Self-intersection of complex
asymptotic manifolds of hyperbolic equilibrium, position and non-existence of
first integrals in the systems of the type of Hamiltonian with two degrees of
freedom
S. L. Ziglin
Kotelnikov
Institute of Radioengineering and Electronics of Russian Academy of Sciences,
Mokhovaya 11-7, Moscow 125009, Russia
The paper is
received on May 13, 2019
Abstract. We
investigate the sufficient conditions of infinite to one transversal on the
level surface of known first integrals self-intersection of complex asymptotic
manifolds of hyperbolic equilibrium position of the system of ordinary
differential equations of the type of Hamiltonian with two degrees of freedom
(the number of known first integrals is three less than the dimension of the
system). The obtained results are applicable to the problem on motion of
dynamically symmetric rigid heavy body around a fixed point, Suslov’s problem
on the motion of a rigid body around a fixed point with non-holonomic
constraint, to the Henon-Heiles system, to the system, describing the stationary
flow of ideal non-compressive liquid, called ABC-dynamo.
Keywords:
self-intersection of complex asymptotic manifolds of hyperbolic equilibrium
position, nonexistence of first integrals.
References
1.
Arnold
V.I. Obyknovennye differentsial’nye uravneniya [Ordinary differential
equations]. Moscow, Nauka Publ., 1974 (In Russian)
2.
Arnold
V.I., Ilyashenko Yu.S. Obyknovennye differentsial’nye uravneniya [Ordinary
differential equations]. In the book: Itogi nauki i tekhniki. Sovremennye
provkemy matematiki . Tom 1 [Results of science and technology. Modern
problems of mathematics. Volume 1]. Moscow, VINITI, 1985, pp.7-49 (In Russian)
3.
Kobayashi
S., Nomizu K.
Foundations
of differential geometry. John
Wiley & Sons, Inc. 1996
4.
Ziglin S.L.
Branching
of solutions and nonexistence of first integrals in Hamiltonian mechanics. I.
Functional
Analysis and its applications, 1982, Vol. 16, No. 3, pp.181-189
5.
Postnikov
M.M., Rudak Yu.B. Morse Lemma. In the book: Matamaticheskaya entsiklopediya,
Tom 3 [Mathematical encyclopedia, Vol. 3] Moscow, Sovetskaya entsiklopediya
Publ., 1982, pp. 824-825. (In Russian)
6.
Shabat
B.V. Vvedenie v kompleksnyi analiz [Introduction into complex analysis].
Moscow, Nauka Publ., 1969 (In Russian)
7.
Arnold
V.I. Matematicheskie metody klassicheskoy mekhaniki [Mathematical
methods of classical mechanics]. Moscow, Nauka Publ., 1974 (In Russian)
8.
Henon
M. Sur la topologie de lignes de courant dans un cas particulier. C.R. Acad.
Sci. Paris, 1966, Vol. 262, pp. 312-314
9.
Zaslavskiy
G.M., Sagdeev P.Z., Usikov U.A., Chernikov A.A. Slabyi khaos i
kvaziregulyarnye struktury [Weak chaos and quasi-regular structures].
Moscow, Nauka Publ., 1991 (In Russian)
10. Ziglin S.L.
The
ABC-flow is not integrable for
A=B.
Functional
Analysis and its applications, 1996, Vol. 30, No. 2, pp.137-138
11.
Ziglin S.L., Cherepenin V.A. Nonintegrability of an
ABC-flow with A=B=C.
Doklady
Mathematics, 1999, Vol. 60, No. 3,
ñ. 441-442.
12. Ziglin
S.L.
An Analytic Proof of the
Nonintegrability of the ABC-flow for A=B=C.
Functional Analysis and its applications,
2003,
Vol. 37, No. 3, pp. 225-227
13.
Henon M., Heiles C. The applicability of the third integral of motion some
numerical experiments. Astron J., 1964, Vol. 69, pp. 73-79.
14. Ziglin
S.L.
Branching of solutions and the
nonexistence of first integrals in Hamiltonian mechanics. II.
Functional Analysis and its applications 1983, Vol. 17, No. 1,
pp.
6–17.
15. Suslov G.K. Teoreticheskaya mekhanika
[Theoretical mechanics]. Moscow,-Leningrad, Gostekhizdat Punl., 1946 (In
Russian)
16. Kozlov V.V. On the theory of integration of the
equations of nonholonomic mechanics. Uspekhi mekhaniki – Advances of
mechanics, 1985, Vol. 8, No. 3, pp. 85-107 (In Russian).
17. Ziglin S.L.
On the absence of a complementary first integral in a special case
of Suslov’s problem.
Russian
Mathematical Surveys, 1997,
Vol. 52, No. 2, pp.
434-436.
18. Ziglin S.L. On the absence of an additional first integral in one
problem of the dynamics of a rigid body. Doklady Academii Nauk SSSR –
Doklady Mathematics, 1987, Vol. 293, No. 4, pp. 804-807. (In Russian)
19. Ziglin S.L. On the absence of a real-analytic first integral for
ABC flow when A=B. Chaos, 1998, Vol. 8, No. 1, pp. 272-273.
20.
Ziglin S.L. The
absence
of an additional real-analytic first integral in some problems of dynamics.
Functional
Analysis and its applications, 1997, Vol.
31, No. 1,
pp.
3-11.
For citation:
S. L. Ziglin.
Self-intersection of complex asymptotic manifolds of hyperbolic
equilibrium, position and non-existence of first integrals in the systems of
the type of Hamiltonian with two degrees of freedom. Zhurnal
Radioelektroniki - Journal of Radio Electronics.
2019. No. 5. Available at http://jre.cplire.ru/jre/may19/5/text.pdf