New 2D integrable families with a quartic second invariant
The method introduced by the present author in 1986 still proves most effective in constructing integrable 2-D Lagrangian systems, which admit in addition to the energy another integral of motion that is polynomial in velocities. In a previous article (J. Phys. A: Math. Gen., 39, 5807– 5824, 2006) w...
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Yehia, H.M. 2014-12-06T21:06:29Z 2014-12-06T21:06:29Z 2011 New 2D integrable families with a quartic second invariant / H.M. Yehia // Механика твердого тела: Межвед. сб. науч. тр. — 2011. — Вип 41. — С. 233-243. — Бібліогр.: 20 назв. — англ. 0321-1975 https://nasplib.isofts.kiev.ua/handle/123456789/71597 531.38 The method introduced by the present author in 1986 still proves most effective in constructing integrable 2-D Lagrangian systems, which admit in addition to the energy another integral of motion that is polynomial in velocities. In a previous article (J. Phys. A: Math. Gen., 39, 5807– 5824, 2006) we constructed a system, which admits a quartic complementary integral. This system, called by us “master”, is the largest known, as it involves 21 parameters, and contains, as special cases of it, almost all previously known systems of the same type that admit a quartic integral. In the present note we generalize the method we used before to construct new severalparameter systems that are not special cases of the master system. A new system involving 16 parameters is introduced and a special case of it admits interpretation in a problem of rigid body dynamics. It gives a unification of certain special versions of known classical integrable cases due to Kovalevskaya, Chaplygin and Goriatchev and other cases recently introduced by the present author. Продолжены исследования, начатые автором в 1986 году, и посвященные изучению условий существования у лагранжевых систем первых интегралов четвертого порядка. Рассматриваемая система характеризуется 16 параметрами. Получена структура лагранжиана, для которой дифференциальные уравнения движения допускают решения, характеризующиеся первым полиномиальным интегралом четвертого порядка. Это позволило обобщить известные случаи интегрируемости Ковалевской, Чаплыгина и Горячева классической задачи о движении твердого тела, имеющего неподвижную точку. Продовжено дослiдження, початi автором у 1986 роцi, i присвяченi вивченню умов iснування у лагранжевих систем перших iнтегралiв четвертого порядку. Розглядувана система характеризується 16 параметрами. Одержано структуру лагранжиана, для якої диференцiальнi рiвняння руху припускають розв’язки, що характеризуються першим полiномiальним iнтегралом четвертого порядку. Це дозволило узагальнити вiдомi випадки iнтегровностi Ковалевської, Чаплигiна i Горячева класичної задачi про рух твердого тiла, яке має нерухому точку. en Інститут прикладної математики і механіки НАН України Механика твердого тела New 2D integrable families with a quartic second invariant Новые интегрируемые случаи уравнений динамики с интегралами четвертой степени Новi iнтегровнi випадки рiвнянь динамiки з iнтегралами четвертого степеня Article published earlier |
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New 2D integrable families with a quartic second invariant |
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New 2D integrable families with a quartic second invariant Yehia, H.M. |
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New 2D integrable families with a quartic second invariant |
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New 2D integrable families with a quartic second invariant |
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New 2D integrable families with a quartic second invariant |
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New 2D integrable families with a quartic second invariant |
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new 2d integrable families with a quartic second invariant |
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Yehia, H.M. |
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Yehia, H.M. |
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Механика твердого тела |
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Інститут прикладної математики і механіки НАН України |
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Новые интегрируемые случаи уравнений динамики с интегралами четвертой степени Новi iнтегровнi випадки рiвнянь динамiки з iнтегралами четвертого степеня |
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The method introduced by the present author in 1986 still proves most effective in constructing integrable 2-D Lagrangian systems, which admit in addition to the energy another integral of motion that is polynomial in velocities. In a previous article (J. Phys. A: Math. Gen., 39, 5807– 5824, 2006) we constructed a system, which admits a quartic complementary integral. This system, called by us “master”, is the largest known, as it involves 21 parameters, and contains, as special cases of it, almost all previously known systems of the same type that admit a quartic integral. In the present note we generalize the method we used before to construct new severalparameter systems that are not special cases of the master system. A new system involving 16 parameters is introduced and a special case of it admits interpretation in a problem of rigid body dynamics. It gives a unification of certain special versions of known classical integrable cases due to Kovalevskaya, Chaplygin and Goriatchev and other cases recently introduced by the present author.
