On exact solutions of a system of quasi-linear equations describing integrable geodesic flows on a surface

Abdikalikova, Gulshat; Mironov, Andrey E.

doi:10.33048/semi.2019.16.063

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2023

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Cited by 3 publications

(2 citation statements)

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“…We reduce this problem to a certain quasi-linear system of PDEs and apply the generalized hodograph method to construct its exact solutions. Recently the same approach has yielded new integrable examples in [16,29] in slightly different situations.…”

Section: Rational Integralsmentioning

confidence: 97%

New examples of non-polynomial integrals of two-dimensional geodesic flows ^*

Agapov,

Shubin

2023

J. Phys. A: Math. Theor.

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Section: Rational Integralsmentioning

confidence: 97%

New examples of non-polynomial integrals of two-dimensional geodesic flows ^*

Agapov,

Shubin

2023

J. Phys. A: Math. Theor.

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“…Notice that the implementation of this method is usually associated with significant difficulties. In application to the problem of geodesic flows, as far as we know, the only example when one managed to construct explicit solutions via this method (they relate to an integral of the fourth degree) is presented in [14].…”

Section: Introductionmentioning

confidence: 99%

Integrable magnetic geodesic flows on 2-surfaces ^*

2023

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We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta integral, and also the case of a rational in momenta integral with a linear numerator and denominator. In both cases certain semi-Hamiltonian systems of partial differential equations (PDEs) appear. In this paper we construct exact solutions (generally speaking, local ones) to these systems: in the first case via the generalized hodograph method, in the second case via the Legendre transformation and the method of separation of variables.

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Integrable magnetic geodesic flows on 2-surfaces

Agapov¹,

Alexey²,

Shubin³

2022

Preprint

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We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta integral, and also the case of a rational in momenta integral with a linear numerator and denominator. In both cases certain semi-Hamiltonian systems of PDEs appear. In this paper we construct exact solutions (generally speaking, local ones) to these systems: in the first case via the generalized hodograph method, in the second case via the Legendre transformation and the method of separation of variables.

show abstract

On exact solutions of a system of quasi-linear equations describing integrable geodesic flows on a surface

Cited by 3 publications

References 4 publications

New examples of non-polynomial integrals of two-dimensional geodesic flows ^*

New examples of non-polynomial integrals of two-dimensional geodesic flows ^*

Integrable magnetic geodesic flows on 2-surfaces ^*

Integrable magnetic geodesic flows on 2-surfaces

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On exact solutions of a system of quasi-linear equations describing integrable geodesic flows on a surface

Cited by 3 publications

References 4 publications

New examples of non-polynomial integrals of two-dimensional geodesic flows *

New examples of non-polynomial integrals of two-dimensional geodesic flows *

Integrable magnetic geodesic flows on 2-surfaces *

Integrable magnetic geodesic flows on 2-surfaces

Contact Info

Product

Resources

About

New examples of non-polynomial integrals of two-dimensional geodesic flows ^*

New examples of non-polynomial integrals of two-dimensional geodesic flows ^*

Integrable magnetic geodesic flows on 2-surfaces ^*