# Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

Authors: Ognyan Christov
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

The low-dimensional periodic Klein-Gordon lattices are studied for integrability. We prove that the periodic lattice with two particles and certain nonlinear potential is nonintegrable. However, in the cases of up to six particles, we prove that their Birkhoff-Gustavson normal forms are integrable, which allows us to apply KAM theory in most cases.


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## Body

## 1. Introduction

In this article we deal with the periodic Klein-Gordon (KG) lattice (see, e.g., [1] and references therein) described by the Hamiltonian(1)H=∑j∈Z/nZpj22+C2qj+1-qj2+Vqj,pj=q˙j.The constant C>0 measures the interaction to nearest neighbor particles (with unit masses) and V(x) is a nonlinear potential.We study the integrability of (1). When C=0 the Hamiltonian is separable and, hence, integrable. There exist plenty of periodic or quasi-periodic solutions in the dynamics of (1). It is natural to investigate whether this behavior persists for C small enough (see, e.g., [2]). Here we assume that C is neither very small nor too large.We are interested in the behavior at low energy; that is why the following main assumptions are in order:(2)Vx=a2x2+b2x4,(3)a>0irrational.Remark 1.
Such types of potentials are frequently used in the literature [1]. As it can be seen below the choice of a simplifies considerably the calculations.We can also assume thatC=1 which can be achieved by rescaling of t. Then our Hamiltonian takes the form(4)H=∑j∈Z/nZpj22+12qj+1-qj2+a2qj2+b2qj4,pj=q˙j.Our first result concerns the Hamiltonian with two degrees of freedom (q2=q0); that is,(5)H=12p12+p22+122q12-4q1q2+2q22+a2q12+q22+b2q14+q24.It simply says that the corresponding Hamiltonian system is integrable only when it is linear.Theorem 2.
The periodic KG lattice withn=2 is nonintegrable unless b=0.The above result suggests that generically the periodic KG lattice is nonintegrable forn≥2. Still there might be some special cases of integrability, while the opposite is not proven rigorously.Motivated by the works of Rink and Verhulst [3, 4], who presented the periodic FPU chain as a perturbation of an integrable and KAM nondegenerated system, namely, the truncated Birkhoff-Gustavson normal form of order 4 in the neighborhood of an equilibrium, our aim is to verify whether this can be done for the low-dimensional KG lattices.One should note that Rink and Verhulst’s result is due to the special symmetry and resonance properties of the FPU chain and should not be expected for lower-order resonant Hamiltonian systems (see, e.g., [5]).We summarize our second result concerning the integrability of truncated resonant normal forms of the periodic KG lattices up to six particles in the following.Theorem 3.
The truncated normal formH¯=H2+H¯4 of the periodic KG lattice is(i)
completely integrable and KAM nondegenerated forn=2,3,4;(ii)
completely integrable and KAM nondegenerated whenn=5 for all a but one;(iii)
completely integrable forn=6.As a consequence from this result, we may conclude for the low-dimensional KG lattices when KAM theory applies that there exist many quasi-periodic solutions of small energy on a long time scale (see Section2 and for more detailed explanation [3]) and chaotic orbits are of small measure.The paper is organized as follows. In Section2 some notions and facts used in the paper are given. We discuss also the appearance of some additional resonant relations. In Section 3 we calculate the Birkhoff-Gustavson normal forms for the cases up to six particles and show that they are integrable in most cases. We finish with some concluding remarks as well as some possible lines of further study.The proof of Theorem2 is based on the Ziglin-Morales-Ramis theory and since it is more algebraic in nature, it is carried out in the Appendix.