Продолжены исследования, начатые автором в 1986 году, и посвященные изучению условий существования у лагранжевых систем первых интегралов четвертого порядка. Рассматриваемая система характеризуется 16 параметрами. Получена структура лагранжиана, для которой дифференциальные уравнения движения допускают решения, характеризующиеся первым полиномиальным интегралом четвертого порядка. Это позволило обобщить известные случаи интегрируемости Ковалевской, Чаплыгина и Горячева классической задачи о движении твердого тела, имеющего неподвижную точку.
Продовжено дослiдження, початi автором у 1986 роцi, i присвяченi вивченню умов iснування у лагранжевих систем перших iнтегралiв четвертого порядку. Розглядувана система характеризується 16 параметрами. Одержано структуру лагранжиана, для якої диференцiальнi рiвняння руху припускають розв’язки, що характеризуються першим полiномiальним iнтегралом четвертого порядку. Це дозволило узагальнити вiдомi випадки iнтегровностi Ковалевської, Чаплигiна i Горячева класичної задачi про рух твердого тiла, яке має нерухому точку.
|
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0321-1975 |
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https://nasplib.isofts.kiev.ua/handle/123456789/71597 |
| citation_txt |
New 2D integrable families with a quartic second invariant / H.M. Yehia // Механика твердого тела: Межвед. сб. науч. тр. — 2011. — Вип 41. — С. 233-243. — Бібліогр.: 20 назв. — англ. |
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2025-11-27T01:16:32Z |
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| fulltext |
ISSN 0321-1975. Механика твердого тела. 2011. Вып. 41
УДК 531.38
c©2011. H.M. Yehia
NEW 2D INTEGRABLE FAMILIES WITH
A QUARTIC SECOND INVARIANT
The method introduced by the present author in 1986 still proves most effective in constructing
integrable 2-D Lagrangian systems, which admit in addition to the energy another integral of
motion that is polynomial in velocities. In a previous article (J. Phys. A: Math. Gen., 39, 5807–
5824, 2006) we constructed a system, which admits a quartic complementary integral. This
system, called by us “master”, is the largest known, as it involves 21 parameters, and contains,
as special cases of it, almost all previously known systems of the same type that admit a quartic
integral. In the present note we generalize the method we used before to construct new several-
parameter systems that are not special cases of the master system. A new system involving 16
parameters is introduced and a special case of it admits interpretation in a problem of rigid
body dynamics. It gives a unification of certain special versions of known classical integrable
cases due to Kovalevskaya, Chaplygin and Goriatchev and other cases recently introduced by
the present author.
Keywords: integrable families, invariant, rigid body dinamics.
Introduction. Historical The famous Kovalevskaya’s integrable case of
rigid body dynamics was the first example of a mechanical system that admits
an integral of motion quartic in velocities [1]. For more than a century this case
attracted attention of many specialists, who treated explicit solution in terms of
time and gave several modifications and generalizations. Only in the last three
decades there appeared a few new integrable systems with a quartic integral, but
mainly concerning the motion of a particle in the Euclidean plane under the action
of certain potential forces. A short, but nearly complete up to its date, list of those
systems can be found in Hietarinta’s review paper [2]. A few cases of the same
type were obtained later in [3].
In virtue of Maupertuis principle, the motion of a natural mechanical system
can be brought into equivalence (in the orbital sense) with the geodesic flow
on some Riemannian metric. Metrics on the Riemannian sphere associated wi-
th known integrable cases of rigid body dynamics were constructed in [4]. Two
families of integrable systems with a quartic integral on S2 were obtained in [5]
and [6]. Few more works discussed possible integrable systems with low-degree
polynomials on S2 and the hyperbolic plane H2 (see e.g. [7–9]).
