Nonintegrability theorem on perturbed integrable Hamiltonian dynamical systems with two independent resonances

Introduction

A very important qualitative feature of Hamiltonian systems is integrability [8]. According to the Liouville–Arnold theorem ([1], p. 270–272), if an autonomous Hamiltonian system with $n$ degrees of freedom possesses $n$ functionally independent first integrals in involution, then every regular, compact, and connected common level set of these integrals is diffeomorphic to an invariant $n$-dimensional torus $\mathbb{T}^{n}$. Moreover, in a neighbourhood of such a torus, action–angle coordinates can be introduced.

Let us begin with a perturbed, autonomous, two-degree-of-freedom system given by a Hamiltonian of the general form. Given that the perturbed Hamiltonian is an analytic function in terms of the parameter $\varepsilon$, on an open interval around $\varepsilon=0$, it can consequently be expressed as a power series in $\varepsilon$ ([2], p. 40),

The perturbed Hamiltonian is written in curvilinear symplectic coordinates called action-angle variables in an open region of the phase space:

More specifically, $H_{1}(J_{1},J_{2},w_{1},w_{2})$ is the perturbation, which is periodic in $w_{1},w_{2}$ and can be expanded as a double Fourier series with period $2\pi$. The function $H_{0}(J_{1},J_{2})$ is the Hamiltonian of the unperturbed problem. It is Liouville–Arnold integrable, with finite motions; that is, in the corresponding regular open region of phase space, the unperturbed system is locally foliated by invariant $n$-dimensional tori. These tori are invariant with respect to the phase flow of the unperturbed system. Furthermore, $H_{0}$ satisfies the nondegeneracy condition ([2], p. 40):

The motion of a phase point is defined by the solution of the system

That is,

Each torus is determined by the value $J_i = J_{i0}$, while $w_i \bmod2\pi$ are the angular coordinates .

In an abstract $n$-dimensional system, a torus $\mathbb{T}^n$ with frequencies

is called non-resonant if $\mathbf{k}\cdot\boldsymbol{\omega} \ne0$ $\forall\mathbf{k} \in\mathbb{Z}^n \setminus\{0\}$,

and resonant if there exists a nonzero integer vector

such that

Obviously, in a two-dimensional system, a torus $\mathbb{T}^2$ is said to be resonant if

For $k_1 \ne0$ and $\omega_2 \ne0$, this gives

Generally, the nondegenerate condition corresponds to the case in which the frequencies are functionally independent. In nondegenerate systems the non-resonant tori form an everywhere dense set of full measure. The resonant tori form a set of measure zero, which is also everywhere dense ([3], p. 273).

A phase trajectory on a non-resonant torus covers the torus densely. One can simply say that when the frequency ratio

the invariant tori contain periodic orbits that close after a finite time, whereas when the frequency ratio is irrational,

the invariant tori have quasi-periodic orbits that cover the tori densely.

Poincaré's theorem on nonintegrability

Obviously,

is an integral of motion. Let us suppose that $H$ has a second integral of motion

in an open neighborhood of $J_1,J_2$, for $w_1,w_2$ from 0 to $2\pi$, and for $\varepsilon$ in a neighborhood of $\varepsilon=0$. Then $\Phi$ can be expanded in a convergent series around $\varepsilon=0$:

If we take a non-resonant torus, note that $\Phi_0$ is independent of the angles. A detailed explanation can be found in ([3] p. 353).

Poincaré proves that $\Phi_0$ is an integral of $H_0$, independent of it. Furthermore, up to order $\varepsilon$, since $\Phi$ is an integral of $H$, it holds that

That leads to

The first relation in (5) states that $\Phi_0$ is an integral of $H_0$. We shall examine whether $\Phi_0$ is independent of $H_0$. The second relation in (5) yields the relation between the Fourier coefficients $B_{k_1,k_2}$ and $A_{k_1,k_2}$:

Here,

and

are the Fourier series of $H_1$ and $\Phi_1$ in $w_1$ and $w_2$.

Relation (6) is an identity on the phase space; that is, it holds for every $J_i,w_i$ in an open region $d\times\mathbb{T}^n$, at least ([2] p. 47).

According to equation (6), its left-hand side becomes zero on the dense set of resonant tori of the unperturbed system $H_0$, since it is nondegenerate:

If an infinite set of Fourier coefficients $A_{k_1,k_2} \ne0$ of $H_1$ are different from zero on these dense resonant tori, then according to equation (6),

If we combine (7) and (8), we get

Therefore, $\Phi_0$ is functionally dependent on $H_0$, which is a contradiction. We point out that (9) holds for the entire domain of the phase space. Since $D$ is continuous, if it is zero on a dense set, then it is zero everywhere. Indeed, take any $J^*$ in the domain. Since the resonant set of tori is dense, there exists a sequence

such that

Therefore,

so (9) holds throughout the entire domain. Thus, Poincaré’s theorem [7] states:

If an infinite set of Fourier coefficients $A_{k_1,k_2} \ne0$ of $H_1$ are nonzero on a dense set of resonant tori of the unperturbed, integrable, nondegenerate system, then $H$ does not possess a second integral of motion analytic in the real variables in an open neighborhood of $J_1,J_2$, for $w_1,w_2$ from 0 to $2\pi$, and for $\varepsilon$ in a neighborhood of $\varepsilon=0$.

