On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom
The completely singular dynamical systems of the Liouville type are studied. The motion paths of these systems are closed graphs if the Liouville tori are compact. The conditions under which a dynamical system of the Liouville type is strongly singular are obtained in the paper. These conditions hav...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2010
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Lisitsa, V.T. 2016-10-01T16:48:36Z 2016-10-01T16:48:36Z 2010 On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom / V.T. Lisitsa // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 295-304. — Бібліогр.: 13 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106647 The completely singular dynamical systems of the Liouville type are studied. The motion paths of these systems are closed graphs if the Liouville tori are compact. The conditions under which a dynamical system of the Liouville type is strongly singular are obtained in the paper. These conditions have a form of the system of integral equations. It is proved that the obtained system is solvable. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom Article published earlier |
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On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom |
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On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom Lisitsa, V.T. |
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On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom |
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On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom |
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On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom |
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on the conditions of total resonance of liouville type hamiltonian systems with n degrees of freedom |
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Lisitsa, V.T. |
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Lisitsa, V.T. |
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2010 |
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Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The completely singular dynamical systems of the Liouville type are studied. The motion paths of these systems are closed graphs if the Liouville tori are compact. The conditions under which a dynamical system of the Liouville type is strongly singular are obtained in the paper. These conditions have a form of the system of integral equations. It is proved that the obtained system is solvable.
|
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1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/106647 |
| citation_txt |
On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems with n Degrees of Freedom / V.T. Lisitsa // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 295-304. — Бібліогр.: 13 назв. — англ. |
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AT lisitsavt ontheconditionsoftotalresonanceofliouvilletypehamiltoniansystemswithndegreesoffreedom |
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2025-11-25T22:46:26Z |
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2025-11-25T22:46:26Z |
| _version_ |
1850575532399263744 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 3, pp. 295–304
On the Conditions of Total Resonance of Liouville Type
Hamiltonian Systems with n Degrees of Freedom
V.T. Lisitsa
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:lisitsa@univer.kharkov.ua
Received March 19, 2010
The completely singular dynamical systems of the Liouville type
are studied. The motion paths of these systems are closed graphs if the
Liouville tori are compact. The conditions under which a dynamical system
of the Liouville type is strongly singular are obtained in the paper. These
conditions have a form of the system of integral equations. It is proved that
the obtained system is solvable.
Key words: Hamiltonian system, resonance, integrable system, action-
angle variables.
Mathematics Subject Classification 2000: 70H06; 53B21.
1. Introduction
It is well known that many fundamental physical phenomena are described in
the terms of the Hamiltonian formalism. The Hamiltonian systems integrable in
the Liouville–Arnold terms [1] are an important class of the Hamiltonian systems.
In the recent years, the interest towards the study of these systems renewed as
new methods for the study of integrable systems [2] had appeared. Introduce the
necessary definitions and recall the well-known facts.
Consider a Hamiltonian system. It is given by an even-dimensional manifold
M2n with nondegenerate exterior closed 2-form ω (the symplectic structure) and
differentiable function H (the Hamiltonian) [1, 3].
Definition 1. A differentiable vector field sgradH given on M2n is said to be
the skew gradient of a function H if it satisfies the equality ω (v̄; sgrad H) = v̄(H)
for all differentiable vector fields v̄ where v̄(H) is the derivative of the function
H in the direction of v̄. The vector field sgradH is called also the Hamiltonian
vector field with Hamiltonian H [1, 3].
c© V.T. Lisitsa, 2010
V.T. Lisitsa
Definition 2. Let F and H be differentiable functions on symplectic mani-
fold M2n with symplectic structure ω. The differentiable function {F ; H} =
ω(sgradF ; sgradH) is said to be the Poisson bracket of the functions F and H.
Definition 3. A differentiable function F is said to be the integral of the
Hamiltonian system with Hamiltonian H if F is constant along the integral tra-
jectories of the given Hamiltonian system.
Note that the function F is an integral of the system with Hamiltonian H if
and only if {F ;H} ≡ 0. In particular, the Hamiltonian H is an integral.
Definition 4. Two functions F1 and F2 are said to be involutive (or to be in
involution) if {F1; F2} ≡ 0.
Definition 5. A Hamiltonian system with n degrees of freedom with Hamilto-
nian H is said to be completely integrable in the Liouville–Arnold terms if it has
n functionally independent, pairwise involutive integrals, H = F1, F2, . . . , Fn,
[1, 3]. The independence of the integrals signifies that n forms of dFi, i = 1, . . . , n,
are linearly independent.