The method introduced in our work [10] and developed in several later
works, has led to construction of a large number of several-parameter famili-
es of integrable systems with a complementary integral ranging from second to
fourth degree (see e.g. [13,14]). This method leads in a natural way to integrable
systems on various flat and curved 2-D manifolds. Although our primary interest is
in systems on Riemannian manifolds, some of the constructed integrable systems
live on pseudo-Euclidean or pseudo-Riemannian manifolds.
The culmination of this method was the construction of the so-called “master”
233
H.M. Yehia
system with a quartic integral [15]. It involves the largest ever number of 21
parameters and covers almost all systems of that type that were known earlier.
In the present article we extend the metod used in [15] to construct certain
systems with more general structures. One of the resulting systems that involves
15 parameters and is investigated in some detail. A special case introduces a new
integrable problem in rigid body dynamics.
Construction of integrable systems. According to a result of Birkhoff
[16], the general natural mechanical system (on an arbitrary 2D Riemannian confi-
guration space) can always be reduced in certain (isometric) coordinates ξ, η and
time parametrization τ to the form of a ficticious plane system described by the
Lagrangian
L =
1
2
[ξ
′2 + η
′2] + U,U = U(ξ, η) (1)
restricted to its zero-energy level
ξ′2 + η′2 − 2U = 0. (2)
The energy constant h for the original system enters linearly as a parameter in
the function U, which has the structure
U = Λ(h− V ) (3)
where V is the potential of the original system and Λ is a function that depends
on the metric of the configation space.
In [15] (see also [10]), it was proved that if an integral of motion of the mechani-
cal system exists in the form of a polynomial of the fourth degree in velocities,
this integral can be reduced while preserving the form of the Lagrangian (1) to
the form
I = ξ′4 + Pξ′2 +Qξ′η′ +R = I0(const) (4)
where R is given by the quadrature
R = −
∫
Q
∂U
∂ξ
dη −
∫
[2P
∂U
∂ξ
+Q
∂U
∂η
+ 2U
∂Q
∂η
]0dξ (5)
in which [ ]0 means that the expression in the bracket is computed for η taking
an arbitrary constant value η0 (say), and the other three functions involved are
expressed as
P =
∂2F
∂ξ2
, Q = − ∂2F
∂ξ∂η
, U = −1
4
(
∂2F
∂ξ2
+
∂2F
∂η2
) (6)
in terms of a single auxiliary function F which satisfies the nonlinear partial
differential equation
∂2F
∂ξ∂η
(∂4F
∂ξ4
−∂4F
∂η4
)
+3
(∂3F
∂ξ3
∂3F
∂ξ2∂η
−∂3F
∂η3
∂3F
∂η2∂ξ
)
+2
(∂2F
∂ξ2
∂4F
∂ξ3∂η
−∂2F
∂η2
∂4F
∂η3∂ξ
)
= 0.
(7)
234
New 2D integrable families with a quartic second invariant
The set of solutions of this equation generates all systems of the type (1) having
an integral of the form (4) on the zero level of their energy integral. Affecting all
possible conformal mappings of the complex ζ = ξ+iη plane followed by a general
point transformation to the generalized coordinates q1, q2 with a suitable change
of the time variable we obtain all systems of the general form on 2D Riemannian
manifolds, having a quartic integral on the zero level of their energy integral.
To construct a system that would be a priori integrable on all levels of energy,
we should receive a function U from (6) that has the structure (3), i.e. involving in
a linear way an arbitrary parameter h. In that case The auxiliary function should
have the form F = hF0 + F1, where F0, F1 are functions not depending on h. It
is shown in [15] how the resulting conditional integrable system can be used to
construct another integrable system valid on arbitrary energy level.