## 2. Resonances and Normalization

In this section we recall briefly some notions and facts about integrability of Hamiltonian systems, action-angle variables, perturbation of integrable systems, and normal forms. More complete exposition can be found in [6–8].LetH be an analytic Hamiltonian defined on a 2n-dimensional symplectic manifold. The corresponding Hamiltonian system is(6)x˙=XHx.It is said that a Hamiltonian system is completely integrable if there exist n independent integrals F1=H,F2,…,Fn in involution, namely, {Fi,Fj}=0 for all i and j, where {,} is the Poisson bracket. On a neighborhood U of the connected compact level sets of the integrals Mc={Fj=cj,j=1,…,n} by Liouville-Arnold theorem one can introduce a special set of symplectic coordinates, Ij,φj, called action-angle variables. Then, the integrals F1=H,F2,…,Fn are functions of action variables only and the flow of XH is simple(7)I˙j=0,φ˙j=∂H∂Ij,j=1,…,n.Therefore, near Mc, the phase space is foliated with XFi invariant tori over which the flow of XH is quasi-periodic with frequencies (ω1(I),…,ωn(I))=(∂H/∂I1,…,∂H/∂In).The map(8)I1,I2,…,In⟶∂H∂I1,∂H∂I2,…,∂H∂Inis called frequency map.Consider a small perturbation of an integrable HamiltonianH0. According to Poincaré the main problem of mechanics is to study the perturbation of quasi-periodic motions in the system given by the Hamiltonian (9)H=H0I+εH1I,φ,ε≪1. KAM theory [9–11] gives conditions on the integrable Hamiltonian H0 which ensures the survival of the most of the invariant tori. The following condition, usually called Kolmogorov’s condition, is that the frequency map should be a local diffeomorphism, or equivalently(10)det⁡∂2H0∂Ii∂Ij≠0on an open and dense subset of U. We should note that the measure of the surviving tori decreases with the increase of both perturbation and the measure of the set where above Hessian is too close to zero.In a neighborhood of the equilibrium(q,p)=(0,0) we have the following expansion of H: (11)H=H2+H3+H4+⋯,H2=∑ωjqj2+pj2,ωj>0. We assume that H2 is a positively defined quadratic form. The frequency ω=(ω1,…,ωn) is said to be in resonance if there exists a vector k=(k1,…,kn),kj∈Z,j=1,…,n, such that (ω,k)=∑kjωj=0, where |k|=∑|kj| is the order of resonance.With the help of a series of canonical transformations close to the identity,H simplifies. In the absence of resonances the simplified Hamiltonian is called Birkhoff normal form and, otherwise, Birkhoff-Gustavson normal form, which may contain combinations of angles arising from resonances.Often to detect the behavior in a small neighborhood of the equilibrium, instead of the HamiltonianH one considers the normal form truncated to some order (12)H¯=H2+⋯+H¯m. It is known that the truncated to any order Birkhoff normal form is integrable [8]. The truncated Birkhoff-Gustavson normal form has at least two integrals—H2 and H¯. Therefore, the truncated normal form of two-degree-of-freedom Hamiltonian is integrable.In order to obtain estimates of the approximation by normalization in a neighborhood of an equilibrium point we scaleq→εq~,p→εp~. Here ε is a small positive parameter and ε2 is a measure for the energy relative to the equilibrium energy. Then, dividing by ε2 and removing tildes we get (13)H¯=H2+εH¯3+⋯+εm-2H¯m. Provided that ωj>0 it is proven in [12] that H¯ is an integral for the original system with error O(εm-1) and H2 is an integral for the original system with error O(ε) for the whole time interval. If we have more independent integrals, then they are integrals for the original Hamiltonian system with error O(εm-2) on the time scale 1/ε.The first integrals for the normal formH¯ are approximate integrals for the original system; that is, if the normal form is integrable, then the original system isnear integrable in the above sense.Returning to the Hamiltonian of the periodic KG lattice (4) we see that its quadratic part H2 is not in diagonal form:(14)H2=12pTp+12qTLnq.Here Ln is the following n×n matrix:(15)Ln≔2+a-1-1-12+a-1⋱⋱⋱-12+a-1-1-12+a.The eigenvalues of Ln are of the form Ωk2=a+ωk2,ωk=2sin⁡kπ/n. In order to obtain the corresponding eigenvectors yk, following [3] we define(16)yn≔1n1,1,…,1Tand if n is even,(17)yn/2≔1n1,-1,1,-1,…,-1T.Further, for 1≤k<n/2, we define yk and yn-k via their coordinates(18)yjk≔2ncos⁡2kjπn,yjn-k≔2nsin⁡2kjπn.It is easily checked that {y1,…,yn} is an orthonormal basis of Rn, consisting of eigenvectors of Ln. Let Y be the n×n matrix formed by the vectors yk as columns; then YTY=Id,Y-1LnY=Ω≔diag(Ω12,…,Ωn2). The symplectic Fourier-transformation q=Yq¯,p=Yp¯ brings H2 in diagonal form(19)H2=12pTp+12qTΩq.The variables (q¯,p¯) are known asphonons.We need one more definition.Definition 4 (see [3]).
It is said thatω∈Rn satisfies the property ofinternal resonance if for any k∈Zn with (k,ω)=0,  |k|=4, we have kj=-kn-j when 1≤j<n/2.In Table1 we list the frequencies Ωk of some low-dimensional periodic KG lattice.Table 1
n
Ω
k
2
a
+
4
,
a
3
a
+
3
,
a
+
3
,
a
4
a
+
2
,
a
+
4
,
a
+
2
,
a
5
a
+
(
5
-
5
)
/
2
,
a
+
(
5
+
5
)
/
2
,
a
+
(
5
+
5
)
/
2
,
a
+
(
5
-
5
)
/
2
,
a
6
a
+
1
,
a
+
3
,
a
+
4
,
a
+
3
,
a
+
1
,
aAs it is seen from the table we almost always have internal resonances. In general,Ωk=Ωj holds if only if j=n-k. There are plenty of resonances when a∈Q.Nevertheless, the assumption ona (3) does not prevent the appearance of more complicated resonances. Indeed, in the case of 4 particles there exists unique a>0 irrational, for which the following resonant relations are fulfilled (Ω1=Ω3):(20)Ω2-Ω3-Ω4=0,Ω2-Ω1-Ω4=0.There are many like them and in particular for n=5 and n=6. Luckily, these resonances are of no importance for our considerations, because they produce third-order resonant terms and none of them appears due to the assumption on V(x) (2).The resonances we have to worry about are the forth-order ones. Trivial inspection shows that fora in the domain (3), there are no such resonances when n=2,3,4,6. However, there are some of that kind when n=5 (Ω1=Ω4,Ω2=Ω3); namely,(1)
fora1=(5-5)/16 the following resonant relations are fulfilled:(21)Ω1-3Ω5=0,Ω4-3Ω5=0;(2)
fora2=(5+5)/16 the following resonant relations are fulfilled:(22)Ω2-3Ω5=0,Ω3-3Ω5=0,(3)
fora3=5(35-5)/16 the following resonant relations are fulfilled:(23)2Ω1-Ω2-Ω5=0,2Ω4-Ω2-Ω5=0,(24)2Ω1-Ω3-Ω5=0,2Ω4-Ω3-Ω5=0.We will see how these additional resonances affect H¯4 in the end of Section 3.4.


