# 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 $H_2 + H_3$