1. The auxiliary equation and certain forms of solutions. However, it
became clear that the original isometric variables ξ, η are not practically suitable
coordinates for the description of the solution, and that a symmetric separation
solution can be more conveniently expressed in terms of the pair of variables p, q
related to ξ, η by the relations
ξ =
∫ p dp
4
√
a4p4 + a3p3 + a2p2 + a1p+ a0
, η =
∫ q dq
4
√
a4q4 + b3q3 + b2q2 + b1q + b0
.
(8)
This solution, first announced in [15], can be written as
F = F0 + νpq, (9)
where F0 =
=
∫ p dp
4
√
a4p4 + a3p3 + a2p2 + a1p+ a0
∫ p (4C0 + 4C1p+ 4Ap2 + 1
4b3p
3)dp
(a4p4 + a3p3 + a2p2 + a1p+ a0)3/4
+
+
∫ q dq
4
√
a4q4 + b3q3 + b2q2 + b1q + b0
∫ q (4D0 + 4D1p+ 4Ap2 + 1
4a3p
3)dq
(a4q4 + b3q3 + b2q2 + b1q + b0)3/4
. (10)
It involves 15 free parametersa0, a1, a2, a3, a4, b0, b1, b2, b3, A,C0, C1,D0,D1, ν. It
should be noted that this solution determine the Lagrangian and the quartic
integral valid only on the zero level of its energy.
In the present article we try some deformations of the last system to
accomodate more terms of the product type, probably at the expense of enforcing
certain restrictions on some of the parameters figuring in (10). In fact, we assume
F in the form
F = F0 +
4
∑
i+j=2
Fijp
iqj (11)
so that we have replaced the monomial term pq with a fourth-degree polynomial.
The limit 4 for the degree was chosen on an experimental basis, as it seemed
235
H.M. Yehia
that fifth-degree polynomials would take incomparable time and computation
resources.
The expression (11) is inserted in equation (7). Using the relations (8) and
making some manipulations, we obtain a polynomial expression of the sixth degree
in the two variables p, q that must vanish, whose coefficients constitute a system
of 27 polynomial equations in the 26 parameters involved {A,C0, C1,D0,D1, ai,
i = 0, ..., 4; bj, j = 0, ..., 3, Fmn, 2 ≤ m ≤ n ≤ 4}. We have solved this system
using the MAPLE computer algebra package. The result was 40 distinct solutions,
i.e. 40 working combinations of the parameters that lead to the construction of
integrable systems with a quartic integral. One of those solutions reproduces the
“master” system. This system is characterized by the preservation of all the 15
parameters ai, bi, A,Cj ,Dj and ν. The remaining 39 cases are not expected to be
all different. In view of the symmetric way in which groups of parameters enter
with the two variables p and q, some solutions lead to rewriting one and the same
system. The final number of different systems turns out to be much less than
40. In a forthcoming article those systems will be classified and put in a form as
simple as possible.
In only one more of the fourty cases a4 6= 0, so that the two polynomials that
occur under the fourth degree root signs are of the fourth degree. This case, which
differs from the master system, will be considered in detail in the next sections.