## 3. Low-Dimensional Lattices

In this section we calculate the normal forms for the periodic KG lattices with particles up to six and in this way we prove Theorem3. We do not use ε in the fourth-degree expression, but keep in mind that we are close to the equilibrium. Of course, it is assumed that b≠0.


### 3.1. Two Particles

This case is easy. It is well known that the truncated to any order normal form of a two-degree-of-freedom Hamiltonian is integrable. It remains only to verify the KAM condition.We have already brought the quadratic part of the Hamiltonian in diagonal form. It is important that it is written in the phonons(q¯,p¯).(25)H=12p¯12+p¯22+124+aq¯12+aq¯22+b4q¯14+6q¯12q¯22+q¯24.Further, we perform a scaling (26)q¯1⟶1a+44q¯1,p¯1⟶a+44p¯1,q¯2⟶1a4q¯2,p¯2⟶a4p¯2, which preserves the symplectic form. The Hamiltonian (25) becomes (27)H=a+42p¯12+q¯12+a2p¯22+q¯22+b4q¯14a+4+6q¯12q¯22aa+4+q¯24a. Usually at this place one makes the following change of variables:(28)q¯j=12zj+wj,p¯j=12izj-wj,j=1,2.Since the frequencies Ω1=a+4,Ω2=a are incommensurable, the only resonant terms which remain arezjwj,  zjwj2,  j=1,2,  z1w1z2w2.The other terms can be removed via symplectic near-identity change. Therefore, the normal form of (25) up to order four H¯=H2+H¯4 is (29)H¯=a+42z1w1+a2z2w2+3b32z1w12a+4+4z1w1z2w2aa+4+z2w22a or, returning to the (q¯,p¯) coordinates, we get (30)H¯=a+42p¯12+q¯12+a2p¯22+q¯22+3b32p¯12+q¯122a+4+4p¯12+q¯12p¯22+q¯22aa+4+p¯22+q¯222a. This normal form is clearly integrable with quadratic first integrals Ij=p¯j2+q¯j2,j=1,2.Finally, introducing symplectic polar coordinates, which are action-angle variables(31)q¯j=2Ijcos⁡φj,p¯j=2Ijsin⁡φj,j=1,2 we get(32)H¯=a+4I1+aI2+3b8I12a+4+4I1I2aa+4+I22a.It can easily be checked that Kolmogorov’s condition is valid.


### 3.2. Three Particles

First, we make use of the phonons(q¯,p¯) (as explained in Section 2) to transform the quadratic part of the Hamiltonian (4) n=3 in diagonal form(33)H=12p¯12+p¯22+p¯32+a+32q¯12+q¯22+a2q¯32+b1892q¯14+q¯24+3q¯34+62q¯13q¯3+9q¯12q¯22+18q¯12+q¯22q¯32-182q¯1q¯22q¯3.Further, the scaling (34)q¯1,2⟶1a+34q¯1,2,p¯1,2⟶a+34p¯1,2,q¯3⟶1a4q¯3,p¯3⟶a4p¯3 results in(35)H=a+32p¯12+q¯12+p¯22+q¯22+a2p¯32+q¯32+b189q¯14+q¯242a+3+3q¯34a+62q¯13q¯3aa+334+9q¯12q¯22a+3+18q¯12+q¯22q¯32aa+3-182q¯1q¯22q¯3aa+334. Passing to the variables (zj,wj),  j=1,2,3 (28), we notice that there is an internal resonance between the frequencies Ω1 and Ω2. Therefore, the generators of the normal form arezjwj,  j=1,2,3,  and  z1w2,z2w1.After removing the nonresonant terms, the normal form of the (33) up to order four H¯=H2+H¯4 is(36)H¯=a+32z1w1+z2w2+a2z3w3+b1827z1w1+z2w2216a+3+9z3w328a+9z1w1+z2w2z3w32aa+3+9z1w2-z2w1216a+3, or, returning to (q¯,p¯), we get(37)H¯=a+32p¯12+q¯12+p¯22+q¯22+a2p¯32+q¯32+b23p¯12+q¯12+p¯22+q¯22216a+3+p¯32+q¯3228a+p¯12+q¯12+p¯22+q¯22p¯32+q¯322aa+3-p¯1q¯2-q¯1p¯224a+3. This normal form is integrable with the following quadratic first integrals:(38)F1≔p¯12+q¯12+p¯22+q¯22,G1≔p¯1q¯2-q¯1p¯2,I3≔p¯32+q¯32.In order to introduce action-angle variables, we need to find the set of regular values of the energy momentum map (39)EM:q¯,p¯⟶F1,G1,I3. In fact, this is done in [4]. Denote by Ur={(F1,G1,I3)∈R3,F1>0,|G1|<F1,I3>0}. Then for all (F1,G1,I3)∈Ur the level sets of EM-1(F1,G1,I3) are diffeomorphic to 3-tori.Letarg:R2∖{(0,0)}→R/2πZ be the argument function arg⁡(rcos⁡Φ,rsin⁡Φ)→Φ. Define the following set of variables (F1,G1,I3,ϕ1,ψ1,φ3)F1,G1,I3 as above and(40)ϕ1≔12arg⁡-p¯2-q¯1,p¯1-q¯2+12arg⁡p¯2-q¯1,p¯1+q¯2,ψ1≔12arg⁡-p¯2-q¯1,p¯1-q¯2-12arg⁡p¯2-q¯1,p¯1+q¯2,φ3≔arctan⁡p¯3q¯3.Using the formula darg⁡(x,y)=xdy-ydx/x2+y2, one can verify that (F1,G1,I3,ϕ1,ψ1,φ3) are indeed canonical coordinates ∑dp¯j∧q¯j=dF1∧dϕ1+dG1∧dψ1+dI3∧dφ3.Then the truncated up to order 4 normal form as a function of actions is(41)H¯=a+32F1+a2I3+b23F1216a+3+F1I32aa+3-G124a+3+I328a.Then one can easily check up that Kolmogorov’s condition is valid.Remark 5.
If we seta=1, we get(42)H¯=22p¯12+q¯12+p¯22+q¯22+12p¯32+q¯32+b23p¯12+q¯12+p¯22+q¯22264+p¯32+q¯3228+p¯12+q¯12+p¯22+q¯22p¯32+q¯324-p¯1q¯2-q¯1p¯2216, which is an example of an integrable KAM nondegenerate normal form of 2 : 2 : 1 (or 1 : 2 : 2) Hamiltonian resonance (see, e.g., [13]).