Meletlidou–Ichtiaroglou nonintegrability criterion and equivalence to Poincaré's theorem on nonintegrability

A more detailed presentation is given in [4]. From the second relation in (5), we obtain

and therefore

Consider a member of the family of resonant tori for which

By integrating (10) over one period, we obtain the following. The right-hand side of this equation gives zero upon integration, since the corresponding term is the time derivative of $\Phi_1$ integrated over one period $T$ on the resonant torus.

The period $T$ is determined by

The left-hand side results in

where

We note that every orbit on a resonant torus is periodic and differs from the others only in terms of $w_{10}$, $w_{20}$. Thus, we obtain an entire family of periodic orbits on a specific resonant torus.

Along the unperturbed motion,

Therefore, $\varphi$ is constant on every periodic orbit. On a resonant torus of $H_0$ there are infinitely many periodic orbits parametrized by $\varphi$.

The mean value of $H_1$ is

It can either be the same for all periodic orbits of the torus, or it can vary as $\varphi$ changes. In the second case, there exists a trajectory for which

On a resonant torus, this leads to

This relation concerns the entire resonant torus. It means that $\Phi_0$ depends on $H_0$ on this torus. This occurs for a dense set of resonant tori if

on this dense set. But this is not possible according to Poincaré’s theorem, since $\Phi_0$ is independent of $H_0$. Therefore, $H$ is nonintegrable.

More specifically, the exact proposition of [4] is that if there exists a dense set of resonant tori of $H_0$ such that, on each of them, the average value of $H_1$ depends on the particular periodic orbit, then the perturbed Hamiltonian is nonintegrable for $\varepsilon\in(-\varepsilon_0,\varepsilon_0)\setminus\{0\}$ in an open neighborhood of $\varepsilon=0$.

It is proven that the Meletlidou–Ichtiaroglou criterion and Poincaré’s theorem are equivalent.

A new theorem on nonintegrability in the case of two independent resonances

We are going to establish a new theorem on the nonintegrability of a perturbed Hamiltonian system with two resonances. We begin with the Hamiltonian

Here $J_1,J_2,w_1,w_2$ are the action-angle variables, $H_0$ is the integrable, nondegenerate, unperturbed part in action-angle variables, and $H_1$ is the perturbation with two resonances, $k_1,k_2,l_1,l_2 \in\mathbb{Z}$. and arbitrary. The vectors

are independent.

We suppose, as in Poincaré’s theorem of nonintegrability, that there exists a second integral of motion independent of $H$, and we are going to use Poincaré’s lemma that $\Phi_0$ is independent of $H_0$. To study the nonintegrability generated by the interaction of these two resonances, we apply a sequence of Lie-series symplectic transformations ([6]).

The Lie operator is defined by the Poisson bracket:

Our Lie generators $W_{m_k,n_k}$ is a sequence of function in angle variables

where

with

Then the transformed Hamiltonian is

According to Cauchy’s theorem for Lie series, the series for $H'$ converges, since it does not have any small divisors[5].

At each step we apply $W_{m_k,n_k}$ to the initial Hamiltonian $H$. At each step we shall prove that eq. (16) is non-zero at the resonant torus with $\frac{\omega_1}{\omega_2}=\frac{k_2+n_k}{k_1+m_k}$. After applying this infinite sequence of Lie generators, we shall prove that (16) is non-zero at the resonant tori $\frac{\omega_1}{\omega_2}=\frac{k_2+n_k}{k_1+m_k}$ where $p_k,q_k,j_k,s_k$ take infinite values and $\frac{\omega_1}{\omega_2}$ at these resonant tori form a dense set in action space.

We also apply the Poisson bracket with $W_{m_k,n_k}$ to equation (5):

Using again Jacobi’s identity for

and

we obtain

According to the criterion by E. Meletlidou and S. Ichtiaroglou, we integrate (16) on the periodic orbits of the unperturbed invariant resonant tori of $H_0$, where the resonance condition holds:

And also

The results of the analysis using $(\alpha)$ also hold for $(\beta)$. Let us consider the resonance condition $(\alpha)$. The period $T$ is

We integrate over one period:

Thus,

Consider

with frequencies of the form

For the first term of eq. (16) which is

and selecting the term $\cos((k_1+m_1)\omega_1-(k_2+m_2)\omega_2)$ as a general term, we obtain

Provided that $\Phi_0$ is independent of $H_0$, the term

is different from zero

Also the other coefficient is nonzero if

as well as

The integral is nonzero since, on the above resonance $(\alpha)$,

where $\phi_k=(k_1+m_k)\omega_{10}-(k_2+n_k)\omega_{20}$ since the resonance does not vanish the initial conditions $\omega_{i0}$.