2. The generic restricted case. We write the Lagrangian and the
complementary integral for this case after some transformation to more symmetric
form that does not affect the generality of the system
L =
1
2
[
ú2√
au4 + k1u2 + k0
+
v́2√
av4 +m1v2 +m0
] + U, (12)
U = −N [m1u
4 + v2(4au4 + 3k1u
2 + 2k0)] + νuv(2au2 + k1) +Ku2 +D√
au4 + k1u2 + k0
−
−N [k1v
4 + u2(4av4 + 3m1v
2 + 2m0)] + νuv(2av2 +m1) +Kv2 + E√
av4 +m1v2 +m0
, (13)
I =
ú4
au4 + k1u2 + k0
+
+
4{N [m1u
4 + v2(4au4 + 3k1u
2 + 2k0)] + νuv(2au2 + k1) +Ku2 +D}
au4 + k1u2 + k0
ú2 −
−8(2Nuv + ν)úv́ − 8(2Nuv + ν)2
√
au4 + k1u2 + k0
√
av4 +m1v2 +m0 +
+16N(Dv2 − Eu2)− 4ν(ν + 4Nuv)(2au2v2 +m1u
2 + k1v
2)−
−16N2[(k0 + k1u
2)v4 + (m0 +m1v
2)u4 + 2au4v4] +
236
New 2D integrable families with a quartic second invariant
+
4
a3(au4 + k1u2 + k0)
×
×{N2{m2
1k0(ak0 − k21) +m2
1[a
3u8 + k0a
2u4 − k21u
2(k1 + au2) + ak0k1u
2] +
+a3v2(2k0 + k1u
2)(2k0v
2 + k1u
2v2 − 2m1u
4)} −
−2Nνa3u3v[m1(2k0 + k1u
2) + v2(4ak0 − k21)]−
−2NDa2[av2(2k0 + k1u
2) +m1(k0 + k1u
2)] +
+ν2a3(k21 − 4ak0)u
2v2 +Da3[D + 2Ku2 + 2νuv(2au2 + k1)]−
−K2a2(k0 + k1u
2)− 2Kνa3uv(2k0 + k1u
2) + 2KNa[m1a
2u6 −
−ak1(−m1 + av2)u4 − (2k0a
2v2 − k21m1)u
2 + k0k1m1]} (14)
in which the prime represents derivative with respect to the independent variable
τ (ficticious time).
The system with the Lagrangian (12) admits the integral (14) only on the
zero-energy level of this system
1
2
[ ú2√
au4 + k1u2 + k0
+
v́2√
av4 +m1v2 +m0
]
− U = 0.
It depends on 10 parameters a, k0, k1,m0,m1,D,E,K, ν,N , of which the first five
enter in both the kinetic energy and potential terms of the Lagrangian and the last
five ones enter only in the potential terms and moreover they enter only linearly.
The last four parameters constitute a set of energy-like parameters (see e.g. [15]).
They are essential in building the unrestricted integrable system. Comparing (12)
to its counterpart in the “master” system [15], we find that (12) involves only one
new parameter N , which is not present in the master system. When N is set equal
to zero (12) turns out to be a special case of the master system resulting from
one restriction on the coefficients of each of the two fourth-degree polynomials
entering under the root sign and two restrictions on the energy -like parameters,
namely C1 = D1 = 0.
3. Dynamics- The unrestricted generalization. We now proceed to
use those parameters to construct a general integrable system valid on arbitrary
energy level out of the restricted one. Introducing new parameters by the relations
D = h1 − hα1, E = h2 − hα2,
K = h3 − hα3, ν = h4 − hα4,
N = h5 − hα5
(15)
and performing the change of independent variable to the actual-time parametri-
zation by using the relation
dτ =
dt
Λ
, (16)
237
H.M. Yehia
where
Λ =
α1 + α3u
2 + α4uv(2au
2 + k1) + α5[m1u
4 + v2(4au4 + 3k1u
2 + 2k0)]√
au4 + k1u2 + k0
+
+
α2 + α3v
2 + α4uv(2av
2 +m1) + α5[k1v
4 + u2(4av4 + 3m1v
2 + 2m0)]√
au4 + k1u2 + k0
(17)
we arrive at the new Lagrangian
L =
1
2
Λ[
u̇2√
au4 + k1u2 + k0
+
v̇2√
av4 +m1v2 +m0
]− V + h,
V =
1
Λ
{h1 + h3u
2 + h4uv(2au
2 + k1) + h5[m1u
4 + v2(4au4 + 3k1u
2 + 2k0)]√
au4 + k1u2 + k0
+
+
h2 + h3v
2 + h4uv(2av
2 +m1) + h5[k1v
4 + u2(4av4 + 3m1v
2 + 2m0)]√
au4 + k1u2 + k0
} (18)
which admits on an arbitrary energy level h the integral resulting from (14) by the
substitutions (15) and (16), i.e. ú → Λu̇, v́ → Λv̇. The integral will depend on the
parameters occuring in the Lagrangian and also on the energy constant h. The
last constant may be substituted by its expression in terms of the coordinates and
velocities to get the final form free of the energy restriction. The resulting system
depends on 16 parameters a, k0, k1,m0,m1, α1, α2, α3, α4, α5, h1, h2, h3, h4, h5 and
h, of which the first nine enter in both the kinetic energy and potential terms of
the Lagrangian and the last five ones enter only in the potential.
