1:2:2 resonance revisited

1:2:2 resonance was studied by Van der Aa and Verhulst who showed its asymptotic integrability, i.e. if H = H_2 + H_3 + …  is in Birkhoff normal form such that H_2 is in 1:2:2 resonance, then H_2 + H_3 is integrable. They also generalized the result to the n degrees of freedom 1:2:2:…:2 resonance case.

The original proof of Van der Aaa of asymptotic integrability, using real symplectic coordinates, is quite lengthy. Here I want to write down a simpler proof.

Instead of using real symplectic coordinates and write H_2 = (p_1^2 + p_2^2) + 2(p_2^2 + q_2^2) + 2(p_3^2 + q_3^2), we will use complex symplectic coordinates and write

H_2 = p_1 q_1 + 2 p_2 q_2 + 2 p_3 q_3

(The problem of finding formal or polynomial first integrals is a purely algebraic problem, so if one can do it over C one can also do it over R).

Then the only monomial functions of degree 3 which Poisson-commute with H_2 are p_1^2q_2, p_1^2 q_3, q_1^2p_2, q_1^2p_3. Thus one can write

H_3 = p_1^2 (a_2 q_2 + a_3 q_3) + q_1^2 (b_2 p_2 + b_3 p_3)

where a_2,a_3,b_2,b_3 are constants.

The quadratic  Hamiltonian H_2 admits a natural gl(2,C) symmetry: On the 4-dimensional supspace spanned by p_2,q_2,p_3,q_3, the Hamiltonian vector field of H_2 can be viewed as the cotangent lifting of the radial vector field on the plane (q_2,q_3), and thus it commutes with the cotangent lifting of any linear vectore field on the plane (q_2,q_3). With this symmetry, one can further normalize H_3. This symmetry also implies that all the quadratic functions p_2q_2, p_2q_3,p_3q_2, p_3q_3 are first integrals of H_2

As a consequence, the quadratic function

F = (a_3 p_2 - a_2 p_3)(b_3 q_2 - b_2 q_3) commutes with both H_2 and H_3, and so this function is an additional first integral (of degree 2) for the system.

(If, say a_3 = a_2 = 0 then in the above formula for F can take any other non-zero coefficients a'_2, a'_3).

The 1:2:…:2 case is similar. One can write H_2 as

H_2 = p_1q_1 + 2 \sum_{i=2}^n p_iq_i

Then H_3 has the form:

H_3 = p_1^2 (\sum_{i=2}^n a_i q_i) + q_1^2 (\sum_{i=2}^n b_i p_i)

where q_i, b_i are constants. This resonance H_2 admits a natural gl(n-1,C) symmetry, from which one can find (n-2) additional quadratic commuting first integrals for H_2 + H_3. In fact, one doesn’t need the whole gl(n-1) algebra, but only some special elements :

Denote by \phi_2 =(a_i), \psi_2 =(b_i). One can find independent (n-1)-dimensional vectors \phi_3, \hdots, \phi_n and independent vectors \psi_3, \hdots, \psi_n such that

\langle \phi_i, \psi_j \rangle = 0 for any i \neq j

Then define

F_j =\langle \phi_j, p \rangle \langle \psi_i, q \rangle

where p =(p_2,\hdots,p_n) and q = (q_2,\hdots,q_n)

Then these  functions F_j are additional quadratic commuting first integrals for H_2 + H_3

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