If the Hamiltonian system H is integrable in the Liouville – Arnold terms, then
it is possible to introduce ”action-angle” variables (I;ϕ) so that the corresponding
Hamiltonian differential equations have the form [1]
dI
dt
= 0,
dϕ
dt
= ω(I) =
∂H
∂I
,
I = I1, . . . , In; ϕ = ϕ1, . . . , ϕn; ω = ω1, . . . , ωn.
(1)
The motion described by the equations of (1) is said to be conditionally periodic
and the values ωi =
∂H
∂Ii
are said to be the frequencies of a conditionally periodic
motion.
Consider the submanifolds Fc = {(q, p) : Fi = ci}, where ci are constants. The
submanifolds Fc are the level surfaces of the integrals Fi, i = 1, . . . , n. In the
general case, the submanifolds Fc are homeomorphic to the product Tn−k ×Ek,
where Tn−k is the torus of the dimension n − k, Ek is the Euclid space. If Fc
are compact, then they are homeomorphic to the torus Tn (the Liouville–Arnold
torus).
Note that in the general case the frequencies ωi are rationally independent,
i.e., the equality k1ω1 + · · · + knωn = 0, where ki are rational numbers, yields
k1 = · · · = kn = 0. Besides, if the level surfaces Fc are the tori Tn, then the
trajectories of the conditionally periodic movement fill Tn everywhere densely.
There often arise special cases when some frequencies are rationally depen-
dent. Besides, the dimensions of the tori filled with conditionally periodic tra-
jectories decrease. If all the frequencies are pairwise rationally dependent, then
296 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems
the motion trajectories become closed curves. In this case the movement is com-
pletely singular.
Definition 6. A completely singular Hamiltonian system is called the totally
resonance Hamiltonian system.
Completely singular systems are of interest from different points of view.
First of all, completely singular systems have (2n − 1) of independent integrals
of movement. Of course, not all of them are in the involution. If the Liouville
tori are compact, they are circles, i.e., the trajectories of the Hamiltonian system
are closed [4]. From the geometric point of view, the manifolds, all of whose
geodesics are closed, belong to these systems. In [5], a completely integrable
Hamiltonian system sgradH with two degrees of freedom is considered and the
following proposition is proved.
Consider the restriction H to a nonsingular leaf Q3 : {x ∈ Mn | H(x) = c}.
Then the following two statements are equivalent:
i) any trajectory on Q3 is periodic;
ii) there exist two functionally independent functions f1, f2 on Q3 that are
Bott integrals of the system v = sgradH on Q3.
Here are some examples below:
1) The Kepler problem. In the spherical coordinates, the Kepler Hamilto-
nian of the motion has the form
H =
1
2
(
p2
r +
p2
θ
r2
+
p2
ϕ
r2 sin2 θ
)
− k
r
, k > 0.
It is well known that the complete integral of the corresponding Hamilton–Jacobi
equation for the given problem allows the separation of variables and has the form
[4, 5]
w = βϕ · ϕ +
∫ √
2h +
2k
r
− β2
θ
r2
dr +
∫ √
β2
θ −
β2
ϕ
sin2 θ
dθ.
It is proved in [4] that all frequencies of the given motion are equal and the
trajectories of this system are closed.
2) The manifolds, all of whose geodesics are closed. Consider the
two-dimensional manifolds as an example. As known, all geodesics of the sphere
S2 are closed. In [8], the problems on the manifolds with closed geodesics are
studied in detail. In particular, the surfaces of revolution, all of whose geodesics
are closed, are analyzed. With a certain choice of coordinates, one can reduce
the metric of the revolution surface to the form of
ds2 = [f(cos r)]2dr2 + sin2 rdθ2, 0 ≤ θ ≤ 2π, 0 ≤ r ≤ π.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 297
V.T. Lisitsa
The Hamiltonian of geodesic flow of the given surface is
H =
1
2
(
p2
1
(f(cos r))2
+
p2
2
sin2 r
)
.
The complete integral of the corresponding Hamilton–Jacobi equation allows the
separation of variables and can be written as
w =
√
2
∫
f(cos r)
√
β1
sin2 r
+ 2hdr +
√
2
∫ √
−β1dθ.
In this case, the condition for the rational dependence of frequencies is
π−c∫
c
sin c · f(cos r)
sin r
√
sin2 r − sin2 c
dr =
p
q
π,
for all c ∈
(
0;
π
2
)
, where p, q are coprime integer numbers. The last equality is
known as the Darboux condition for closedness of all geodesics on the revolution
surface [8, 9].