4. Special cases. Generalization of the cases of Bozis and Wojci-
echowski. Let a = k0 = m0 = 1, k1 = m1 = −2. Under the coordinate
transformation p = sin y, q = sinx the Lagrangian (18) takes the form
L =
1
2
[α+ β sinx sin y + γ(2 cos2 x cos2 y − cos2 x− cos2 y) +
+
δ1
cos2 x
+
δ2
cos2 y
](ẋ2 + ẏ2)− (19)
−
a+ b sinx sin y + c(2 cos2 x cos2 y − cos2 x− cos2 y) +
d1
cos2 x
+
d2
cos2 y
α+ β sinx sin y + γ(2 cos2 x cos2 y − cos2 x− cos2 y) +
δ1
cos2 x
+
δ2
cos2 y
.
When β = γ = δ1 = δ2 = 0 we have, after ignoring an insignificant additive
constant
L =
1
2
(ẋ2+ ẏ2)− [b sinx sin y+c(2 cos2 x cos2 y−cos2 x−cos2 y)+
d1
cos2 x
+
d2
cos2 y
].
(20)
238
New 2D integrable families with a quartic second invariant
This system is new. It describes plane motion of a particle in a 4-parameter
potential. The complementary integral of this system can be written as
I = (ẋ2 +
2d1
cos2 x
)(ẏ2 +
2d2
cos2 y
)− 2 cos x cos y(b+ 2c sin x sin y)ẋẏ +
+cos2 x cos2 y(b+ 2c sin x sin y)2 + 4c(d1 cos
2 y + d2 cos
2 x).
When c = 0 this case reduces to a special version of that found by Bozis [17]
and when c = b = 0 the system becomes separable and the integral degenerates
into the product of two quadratic integrals. A slight variation of the parameters
in (20) to be k1 = m1 = 2 changes trigonometric functions to hyperbolic (or
exponential) functions, and thus giving a new system like the type of [18]. The
analog of (20) gives a particle in the potential
V = b sinhx sinh y + c(2 cosh2 x cosh2 y − cosh2 x− cosh2 y) +
d1
cosh2 x
+
d2
cosh2 y
.
In a similar way, one can obtain a mix of the two types.by taking k1 = −m1 = 2.
Some other variations lead to combinations of exponential and trigonometric
or hyperbolic functions. For example, let a = 1, k0 = k1 = 0, m0 = 1, m1 = 2.
Under the coordinate transformation p = ex, q = cosh y the Lagrangian takes
the form
L =
1
2
(ẋ2 + ẏ2)− V, V = ae2x cosh 2y + bex cosh y + ce−2x +
d
sinh2 y
. (21)
Generalization of ystems of the Toda type. If in (18) we set
a = 1, k0 = k1 = m0 = m1 = 0, the Lagrangian takes the form
L =
1
2
λ(ẋ2 + ẏ2)− 1
λ
[h0 + ae−2x + be−2y + cex+y + de2(x+y)], (22)
where
λ = α0 + αe−2x + βe−2y + γex+y + δe2(x+y) (23)
and the integral may be written, after using the energy integral to eliminate
h, as
I = λ4ẋ2ẏ2 + 2λ2[be−2y ẋ2 + ae−2xẏ2 + (cex+y + de2x+2y)ẋẏ] +
+e2x+2y(c+ dex+y)2 + 2d(be2x + ae2y) + 4abe−2x−2y . (24)
Application to rigid body dynamics. We now consider the general
problem of motion of a rigid body about a fixed point under the action of a combi-
nation of conservative axisymmetric potential forces. The equations of motion for
this problem can be written in the Euler–Poisson form:
239
H.M. Yehia
Aṗ + (C −B)qr = γ2
∂V
∂γ3
− γ3
∂V
∂γ2
,
Bq̇ + (A− C)pr = γ3
∂V
∂γ1
− γ1
∂V
∂γ3
,
Cṙ + (B −A)pq = γ1
∂V
∂γ2
− γ2
∂V
∂γ1
,
γ̇1 + qγ3 − rγ2 = 0, γ̇2 + rγ1 − pγ3 = 0, γ̇3 + pγ2 − qγ1 = 0, (25)
where A,B,C are the principal moments of inertia, p, q, r are the components of
the angular velocity of the body and γ1, γ2, γ3 are the components of the unit
vector γ fixed in space in the direction of the axis of symmetry of the force fields
applied to the body, all being referred to the principal axes of inertia at the fixed
point.
