrödinger
equations [9], 23 distinct cases with integer positions of all resonances were found, but the
recursion relations turned out to be compatible in only one case out of those 23 cases (see the
table in [10] for more details).
[1] Sokolov V.V., Wolf T., Integrable quadratic classical Hamiltonians on so(4) and so(3, 1), J. Phys. A: Math.
Gen., 2006, V.39, 1915–1926, nlin.SI/0405066.
[2] Ablowitz M.J., Ramani A., Segur H., A connection between nonlinear evolution equations and ordinary
differential equations of P-type. I, J. Math. Phys., 1980, V.21, 715–721.
[3] Ramani A., Grammaticos B., Bountis T., The Painlevé property and singularity analysis of integrable and
non-integrable systems, Phys. Rep., 1989, V.180, 159–245.
[4] Tsiganov A.V., Goremykin O.V., Integrable systems on so(4) related with XXX spin chains with bounda-
ries, J. Phys. A: Math. Gen., 2004, V.37, 4843–4849, nlin.SI/0310049.
[5] Sokolov V.V., On a class of quadratic Hamiltonians on so(4), Dokl. Akad. Nauk, 2004, V.394, 602–605 (in
Russian).
[6] Ramani A., Dorizzi B., Grammaticos B., Painlevé conjecture revisited, Phys. Rev. Lett., 1982, V.49, 1539–
1541.
[7] Grammaticos B., Dorizzi B., Ramani A., Integrability of Hamiltonians with third- and fourth-degree poly-
nomial potentials, J. Math. Phys., 1983, V.24, 2289–2295.
[8] Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, Cambridge,
Cambridge University Press, 1991.
[9] Sakovich S.Yu., Tsuchida T., Symmetrically coupled higher-order nonlinear Schrödinger equations: singu-
larity analysis and integrability, J. Phys. A: Math. Gen., 2000, V.33, 7217–7226, nlin.SI/0006004.
[10] Sakovich S.Yu., Tsuchida T., Coupled higher-order nonlinear Schrödinger equations: a new integrable case
via the singularity analysis, nlin.SI/0002023.
http://arxiv.org/abs/nlin.SI/0405066
http://arxiv.org/abs/nlin.SI/0310049
http://arxiv.org/abs/nlin.SI/0006004
http://arxiv.org/abs/nlin.SI/0002023
1 Introduction
2 The studied quadratic Hamiltonian
3 Demystifying the `mysterious' case
4 Conclusion
|