2. Hamiltonian Systems of the Liouville Type
In [10, 11], the geodesic flows of the Liouville type with two degrees of freedom
were studied and the conditions for closedness of all the trajectories, i.e., the
conditions for complete singularity of these systems, were obtained.
The aim of the paper is to find the conditions for complete singularity of the
Liouville type systems with n degrees of freedom.
A dynamical system is said to be of the Liouville type if its kinetic energy T
and potential energy P are given by
T =
1
2
f
n∑
i=1
ϕi (qi) q2
i , P =
ψ
f
, where f =
n∑
i=1
fi (qi) ,
ψ =
n∑
i=1
ψi (qi) .
Hereinafter we suppose that ϕi > 0. The Hamiltonian of this system is given by
H =
1
2f
n∑
i=1
p2
i
ϕi
+
ψ
f
. (2)
Consider the system with Hamiltonian (2) in the parallelepiped 0 ≤ qi ≤ ai,
i = 1, . . . , n. Denote gi = hfi − ψi. Let gi(0) = gi(ai) = 0, gi0 = max
qi∈[0;ai]
gi = gi (qi0)
298 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems
(or gi0 = min
qi∈[0;ai]
gi = gi (qi0)). Note that the functions gi and gi0 depend on h.
If the functions gi are strictly monotonous on the intervals (0; qi0) and (qi0; ai),
then for every value of ci ∈ (0; gi0) there exist exactly two values of the variable qi
from the interval (0, ai), qi1 (ci), qi2 (ci) ∈ (0; ai) such that gi (qi1) = gi (qi2) = ci.
The following theorem is the main result of the paper.
Theorem. Let there be a dynamical system with Hamiltonian (2). If the func-
tions gi = hfi − ψi satisfy the conditions gi(0) = gi (ai) = 0 and are monotonous
on the intervals (0, qi0) and (qi0; ai), then the conditions for total resonance of
this Hamiltonian system are the following: there exists a constant λ (h) depending
on energy level h and rational numbers ri such that
qi2(ci)∫
qi1(ci)
√
ϕi
gi − ci
dqi = riλ (h) , i = 1, . . . , n− 1, (3)
for all ci ∈ (0; gi0) (if n > 2, then the constants ci satisfy the condition c1 + · · ·+
cn = 0).
P r o o f. As known, the complete integral w (q1, . . . , qn;β1, . . . , βn−1, h) of
a Hamiltonian system of the Liouville type allows the separation of variables and
is given by [7]
w =
∫ √
2ϕ1 (β1 + hf1 − ψ1)dq1
+
n−1∑
i=2
∫ √
2ϕi (−βi−1 + βi + hfi − ψi)dqi
+
∫ √
2ϕn (−βn−1 + hfn − ψn)dqn.
(4)
Note that the expressions under radicals must be positive. If the complete integral
of the Hamilton–Jacobi equation H
(
q1, . . . , qn,
∂w
∂q1
, . . . ,
∂w
∂qn
)
= h is found, then
the canonical action variables can be defined by the formulas
Ii =
1
2π
∮
γi
pidqi =
1
2π
∮
γi
∂w(q; β)
∂qi
dqi, (5)
where the integration is realized by the basic cycles of the Liouville–Arnold
tori [1]. In our case, using (4), (5), we can find action variables
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 299
V.T. Lisitsa
I1 =
1
2π
∮
γ1
√
2ϕ1 (β1 + hf1 − ψ1)dq1,
Ii =
1
2π
∮
γi
√
2ϕi (−βi−1 + βi + hfi − ψi)dqi,
In =
1
2π
∮
γn
√
2ϕn (−βn−1 + hfn − ψn)dqn, i = 2, . . . , n− 1.
(6)
If we proceed to canonical ”action–angle” variables, then the frequencies of con-
ditionally periodic motion are given by ωi =
∂h
∂Ii
. The equation
∂Ii
∂Ij
= δi
j , where
δi
j is the Kronecker symbol, holds true by virtue of independence of canonical
variables Ii, I = 1, . . . , n. In our case, we can put together the equation system
∂I1
∂I1
= 1 =
∂I1
∂h
· ∂h
∂I1
+
∂I1
∂β1
· ∂β1
∂I1
∂I2
∂I1
= 0 =
∂I2
∂h
· ∂h
∂I1
+
∂I2
∂β1
· ∂β1
∂I1
+
∂I2
∂β2
· ∂β2
∂I1
∂I3
∂I1
= 0 =
∂I3
∂h
· ∂h
∂I1
+
∂I3
∂β2
· ∂β2
∂I1
+
∂I3
∂β3
· ∂β3
∂I1
· · ·
∂In−1
∂I1
= 0 =
∂In−1
∂h
· ∂h
∂I1
+
∂In−1
∂βn−2
· ∂βn−2
∂I1
+
∂In−1
∂βn−1
· ∂βn−1
∂I1
∂In
∂I1
= 0 =
∂In
∂h
· ∂h
∂I1
+
∂In
∂βn−1
∂βn−1
∂I1
.