### 3.3. Four Particles

It turns out that the normal form of the periodic KG lattice in the case of four particles is surprisingly simple, no matter the internal resonance.After transforming the quadratic part in diagonal form and scaling(43)q¯1,3⟶1a+24q¯1,3,p¯1,3⟶a+24p¯1,3,q¯2⟶1a+44q¯2,p¯2⟶a+44p¯2,q¯4⟶1a4q¯4,p¯4⟶a4p¯4 the Hamiltonian (4) n=4 takes the form(44)H=a+22p¯12+q¯12+a+42p¯22+q¯22+a+22p¯32+q¯32+a2p¯42+q¯42+b82q¯14+q¯34a+2+q¯24a+4+q¯44a+12q¯2q¯32q¯4aa+4a+224-12q¯12q¯2q¯4aa+4a+224+6q¯22q¯12+q¯32a+2a+4+6q¯42q¯12+q¯32aa+2+6q¯22q¯42aa+4.There is an internal resonance between the frequencies Ω1 and Ω3. In variables (zj,wj),j=1,2,3,4, the generators of the normal form arezjwj,  and  z1w3,z3w1.However, as it is seen from (44), the variables q¯1 and q¯3 do not couple, so the last two generators do not appear in the normal form as if there are no resonances. After removing the nonresonant terms, the normal form of (44) up to order four H¯=H2+H¯4 reads(45)H¯=a+22z1w1+a+42z2w2+a+22z3w3+a2z4w4+b83z1w124a+2+3z2w228a+4+3z3w324a+2+3z4w428a+3z2w2z1w1+z3w32a+2a+4+3z4w4z1w1+z3w32aa+2+3z2w2z4w42aa+4, or, in (q¯,p¯) variables,(46)H¯=a+22p¯12+q¯12+a+42p¯22+q¯22+a+22p¯32+q¯32+a2p¯42+q¯42+b83p¯12+q¯1224a+2+3p¯22+q¯2228a+4+3p¯32+q¯3224a+2+3p¯42+q¯4228a+3p¯22+q¯22p¯12+q¯12+p¯32+q¯322a+2a+4+3p¯42+q¯42p¯12+q¯12+p¯32+q¯322aa+2+3p¯22+q¯22p¯42+q¯422aa+4. This normal form is integrable with integrals Ij=p¯j2+q¯j2,j=1,2,3,4 and the action-angle variables are clear. The truncated normal form up to order 4 as a function of the action variables is(47)H¯=a+22I1+a+42I2+a+22I3+a2I4+b83I12+I324a+2+3I228a+4+3I428a+3I2I1+I32a+2a+4+3I4I1+I32aa+2+3I2I42aa+4, and the verification of the validity of the Kolmogorov’s condition is straightforward.


