Polynomial Integrals of Geodesic Flows on a Two-Dimensional Torus

Козлов, Валерий Васильевич; Денисова, Наталия Викторовна

doi:10.1070/sm1995v083n02abeh003601

Cited by 22 publications

(27 citation statements)

References 9 publications

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“…Sums of the form (23) play the role of Fourier series, and one can operate with them using the same rules as for ordinary Fourier series. By (23), applying the method of [2], we obtain the same geometric relations for the integer vectors w ~ 0, with the ordinary Euclideml metric replaced by the metric (22). The analysis of these relations becomes somewhat more complicated (e.g., in contrast with [2], to analyze the odd-degree integral, we have to introduce the adjoint vertices for the spectrum of M).…”

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 93%

“…By (23), applying the method of [2], we obtain the same geometric relations for the integer vectors w ~ 0, with the ordinary Euclideml metric replaced by the metric (22). The analysis of these relations becomes somewhat more complicated (e.g., in contrast with [2], to analyze the odd-degree integral, we have to introduce the adjoint vertices for the spectrum of M). However, finally we again find that the spectrum of M lies on one or two lines orthogonal in the metric dual to (2).…”

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 93%

“…The problem of polynomial integrals of geodesic flows on the two-dimensional torus for the metric (2) with a = c and b = 0 was considered in [2]. It was established that if M is a trigonometric polynomial (the spectrum of M is finite) and the geodesic flow admits an additional irreducible integral polynomial in the momenta, then the degree of this integral does not exceed 2.…”

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 98%

“…Therefore, an irreducible quadratic integral can exist only in exceptional cases. Theorem 1 can be proved by using the method of [2]. To this end, we reduce the quadratic form a~ + 2b~ + c~…”

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 99%

“…In w we extend the result of [2] to polynomial integrals of degree _> 2 for geodesic flows on the two-dimensional toms.…”

mentioning

confidence: 96%

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Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

Денисова

1998

Math Notes

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ABSTRACT. We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an ~irreducible ~ integral of the geodesic flow on the torus does not exceed 2.KgY WORDS: dynamical systems on the torus, polynomial first integrals, complete integrability. w IntroductionWe consider dynamical systems with two degrees of freedom whose configuration space is the twodimensional toms T 2 and which admit first integrals polynomial in the velocity. This is equivalent to the complete integrability of the system. Polynomial integrals can be represented as polynomials in the velocity with smooth single-valued coefficients on the configuration space. A least-order nontrivial polynomial F independent of energy is called irredncible. Polynomials in the velocity which are integrals only for some given values of the total energy are said to be conditional polynomial integrals.In [1] Birldaoff studied the local existence of conditional integrals polynomial of order <_ 2 in the velocity. It turned out that the presence of a conditional linear integral is related to "hidden" cyclic coordinates and the presence of a conditional quadratic integral permits one to separate the canonical variables. In w and w of this paper, we present the global versions of these statements for the case in which the configuration space of the system is the two-dimensional toms.In w we extend the result of [2] to polynomial integrals of degree _> 2 for geodesic flows on the two-dimensional toms.The problem for polynomial integrals of degree _< 2 was considered in [3,4]. It was shown in [5] that the natural mechanical system on the two-dimensional torus with kinetic energy 1 -2 has no irreducible integrals of degrees 3 and 4; in this case there necessarily exist integrals of degrees 1 and 2, respectively.In [6] the problem of the existence of a complete set of independent polynomial integrals was considered for a system with configuration space T n = {Zl,..., zn rood 27r}, kinetic energy

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Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 93%

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 93%

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 98%

Section: W Polynomial Integrals Of Degree Higher Thanmentioning

confidence: 99%

“…In w we extend the result of [2] to polynomial integrals of degree _> 2 for geodesic flows on the two-dimensional toms.…”

mentioning

confidence: 96%

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Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

Денисова

1998

Math Notes

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Integrable Magnetic Geodesic Flows on 2-Torus: New Examples via Quasi-Linear System of PDEs

V.Agapov¹,

Michael²,

Bialy³

et al. 2017

Commun. Math. Phys.

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The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on one energy level is considered. This problem can be reduced to a remarkable Semihamiltonian system of quasi-linear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville metric with zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of a quadratic in momenta integral. Thus our construction gives a new example of smooth periodic solution to the Semi-hamiltonian (Rich) quasilinear system of PDEs.

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Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

Szumiński

Maciejewski

2021

Nonlinear Dyn

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In the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

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Polynomial Integrals of Geodesic Flows on a Two-Dimensional Torus

Cited by 22 publications

References 9 publications

Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

Integrable Magnetic Geodesic Flows on 2-Torus: New Examples via Quasi-Linear System of PDEs

Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

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