Analytic integrability of Hamiltonian systems with exceptional potentials

Jaume Llibre; Claudia Valls

doi:https://doi.org/10.1016/j.physleta.2015.07.034

Analytic integrability of Hamiltonian systems with exceptional potentials

Jaume Llibre, Claudia Valls

Department of Mathematics

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Abstract

© 2015 Elsevier B.V. All rights reserved. We study the existence of analytic first integrals of the complex Hamiltonian systems of the formH=12 i=12pi2+Vl(q1,q2) with the homogeneous polynomial potentialVl(q1,q2)=α(q2-iq1)l(q2+iq1)k-l,l=0,...,k,αC-{0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials V0, V1, Vk-<inf>1</inf>, Vk and Vk/<inf>2</inf> when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials. Here we prove that the exceptional potentials with k even different from V0, V1, Vk-<inf>1</inf>, Vk and Vk/<inf>2</inf>, have no independent analytic first integral different from the Hamiltonian one.

Original language	English
Pages (from-to)	2295-2299
Journal	Physics Letters, Section A: General, Atomic and Solid State Physics
Volume	379
Issue number	38
DOIs	https://doi.org/10.1016/j.physleta.2015.07.034
Publication status	Published - 22 Aug 2015

Keywords

Exceptional potentials
Hamiltonian system with 2 degrees of freedom
Homogeneous potentials of degree k
Integrability

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title = "Analytic integrability of Hamiltonian systems with exceptional potentials",

abstract = "{\textcopyright} 2015 Elsevier B.V. All rights reserved. We study the existence of analytic first integrals of the complex Hamiltonian systems of the formH=12 i=12pi2+Vl(q1,q2) with the homogeneous polynomial potentialVl(q1,q2)=α(q2-iq1)l(q2+iq1)k-l,l=0,...,k,αC-{0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials V0, V1, Vk-1, Vk and Vk/2 when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials. Here we prove that the exceptional potentials with k even different from V0, V1, Vk-1, Vk and Vk/2, have no independent analytic first integral different from the Hamiltonian one.",

keywords = "Exceptional potentials, Hamiltonian system with 2 degrees of freedom, Homogeneous potentials of degree k, Integrability",

author = "Jaume Llibre and Claudia Valls",

year = "2015",

month = aug,

day = "22",

doi = "https://doi.org/10.1016/j.physleta.2015.07.034",

language = "English",

volume = "379",

pages = "2295--2299",

number = "38",

}

TY - JOUR

T1 - Analytic integrability of Hamiltonian systems with exceptional potentials

AU - Llibre, Jaume

AU - Valls, Claudia

PY - 2015/8/22

Y1 - 2015/8/22

N2 - © 2015 Elsevier B.V. All rights reserved. We study the existence of analytic first integrals of the complex Hamiltonian systems of the formH=12 i=12pi2+Vl(q1,q2) with the homogeneous polynomial potentialVl(q1,q2)=α(q2-iq1)l(q2+iq1)k-l,l=0,...,k,αC-{0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials V0, V1, Vk-1, Vk and Vk/2 when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials. Here we prove that the exceptional potentials with k even different from V0, V1, Vk-1, Vk and Vk/2, have no independent analytic first integral different from the Hamiltonian one.

AB - © 2015 Elsevier B.V. All rights reserved. We study the existence of analytic first integrals of the complex Hamiltonian systems of the formH=12 i=12pi2+Vl(q1,q2) with the homogeneous polynomial potentialVl(q1,q2)=α(q2-iq1)l(q2+iq1)k-l,l=0,...,k,αC-{0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials V0, V1, Vk-1, Vk and Vk/2 when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials. Here we prove that the exceptional potentials with k even different from V0, V1, Vk-1, Vk and Vk/2, have no independent analytic first integral different from the Hamiltonian one.

KW - Exceptional potentials

KW - Hamiltonian system with 2 degrees of freedom

KW - Homogeneous potentials of degree k

KW - Integrability

U2 - https://doi.org/10.1016/j.physleta.2015.07.034

DO - https://doi.org/10.1016/j.physleta.2015.07.034

M3 - Article

VL - 379

SP - 2295

EP - 2299

IS - 38

ER -

Analytic integrability of Hamiltonian systems with exceptional potentials

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