### 3.4. Five Particles

There is a close resemblance between the two cases of tree and five particles.We transform the quadratic part of the Hamiltonian (4) n=5 using the phonons (q¯,p¯) and scale in the usual way. Further, we use the notations Ωj for short. Recall that (48)Ω1=Ω4=a+5-52,Ω2=Ω3=a+5+52,Ω5=a; that is, Ω contains two internal resonances. The generators of the normal form in (z,w) variables arezjwj,  j=1,…,5,  and  z1w4,z4w1,z2w3,z3w2.Then the truncated up to order 4 normal formH¯=H2+H¯4 for (4) n=5 reads(49)H¯=Ω12z1w1+z4w4+Ω22z2w2+z3w3+Ω52z5w5+b10916Ω12z1w1+z4w42+916Ω22z2w2+z3w32+38Ω52z5w52+316Ω12z1w4-z4w12+316Ω22z2w3-z3w22+32Ω1Ω2z1w1+z4w4z2w2+z3w3+32Ω1Ω5z1w1+z4w4z5w5+32Ω2Ω5z2w2+z3w3z5w5, or, written in the coordinates (q¯,p¯), it is(50)H¯=Ω12p¯12+q¯12+p¯42+q¯42+Ω22p¯22+q¯22+p¯32+q¯32+Ω52p¯52+q¯52+b10916Ω12p¯12+q¯12+p¯42+q¯422+916Ω22p¯22+q¯22+p¯32+q¯322+38Ω52p¯52+q¯522-34Ω12p¯1q¯4-q¯1p¯42-34Ω22p¯2q¯3-q¯2p¯32+32Ω1Ω2p¯12+q¯12+p¯42+q¯42p¯22+q¯22+p¯32+q¯32+32Ω1Ω5p¯12+q¯12+p¯42+q¯42p¯52+q¯52+32Ω2Ω5p¯22+q¯22+p¯32+q¯32p¯52+q¯52. As it is seen, the above normal form is integrable with the following first integrals:(51)F1≔p¯12+q¯12+p¯42+q¯42,F2≔p¯22+q¯22+p¯32+q¯32,G1≔p¯1q¯4-q¯1p¯4,G2≔p¯2q¯3-q¯2p¯3,I5≔p¯52+q¯52.In a similar way as in [4] and above treated case of tree particles, in the domain where the first integrals are independent, we can introduce action-angle variables (52)q¯,p¯⟶F1,F2,G1,G2,I5,ϕ1,ϕ2,ψ1,ψ2,φ5, where Fj,Gj,j=1,2 are as above and(53)ϕj≔12arg⁡-p¯5-j-q¯j,p¯j-q¯5-j+12arg⁡p¯5-j-q¯j,p¯j+q¯5-j,j=1,2,ψj≔12arg⁡-p¯5-j-q¯j,p¯j-q¯5-j-12arg⁡p¯5-j-q¯j,p¯j+q¯5-j,j=1,2,φ5≔arctan⁡p¯5q¯5.As above one can verify that (F1,F2,G1,G2,I5,ϕ1,ϕ2,ψ1,ψ2,φ5) are canonical coordinates. Then the truncated up to order 4 normal form as a function of the action variables is(54)H¯=Ω12F1+Ω22F2+Ω52I5+b10916Ω12F12+916Ω22F22+38Ω52I52-34Ω12G12-34Ω22G22+32Ω1Ω2F1F2+32Ω1Ω5F1I5+32Ω2Ω5F2I5. It is straightforward to check up whether the Kolmogorov’s condition is valid.Remark 6.
Now, let us study the normal forms for the exceptional cases, listed in the end of Section2. Surprisingly, the resonant terms corresponding to the resonant relations (21), (22), and (24) are missing in H4. Solely the resonant terms corresponding to the resonant relations (23) appear in H4 in the following way: (55)1528Ω1Ω2Ω5z12w2w5+w12z2z5-w42z2z5-z42w2w5. Therefore, the truncated normal form H¯=H2+H¯4 for (4) n=5 and a=a3 is(56)H¯=Ω12p¯12+q¯12+p¯42+q¯42+Ω22p¯22+q¯22+p¯32+q¯32+Ω52p¯52+q¯52+b10916Ω12p¯12+q¯12+p¯42+q¯422+916Ω22p¯22+q¯22+p¯32+q¯322+38Ω52p¯52+q¯522-34Ω12p¯1q¯4-q¯1p¯42-34Ω22p¯2q¯3-q¯2p¯32+32Ω1Ω2p¯12+q¯12+p¯42+q¯42p¯22+q¯22+p¯32+q¯32+32Ω1Ω5p¯12+q¯12+p¯42+q¯42p¯52+q¯52+32Ω2Ω5p¯22+q¯22+p¯32+q¯32p¯52+q¯52+1524Ω1Ω2Ω52q2p5+p2q5q1p1-q4p4+q2q5-p2p5q12-p12+p42-q42. It can be easily checked that the expressions (51) are no longer first integrals for this normal form. The known integrals are H2 and H¯ and that makes the integrability of this particular resonant normal form unclear.


### 3.5. Six Particles

In the diagonalized Hamiltonian system (4) n=6 we scale (57)q¯1,5⟶1a+14q¯1,5,p¯1,5⟶a+14p¯1,5,q¯2,4⟶1a+34q¯2,4,p¯2,4⟶a+34p¯2,4,q¯3⟶1a+44q¯3,p¯3⟶a+44p¯3,q¯6⟶1a4q¯6,p¯6⟶a4p¯6. Recall that Ω1=Ω5=a+1,Ω2=Ω4=a+3,Ω3=a+4,Ω6=a; that is, in these cases we again have two independent 1 : 1 resonances. In the variables (zj,wj),j=1,…,6, the generators of the normal form arezjwj  and  z1w5,w1z5,z2w4,z4w2.We normalize and find that the normal formH¯=H2+H¯4 up to order 4 is(58)H¯=Ω12z1w1+z5w5+Ω22z2w2+z4w4+Ω32z3w3+Ω62z6w6+b72278Ω12z1w1+z5w52+278Ω22z2w2+z4w42+94Ω32z3w32+94Ω62z6w62+9z3w3z6w6Ω3Ω6+9z3w3Ω1Ω3z1w1+z5w5+9z3w3Ω2Ω3z2w2+z4w4+9z6w6Ω1Ω6z1w1+z5w5+9z6w6Ω2Ω6z2w2+z4w4+98z1w5-w1z5Ω1-z2w4-w2z4Ω22+92Ω1Ω2z1w1z4w4+z2w2z5w5+3z1w1z2w2+3z4w4z5w5-z1w5+w1z5z2w4+w2z4+12z1w5-w1z5z2w4-w2z4. Before returning to the variables (q¯,p¯) we denote(59)F1≔p¯12+q¯12+p¯52+q¯52,F2≔p¯22+q¯22+p¯42+q¯42,I3≔p¯32+q¯32,I6≔p¯62+q¯62.Then the truncated normal form H¯=H2+H¯4 becomes(60)H¯=Ω12F1+Ω22F2+Ω32I3+Ω62I6+b83F128Ω12+3F228Ω22+I324Ω32+I624Ω62+F1F22Ω1Ω2+I3I6Ω3Ω6+I3F1Ω1Ω3+I3F2Ω2Ω3+I6F1Ω1Ω6+I6F2Ω2Ω6-12p¯1q¯5-q¯1p¯5Ω1-p¯2q¯4-q¯2p¯4Ω32+1Ω1Ω2p¯12+q¯12p¯22+q¯22+p¯42+q¯42p¯52+q¯52-2q¯1q¯5+p¯1p¯5q¯2q¯4+p¯2p¯4-p¯1q¯5-q¯1p¯5p¯2q¯4-q¯2p¯4.This normal form is integrable: the independent first integrals areF1,F2,I3,I6,G≔(p¯1q¯5-q¯1p¯5)-(p¯2q¯4-q¯2p¯4)  and  H¯4.However, the construction of all action-angle variables is unclear so far.This finishes the proof of Theorem3.


