On the integrability of Hamiltonian 1:2:2 resonance
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ORIGINAL PAPER
On the integrability of Hamiltonian 1:2:2 resonance Ognyan Christov
Received: 10 July 2020 / Accepted: 16 October 2020 © Springer Nature B.V. 2020
Abstract We study the integrability of the Hamiltonian normal form of 1:2:2 resonance. It is known that this normal form truncated to order three is integrable. The truncated to order four normal form contains many parameters. For a generic choice of parameters in the normal form up to order four, we carry on non-integrability analysis, based on the Morales– Ramis theory using only first variational equations along certain particular solutions. The non-integrability obtained by algebraic proofs produces dynamics illustrated by some numerical experiments.We also isolate a non-trivial case of integrability. Keywords Hamiltonian 1:2:2 resonance · Liouville integrability · Differential Galois groups · Morales– Ramis theory
H = H2 + H 3 + · · · + H m .
Mathematics Subject Classification 70H07 · 70H08 · 70K45
1 Introduction For an analytic Hamiltonian H (q, p) with an equilibrium at the origin, we have the following expansion H = H2 + H3 + H4 + · · · , O. Christov (B) Faculty of Mathematics and Informatics, Sofia University, 5 J. Bouchier blvd., 1164 Sofia, Bulgaria e-mail: [email protected]
where H2 = ω j (q 2j + p 2j ), ω j > 0, that is, H2 is a positive-definite form and H j are homogeneous of degree j. It is said that the frequency vector ω = (ω1 , . . . , ωn ) satisfies a resonant relation if there exists a vector k = , kn ), k j ∈ Z, such that (ω, k) = k j ω j = 0, (k1 . . . | k| = | k j | being the order of the resonance. There exists a procedure called normalization, which simplifies the Hamiltonian function in a neighborhood of the equilibrium and is achieved by means of canonical near-identity transformations [1,22,27]. When resonances appear, this simplified Hamiltonian is called Birkhoff–Gustavson normal form. To study the behavior of a given Hamiltonian system near the equilibrium, one usually considers the normal form truncated to some order
(1)
(2)
Note that by construction {H j , H2 } = 0 ( {, } being the Poisson bracket). This means that the truncated resonant normal form has at least two integrals—H and H2 . The first integrals for the resonant normal form H¯ are approximate integrals for the original system, see Verhulst [27] for the precise statements. If the truncated normal form happens to be integrable, then the original Hamiltonian system is called near-integrable. A recent review of some known results on integrability of the Hamiltonian normal forms can be found in [28]. In this paper, we study the integrability of the semisimple Hamiltonian 1:2:2 resonance. The classical
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O. Christov
water molecule model and concrete models of coupled rigid bodies serve as examples which are described by the Hamiltonian systems in 1:2:2 resonance, see Haller [7]. When studying normal forms, it is natural to introduce the complex coordinates
2 2 1 2 ( p1 + q12 ) + ( p22 + q22 ) + ( p32 + q32 ) 2 2 2 + β1 q12 q2 + β2 q12 q3
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