The system (25) admits three integrals:
I1 =
1
2
(Ap2 +Bq2 + Cr2) + V, (26)
I2 = Apγ1 +Bqγ2 + Crγ3, (27)
I3 = γ21 + γ22 + γ23 = 1. (28)
Equations (25) admit an equivalent representation in the Lagrangian form
(see e.g. [15]). After ignoring the cyclic angle of precession, around the axis of
symmetry of the field on the zero level of the cyclic integral I2 = 0, the Routhian
of this mechanical system expressed in the other two Eulerian angles has the form
R =
1
2
A
[
θ̇2 +
C sin2 θϕ̇2
A− (A− C) cos2 θ
]
− V, (29)
where θ is the angle of nutation and ϕ is the angle of proper rotation (about the
axis of symmetry of the body). Comparing the structure of this Routhian function
to that of the Lagrangian (18) and recalling the procedure followed in a similar
situation in [15], we get convinced that they become identical only in the four
case A = B = 2C.
In fact, setting a = 1, k1 = 1, k0 = 0, m1 = 2, m0 = 1, α1 = α4 =
= α5 = 0, α2 = α3, and affecting the substitution u =
cos2 θ
2 sin θ
, v = cosϕ and
renaming the remaining parameters, we get the Lagrangian
L =
1
2
[θ̇2 +
sin2 θ
1 + sin2 θ
ϕ̇2]− V, (30)
V = 2C[a sin θ sinϕ+ b sin2 θ cos(2ϕ) +
λ
cos2 θ
+ δ
1 + sin2 θ
sin2 θ cos2 ϕ
] (31)
240
New 2D integrable families with a quartic second invariant
and the integral
I =
sin6 θϕ̇4
(1 + sin2 θ)4
+
sin2 θϕ̇2
(1 + sin2 θ)2
[2a sin θ sinϕ+ b(2− 3 cos2 θ)+
+
2λ sin2 θ
cos2 θ
+
4δ
cos2 ϕ
+ sin2 θθ̇2]+
+
2θ̇ϕ̇ sin θ cos θ cosϕ
(1 + sin2 θ)
(a sin θ + 2b cos2 θ sinϕ) + θ̇2[b cos 2ϕ +
2δ
cos2 ϕ
]+
+
4δ2
sin2 θ cos4 ϕ
+ δ[
4λ
cos2 θ cos2 ϕ
+
4a sinϕ
sin θ cos2 ϕ
+ 2b(2 sin2 θ +
1− 3 sin2 θ
sin2 θ cos2 ϕ
)]+
+ (
1
2
a2 − 2ab sin θ sinϕ)(1 − 2 sin2 θ cos2 ϕ)+
+
1
2
b2 sin2 θ[sin2 θ(cos 4ϕ− 1) + 4] + 2λb tan2 θ cos 2ϕ. (32)
To facilitate comparison with other results, we now express the last integrable
case of rigid body dynamics in the Euler-Poisson variables, i.e. as a solution of
the system (25), in the next
Theorem 1. For A = B = 2C and for the potential
V = 2C
[
aγ1 + b(γ21 − γ22) +
λ
γ23
+ δ
2 − γ23
γ22
]
(33)
equations (25) are integrable on the level I2 = 0. The complementary integral has
the form
I = (p2 − q2 − aγ1 + bγ23 −
λ(γ21 − γ22)
γ23
)2 + (2pq − aγ2 −
2λγ1γ2
γ23
)2 +
+
δ
γ22
[
2(p2 + q2)γ23 − 2aγ1γ
2
3 − 2λγ21 + 2b+
δγ43
γ22
]
. (34)
This case is new. For comparison we provide a table of presently known
integrable potentials related to the type (33), which admit a quartic integral
under the condition A = B = 2C:
Author- year Potential
Kovalevskaya [1]1889 V1 = a1γ1 + a2γ2
Chaplygin [19] 1903 V2 = b1(γ
2
1 − γ22) + b2γ1γ2