## 3.1. Two Particles

This case is easy. It is well known that the truncated to any order normal form of a two-degree-of-freedom Hamiltonian is integrable. It remains only to verify the KAM condition.We have already brought the quadratic part of the Hamiltonian in diagonal form. It is important that it is written in the phonons(q¯,p¯).(25)H=12p¯12+p¯22+124+aq¯12+aq¯22+b4q¯14+6q¯12q¯22+q¯24.Further, we perform a scaling (26)q¯1⟶1a+44q¯1,p¯1⟶a+44p¯1,q¯2⟶1a4q¯2,p¯2⟶a4p¯2, which preserves the symplectic form. The Hamiltonian (25) becomes (27)H=a+42p¯12+q¯12+a2p¯22+q¯22+b4q¯14a+4+6q¯12q¯22aa+4+q¯24a. Usually at this place one makes the following change of variables:(28)q¯j=12zj+wj,p¯j=12izj-wj,j=1,2.Since the frequencies Ω1=a+4,Ω2=a are incommensurable, the only resonant terms which remain arezjwj,  zjwj2,  j=1,2,  z1w1z2w2.The other terms can be removed via symplectic near-identity change. Therefore, the normal form of (25) up to order four H¯=H2+H¯4 is (29)H¯=a+42z1w1+a2z2w2+3b32z1w12a+4+4z1w1z2w2aa+4+z2w22a or, returning to the (q¯,p¯) coordinates, we get (30)H¯=a+42p¯12+q¯12+a2p¯22+q¯22+3b32p¯12+q¯122a+4+4p¯12+q¯12p¯22+q¯22aa+4+p¯22+q¯222a. This normal form is clearly integrable with quadratic first integrals Ij=p¯j2+q¯j2,j=1,2.Finally, introducing symplectic polar coordinates, which are action-angle variables(31)q¯j=2Ijcos⁡φj,p¯j=2Ijsin⁡φj,j=1,2 we get(32)H¯=a+4I1+aI2+3b8I12a+4+4I1I2aa+4+I22a.It can easily be checked that Kolmogorov’s condition is valid.


## 3.2. Three Particles

First, we make use of the phonons(q¯,p¯) (as explained in Section 2) to transform the quadratic part of the Hamiltonian (4) n=3 in diagonal form(33)H=12p¯12+p¯22+p¯32+a+32q¯12+q¯22+a2q¯32+b1892q¯14+q¯24+3q¯34+62q¯13q¯3+9q¯12q¯22+18q¯12+q¯22q¯32-182q¯1q¯22q¯3.Further, the scaling (34)q¯1,2⟶1a+34q¯1,2,p¯1,2⟶a+34p¯1,2,q¯3⟶1a4q¯3,p¯3⟶a4p¯3 results in(35)H=a+32p¯12+q¯12+p¯22+q¯22+a2p¯32+q¯32+b189q¯14+q¯242a+3+3q¯34a+62q¯13q¯3aa+334+9q¯12q¯22a+3+18q¯12+q¯22q¯32aa+3-182q¯1q¯22q¯3aa+334. Passing to the variables (zj,wj),  j=1,2,3 (28), we notice that there is an internal resonance between the frequencies Ω1 and Ω2. Therefore, the generators of the normal form arezjwj,  j=1,2,3,  and  z1w2,z2w1.After removing the nonresonant terms, the normal form of the (33) up to order four H¯=H2+H¯4 is(36)H¯=a+32z1w1+z2w2+a2z3w3+b1827z1w1+z2w2216a+3+9z3w328a+9z1w1+z2w2z3w32aa+3+9z1w2-z2w1216a+3, or, returning to (q¯,p¯), we get(37)H¯=a+32p¯12+q¯12+p¯22+q¯22+a2p¯32+q¯32+b23p¯12+q¯12+p¯22+q¯22216a+3+p¯32+q¯3228a+p¯12+q¯12+p¯22+q¯22p¯32+q¯322aa+3-p¯1q¯2-q¯1p¯224a+3. This normal form is integrable with the following quadratic first integrals:(38)F1≔p¯12+q¯12+p¯22+q¯22,G1≔p¯1q¯2-q¯1p¯2,I3≔p¯32+q¯32.In order to introduce action-angle variables, we need to find the set of regular values of the energy momentum map (39)EM:q¯,p¯⟶F1,G1,I3. In fact, this is done in [4]. Denote by Ur={(F1,G1,I3)∈R3,F1>0,|G1|<F1,I3>0}. Then for all (F1,G1,I3)∈Ur the level sets of EM-1(F1,G1,I3) are diffeomorphic to 3-tori.Letarg:R2∖{(0,0)}→R/2πZ be the argument function arg⁡(rcos⁡Φ,rsin⁡Φ)→Φ. Define the following set of variables (F1,G1,I3,ϕ1,ψ1,φ3)F1,G1,I3 as above and(40)ϕ1≔12arg⁡-p¯2-q¯1,p¯1-q¯2+12arg⁡p¯2-q¯1,p¯1+q¯2,ψ1≔12arg⁡-p¯2-q¯1,p¯1-q¯2-12arg⁡p¯2-q¯1,p¯1+q¯2,φ3≔arctan⁡p¯3q¯3.Using the formula darg⁡(x,y)=xdy-ydx/x2+y2, one can verify that (F1,G1,I3,ϕ1,ψ1,φ3) are indeed canonical coordinates ∑dp¯j∧q¯j=dF1∧dϕ1+dG1∧dψ1+dI3∧dφ3.Then the truncated up to order 4 normal form as a function of actions is(41)H¯=a+32F1+a2I3+b23F1216a+3+F1I32aa+3-G124a+3+I328a.Then one can easily check up that Kolmogorov’s condition is valid.Remark 5.
If we seta=1, we get(42)H¯=22p¯12+q¯12+p¯22+q¯22+12p¯32+q¯32+b23p¯12+q¯12+p¯22+q¯22264+p¯32+q¯3228+p¯12+q¯12+p¯22+q¯22p¯32+q¯324-p¯1q¯2-q¯1p¯2216, which is an example of an integrable KAM nondegenerate normal form of 2 : 2 : 1 (or 1 : 2 : 2) Hamiltonian resonance (see, e.g., [13]).