(7)
System (7) may be considered as a linear equation system with n unknown
∂h
∂I1
,
∂βi
∂I1
, i = 1, . . . , n − 1. The main determinant of this system is the Jacobian
∆ =
∂ (I1, . . . , In)
∂ (h, β1, . . . , βn−1)
6= 0. By the Cramer rule, find
∂h
∂I1
=
∂I2
∂β1
· ∂I3
∂β2
· . . . · ∂In
∂βn−1
∆
. (8)
Similarly, can be found
∂h
∂Ii
=
(−1)i+1 ∂I1
∂β1
· ∂I2
∂β2
. . .
∂Ii−1
∂βi−1
· ∂Ii+1
∂βi
. . .
∂In
∂βn−1
∆
. (9)
300 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems
Consider the case of a completely singular system. This implies that all the
frequencies are pairwise rationally dependent, i.e.,
ω1 = r1ω2, ω2 = r2ω3, . . . , ωn−1 = rn−1ωn, (10)
where ri are rational numbers.
Since ωi =
∂h
∂Ii
, then using (8), (9), and (10) find that
∂I1
∂β1
= −r1
∂I2
∂β1
,
∂I2
∂β2
= −r2
∂I3
∂β2
, . . . ,
∂In−1
∂βn−1
= −rn−1
∂In
∂βn−1
. (11)
Equations (6) and (11) yield the conditions for complete singularity of the Hamil-
tonian system of the Liouville type
∮
γ1
√
ϕ1
β1 + hf1 − ψ1
dq1 = r1
∮
γ1
√
ϕ2
β2 − β1 + hf2 − ψ2
dq2
· · ·∮
γi
√
ϕi
βi − βi−1 + hfi − ψi
dqi
= ri
∮
γi+1
√
ϕi+1
βi+1 − βi + h · fi+1 − ψi+1
dqi+1
· · ·∮
γn−1
√
ϕn−1
βn−1 − βn−2 + hfn−1 − ψn−1
dqn−1
= rn−1
∮
γn
√
ϕn
−βn−1 + hfn − ψn
dqn.
(12)
Consider a movement of the given Hamiltonian system on a surface of the fixed en-
ergy level h. If we introduce the designations c1 = −β1, c2 = −β2+β1, . . . , ci+1 =
−βi+1 + βi, cn = βn−1, then we get
∮
γi
√
ϕi
gi − ci
dqi = ri
∮
γi+1
√
ϕi+1
gi+1 − ci+1
dqi+1, i = 1, . . . , n− 1. (13)
It should be noted that the constants ci satisfy the condition c1 + · · · + cn = 0
and cannot have the same sign. Let gi0 = max gi when qi ∈ [0, ai], gi0 = gi (qi0),
where qi0 ∈ (0, ai)). By the supposition of the theorem, the functions gi are
strictly monotonous on the intervals (0; qi0) and (qi0; ai). In this case, for every
value of ci ∈ (0; gi0) there exist exactly two values of the variable qi from the
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3 301
V.T. Lisitsa
interval (0, ai), qi1 (ci), qi2 (ci) ∈ (0; ai) such that gi (qi1) = gi (qi2) = ci. Now,
transform
∮
γi
√
ϕi
gi − ci
dqi. Since the integrand must be nonnegative, gi − ci > 0,
then qi ∈ (qi1 (ci) ; qi2 (ci)). While going through the basic closed cycle of γi, qi
varies from qi1 to qi2, and then from qi2 to qi1. Then
∮
γi
√
ϕi
gi − ci
dqi = 2
qi2(ci)∫
qi1(ci)
√
ϕi
gi − ci
dqi, ci ∈ (0, gi0) . (14)
Represent the integral from the right-hand side of (14) in the form
qi2(ci)∫
qi1(ci)
√
ϕi
gi − ci
dqi =
qi0∫
qi1
√
ϕi
gi − ci
dqi +
qi2∫
qi0
√
ϕi
gi − ci
dqi. (15)
Since the function gi is monotonous on the intervals (qi1; qi0) and (qi0; qi2), then
we can change a variable into the intervals mentioned above. Let θi1 be the
inverse function to gi on the interval (qi1, qi0). Then qi = θi1 (gi), dqi = θ′i1dgi on
(qi1; qi0). In this case, the first term in the right-hand side of (15) is given by
qi0∫
qi1(ci)
√
ϕi
gi − ci
dqi =
gi0∫
ci
√
ϕ̃i1
gi − ci
θ′i1dgi where ϕ̃i1 = ϕi (θi1 (gi)) .