Goriatchev [20] 1916 V3 = aγ1 + a2γ2 + b(γ21 − γ22) + b1γ1γ2 +
λ
γ23
Yehia [15] 2006
V4 = b(γ21 − γ22) +
λ
γ23
+ ρ(
1
γ43
− 1
γ63
) + (2− γ23)(
ν
γ21
+
δ
γ22
)
V5 = aγ1 +
λ
γ23
+
ε
√
γ21 + γ22
+
(2− γ23)
γ22
(δ + µ
γ1
√
γ21 + γ22
)
241
H.M. Yehia
In the potential (33) the parameter a is present in Kovalevskaya’s case (1889),
b in Chaplygin’s case (1903), λ in Goriatchev’s case (1916) and δ in both cases
announced in our work [15] (2006). The combination (33) is new.
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4059.
4. Bolsinov A.V., Kozlov V.V., Fomenko A.T. The Maupertuis principle and geodesic fows
on a sphere that arise from integrable case of the dynamics of a rigid body // Uspekhi
Mat. Nauk. – 1995. – 50. – P. 3–32. = Russian Math. Surveys. – 1995. – 50. – P. 473–501.
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2 possessing an integral of fourth
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// Reg. Chaot. Dyn. – 2006, 11. – P. 103–122.
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Х.М. Яхья
Новые интегрируемые случаи уравнений динамики с интегралами
четвертой степени
Продолжены исследования, начатые автором в 1986 году, и посвященные изучению условий
существования у лагранжевых систем первых интегралов четвертого порядка. Рассматри-
ваемая система характеризуется 16 параметрами. Получена структура лагранжиана, для
которой дифференциальные уравнения движения допускают решения, характеризующиеся
242
New 2D integrable families with a quartic second invariant
первым полиномиальным интегралом четвертого порядка. Это позволило обобщить изве-
стные случаи интегрируемости Ковалевской, Чаплыгина и Горячева классической задачи
о движении твердого тела, имеющего неподвижную точку.
Ключевые слова: лагранжиан, решение, потенциальная функция.
Х.М. Яхья
Новi iнтегровнi випадки рiвнянь динамiки з iнтегралами
четвертого степеня
Продовжено дослiдження, початi автором у 1986 роцi, i присвяченi вивченню умов iснува-
ння у лагранжевих систем перших iнтегралiв четвертого порядку. Розглядувана система
характеризується 16 параметрами. Одержано структуру лагранжиана, для якої диферен-
цiальнi рiвняння руху припускають розв’язки, що характеризуються першим полiномiаль-
ним iнтегралом четвертого порядку. Це дозволило узагальнити вiдомi випадки iнтегров-
ностi Ковалевської, Чаплигiна i Горячева класичної задачi про рух твердого тiла, яке має
нерухому точку.
Ключовi слова: лагранжиан, розв’язок, потенцiальна функцiя.
Mansoura University, Mansoura, Egypt
hyehia@mans.edu.eg
Получено 28.10.11
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