## 3.3. Four Particles

It turns out that the normal form of the periodic KG lattice in the case of four particles is surprisingly simple, no matter the internal resonance.After transforming the quadratic part in diagonal form and scaling(43)q¯1,3⟶1a+24q¯1,3,p¯1,3⟶a+24p¯1,3,q¯2⟶1a+44q¯2,p¯2⟶a+44p¯2,q¯4⟶1a4q¯4,p¯4⟶a4p¯4 the Hamiltonian (4) n=4 takes the form(44)H=a+22p¯12+q¯12+a+42p¯22+q¯22+a+22p¯32+q¯32+a2p¯42+q¯42+b82q¯14+q¯34a+2+q¯24a+4+q¯44a+12q¯2q¯32q¯4aa+4a+224-12q¯12q¯2q¯4aa+4a+224+6q¯22q¯12+q¯32a+2a+4+6q¯42q¯12+q¯32aa+2+6q¯22q¯42aa+4.There is an internal resonance between the frequencies Ω1 and Ω3. In variables (zj,wj),j=1,2,3,4, the generators of the normal form arezjwj,  and  z1w3,z3w1.However, as it is seen from (44), the variables q¯1 and q¯3 do not couple, so the last two generators do not appear in the normal form as if there are no resonances. After removing the nonresonant terms, the normal form of (44) up to order four H¯=H2+H¯4 reads(45)H¯=a+22z1w1+a+42z2w2+a+22z3w3+a2z4w4+b83z1w124a+2+3z2w228a+4+3z3w324a+2+3z4w428a+3z2w2z1w1+z3w32a+2a+4+3z4w4z1w1+z3w32aa+2+3z2w2z4w42aa+4, or, in (q¯,p¯) variables,(46)H¯=a+22p¯12+q¯12+a+42p¯22+q¯22+a+22p¯32+q¯32+a2p¯42+q¯42+b83p¯12+q¯1224a+2+3p¯22+q¯2228a+4+3p¯32+q¯3224a+2+3p¯42+q¯4228a+3p¯22+q¯22p¯12+q¯12+p¯32+q¯322a+2a+4+3p¯42+q¯42p¯12+q¯12+p¯32+q¯322aa+2+3p¯22+q¯22p¯42+q¯422aa+4. This normal form is integrable with integrals Ij=p¯j2+q¯j2,j=1,2,3,4 and the action-angle variables are clear. The truncated normal form up to order 4 as a function of the action variables is(47)H¯=a+22I1+a+42I2+a+22I3+a2I4+b83I12+I324a+2+3I228a+4+3I428a+3I2I1+I32a+2a+4+3I4I1+I32aa+2+3I2I42aa+4, and the verification of the validity of the Kolmogorov’s condition is straightforward.