Looking at the last term of (17), it can be proven that it is zero, since it is the time derivative of $[W_{m_k,n_k},\Phi_1]$:

That is,

The other terms are

The second term is

and has the coefficient

It can be proven that this it is nonzero when $H_0$ does not have a linear integral in $J_1,J_2$, namely $\Phi_0$, that is continued in the perturbed system. Also, the integral (18) and the last term in (19) at the resonance is nonzero.

Also, the terms do not balance.

We shall prove in Appendix A. that the ratios

and

form a dense set in the relevant interval. Therefore, the resonant tori form a dense set in the phase space. After proving that the resonant tori are dense in the action space, the Hamiltonian given by (12) is nonintegrable

For $H$ to possess a second independent first integral analytic in $\varepsilon$, equation (17) must vanish. Hence, the vanishing of (17) is a necessary condition for the existence of such an integral. Under the assumptions stated below, equation (17) is non-zero on a dense set of resonant tori, which yields a contradiction.

Finally,according to the criterion, the Hamiltonian system given by (12) is nonintegrable, except in the cases described above.

We formulate the following theorem:

or, analogously,

generate a dense set of resonant tori in the considered action domain. Assume further that, on this dense resonant set, the second-order equation (16) does not vanish. In particular, either for the relevant resonant contribution,

with analogous non-vanishing condition for $A_2$, or $H_0$ does not possess a linear integral with respect to $J_1,J_2$ or that the remaining averaged Poisson-bracket contributions do not cancel each other we proved that for sufficiently small

the perturbed Hamiltonian $H$ admits no second first integral which is functionally independent of $H$ and analytic in $\varepsilon$ in the considered open domain that is continued in the perturbed system. Consequently $H$ is nonintegrable.

Remark: The theorem is inconclusive in cases:

1. Where all second-order resonant coefficient vanishes, as well as the coefficients,

2. The two Fourier vectors are not independent, i.e.

then it is known that the system is integrable.

Failure of any of these hypotheses does not imply integrability; it only means that the present criterion is inconclusive.

Conclusions

In this work we have investigated a two-degree-of-freedom nearly integrable Hamiltonian system whose perturbation contains two linearly independent Fourier modes. By combining a Lie-series symplectic transformation with the non-integrability approach of Poincaré and the criterion of Meletlidou–Ichtiaroglou, we obtain a second-order resonant obstruction to the existence of an additional analytic first integral.

The Lie transformation generates combination harmonics at order $\mathcal{O}(\varepsilon^2)$. When one of these harmonics becomes resonant with an unperturbed invariant torus, its average over the corresponding periodic orbit does not, in general, vanish and depends on the phase labelling the periodic orbit on that torus.

If this non-vanishing, orbit-dependent averaged contribution occurs on a dense set of resonant tori, and if the remaining second-order averaged terms do not cancel it, the necessary condition for the existence of a second first integral analytic in $\varepsilon$ is violated. Consequently, for sufficiently small $\varepsilon\ne0$, no second first integral functionally independent of $H$ and analytic in the stated class can exist in the considered domain.

Thus, the obstruction is not produced merely by the presence of two independent Fourier modes, but by their second-order interaction together with the density and non-vanishing conditions stated in the theorem.

The Lie-series formulation is particularly convenient for this analysis because it preserves the canonical structure and provides an explicit order-by-order description of the harmonics generated by the interaction of the perturbing terms.

Dense set of Resonant Tori

We prove that the resonance ratios

form a dense set in the relevant positive frequency-ratio interval.

Recall that

where

Let

By Bézout’s identity,

and similarly,

Therefore $m_{k}$ and $n_{k}$ may attain arbitrary integer multiples of $d_{1}$ and $d_{2}$, respectively.

Consider an arbitrary open interval

Since $m_{k}$ may take arbitrarily large values, we can choose $m_{k}$ such that

and

Now choose $n_{k}$ such that

which is a large enough multiple of $d_{2}$ satisfying

Therefore the set of integers

is nonempty. In the above set there is a minimum integer $n_{k}$ such that $S_{k}=\min n_{k},\ n_{k}\in B$ for which it holds that $S_k \in B$.

We will show that for $S_k$ it holds that

Let us suppose the opposite

then,

which is a contradiction, so

since $S_k$ is the minimum of the set $B$, and $D_k$ may attain arbitrary large values,

The interval $(\alpha, \beta)$ is arbitrary, the set

is dense in $\mathbb{R}^{+}$.

The same argument applies to the second family

and therefore it is also dense in $\mathbb{R}^{+}$.

Finally, since the unperturbed Hamiltonian $H_0$ is non-degenerate, the map

is locally invertible. Hence the the above rational frequency ratios imply that the corresponding resonant invariant tori form a dense set in the considered regular action domain.