If θi2 is the inverse function to gi on (gi0; gi2), then in a similar way we can find
qi2(ci)∫
qi0
√
ϕi
gi − ci
dqi =
ci∫
gi0
√
ϕ̃i2
gi − ci
θ′2idgi = −
gi0∫
ci
√
ϕ̃i2
gi − ci
θ′i2dgi.
Denoting Fi (gi) =
√
ϕ̃i1θ
′
i1 −
√
ϕ̃i2θ
′
i2 gives
qi2(ci)∫
qi1(ci)
√
ϕi
gi − ci
dqi =
gi0∫
ci
Fi (gi)√
gi − ci
dgi. (16)
If ci < 0, then we can repeat all previous arguments, but in this case
gi0 = min
qi∈[0;ai]
gi = gi (qi0)), Fi(x) < 0, and it is necessary to substitute the expres-
sion under radical by ci − gi. Taking into account equality (16), the conditions
for the resonance can be written as
gi0∫
ci
Fi(x)√
x− ci
dx = ri
gi+10∫
ci+1
Fi+1(x)√
x− ci+1
dx, i = 1, . . . , n− 1, (17)
302 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 3
On the Conditions of Total Resonance of Liouville Type Hamiltonian Systems
where ci ∈ (0; gi0). When i = 1, the equation of system (18) is
g10∫
c1
F1(x)√
x− c1
dx = r1
g20∫
c2
F2(x)√
x− c2
dx. (18)
If the left-hand side of equation (18) put λ (h) to some function, then for the
function F1(x) one obtains the Abel equation
λ (h) =
gi0∫
c1
F1(x)√
x− c1
dx. (19)
From equations (18) and (19), it follows that
g20∫
c2
F2(x)√
x− c2
= r1λ (h) .
Analogously, it follows that
gi0∫
ci
Fi(x)√
x− ci
dx = riλ (h) , (20)
where ri are rational numbers. The solutions of equations (18)–(20) can be found
explicitly in [12]. Thus, system (3) of the integral equations is solvable. From
equations (16), (20) there follows the statement of the theorem.
The theorem is proved.
Note that in terms of [13] the statement of the theorem means that all numbers
of rotation of the given Hamiltonian system are rational.
References
[1] V.I. Arnold, Mathematical Methods of Classical Mechanics. Nauka, Moscow,
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[2] A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras.
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[3] A.T. Fomenko, Symplectic Geometry. Methods and Applications. MGU, Moscow,
1988. (Russian)
[4] N.N. Nekhoroshev, Action-Angle Variables and Their Generalizations. — Tr.
Moskow Math. Soc. 26 (1972), 181–198. (Russian)
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[5] B.S. Kruglikov, Existence of a Pair of Additional Bott Integrals for a Resonance
Hamiltonian System with Two Degrees of Freedom. — Tr. Mat. Inst. Steklov. 205
(1994), 109–112. (Russian)
[6] G. Goldsteyn, Classical Mechanics. Nauka, Moscow, 1975. (Russian)
[7] A.I. Lurye, Analitical Mechanics. GIFML, Moscow, 1961. (Russian)
[8] A. Besse, Manifolds with Closed Geodesics. Mir, Moscow, 1981. (Russian)
[9] J. Darboux, Leçons sur la Théorie Générale des Surfaces. Chelsea. 3. (3-éme éd.),
1972.
[10] V.N. Kolokoltsov, Geodesic Flows on Two-Dimensional Manifolds with an Addi-
tional First Integral that is Polinomial in the Velocities. — Izv. AN SSSR, Ser.
Math. 46 (1982), No. 5, 994–1010. (Russian)
[11] V.N. Kolokoltsov, New Examples of Manifolds with Closed Geodesics. — Vestnik
Moscow Univ. Ser. I: Math., Mech. No. 4 (1984), 80–82.
[12] S.G. Michlin, Lectures on Linear Integral Equations. GIFML, Moscow, 1959.
(Russian)
[13] A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian Systems. 1. Udmurt.
Univ., Izhevsk, 1999. (Russian)
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