## 3.4. Five Particles

There is a close resemblance between the two cases of tree and five particles.We transform the quadratic part of the Hamiltonian (4) n=5 using the phonons (q¯,p¯) and scale in the usual way. Further, we use the notations Ωj for short. Recall that (48)Ω1=Ω4=a+5-52,Ω2=Ω3=a+5+52,Ω5=a; that is, Ω contains two internal resonances. The generators of the normal form in (z,w) variables arezjwj,  j=1,…,5,  and  z1w4,z4w1,z2w3,z3w2.Then the truncated up to order 4 normal formH¯=H2+H¯4 for (4) n=5 reads(49)H¯=Ω12z1w1+z4w4+Ω22z2w2+z3w3+Ω52z5w5+b10916Ω12z1w1+z4w42+916Ω22z2w2+z3w32+38Ω52z5w52+316Ω12z1w4-z4w12+316Ω22z2w3-z3w22+32Ω1Ω2z1w1+z4w4z2w2+z3w3+32Ω1Ω5z1w1+z4w4z5w5+32Ω2Ω5z2w2+z3w3z5w5, or, written in the coordinates (q¯,p¯), it is(50)H¯=Ω12p¯12+q¯12+p¯42+q¯42+Ω22p¯22+q¯22+p¯32+q¯32+Ω52p¯52+q¯52+b10916Ω12p¯12+q¯12+p¯42+q¯422+916Ω22p¯22+q¯22+p¯32+q¯322+38Ω52p¯52+q¯522-34Ω12p¯1q¯4-q¯1p¯42-34Ω22p¯2q¯3-q¯2p¯32+32Ω1Ω2p¯12+q¯12+p¯42+q¯42p¯22+q¯22+p¯32+q¯32+32Ω1Ω5p¯12+q¯12+p¯42+q¯42p¯52+q¯52+32Ω2Ω5p¯22+q¯22+p¯32+q¯32p¯52+q¯52. As it is seen, the above normal form is integrable with the following first integrals:(51)F1≔p¯12+q¯12+p¯42+q¯42,F2≔p¯22+q¯22+p¯32+q¯32,G1≔p¯1q¯4-q¯1p¯4,G2≔p¯2q¯3-q¯2p¯3,I5≔p¯52+q¯52.In a similar way as in [4] and above treated case of tree particles, in the domain where the first integrals are independent, we can introduce action-angle variables (52)q¯,p¯⟶F1,F2,G1,G2,I5,ϕ1,ϕ2,ψ1,ψ2,φ5, where Fj,Gj,j=1,2 are as above and(53)ϕj≔12arg⁡-p¯5-j-q¯j,p¯j-q¯5-j+12arg⁡p¯5-j-q¯j,p¯j+q¯5-j,j=1,2,ψj≔12arg⁡-p¯5-j-q¯j,p¯j-q¯5-j-12arg⁡p¯5-j-q¯j,p¯j+q¯5-j,j=1,2,φ5≔arctan⁡p¯5q¯5.As above one can verify that (F1,F2,G1,G2,I5,ϕ1,ϕ2,ψ1,ψ2,φ5) are canonical coordinates. Then the truncated up to order 4 normal form as a function of the action variables is(54)H¯=Ω12F1+Ω22F2+Ω52I5+b10916Ω12F12+916Ω22F22+38Ω52I52-34Ω12G12-34Ω22G22+32Ω1Ω2F1F2+32Ω1Ω5F1I5+32Ω2Ω5F2I5. It is straightforward to check up whether the Kolmogorov’s condition is valid.Remark 6.
Now, let us study the normal forms for the exceptional cases, listed in the end of Section2. Surprisingly, the resonant terms corresponding to the resonant relations (21), (22), and (24) are missing in H4. Solely the resonant terms corresponding to the resonant relations (23) appear in H4 in the following way: (55)1528Ω1Ω2Ω5z12w2w5+w12z2z5-w42z2z5-z42w2w5. Therefore, the truncated normal form H¯=H2+H¯4 for (4) n=5 and a=a3 is(56)H¯=Ω12p¯12+q¯12+p¯42+q¯42+Ω22p¯22+q¯22+p¯32+q¯32+Ω52p¯52+q¯52+b10916Ω12p¯12+q¯12+p¯42+q¯422+916Ω22p¯22+q¯22+p¯32+q¯322+38Ω52p¯52+q¯522-34Ω12p¯1q¯4-q¯1p¯42-34Ω22p¯2q¯3-q¯2p¯32+32Ω1Ω2p¯12+q¯12+p¯42+q¯42p¯22+q¯22+p¯32+q¯32+32Ω1Ω5p¯12+q¯12+p¯42+q¯42p¯52+q¯52+32Ω2Ω5p¯22+q¯22+p¯32+q¯32p¯52+q¯52+1524Ω1Ω2Ω52q2p5+p2q5q1p1-q4p4+q2q5-p2p5q12-p12+p42-q42. It can be easily checked that the expressions (51) are no longer first integrals for this normal form. The known integrals are H2 and H¯ and that makes the integrability of this particular resonant normal form unclear.


## 3.5. Six Particles

In the diagonalized Hamiltonian system (4) n=6 we scale (57)q¯1,5⟶1a+14q¯1,5,p¯1,5⟶a+14p¯1,5,q¯2,4⟶1a+34q¯2,4,p¯2,4⟶a+34p¯2,4,q¯3⟶1a+44q¯3,p¯3⟶a+44p¯3,q¯6⟶1a4q¯6,p¯6⟶a4p¯6. Recall that Ω1=Ω5=a+1,Ω2=Ω4=a+3,Ω3=a+4,Ω6=a; that is, in these cases we again have two independent 1 : 1 resonances. In the variables (zj,wj),j=1,…,6, the generators of the normal form arezjwj  and  z1w5,w1z5,z2w4,z4w2.We normalize and find that the normal formH¯=H2+H¯4 up to order 4 is(58)H¯=Ω12z1w1+z5w5+Ω22z2w2+z4w4+Ω32z3w3+Ω62z6w6+b72278Ω12z1w1+z5w52+278Ω22z2w2+z4w42+94Ω32z3w32+94Ω62z6w62+9z3w3z6w6Ω3Ω6+9z3w3Ω1Ω3z1w1+z5w5+9z3w3Ω2Ω3z2w2+z4w4+9z6w6Ω1Ω6z1w1+z5w5+9z6w6Ω2Ω6z2w2+z4w4+98z1w5-w1z5Ω1-z2w4-w2z4Ω22+92Ω1Ω2z1w1z4w4+z2w2z5w5+3z1w1z2w2+3z4w4z5w5-z1w5+w1z5z2w4+w2z4+12z1w5-w1z5z2w4-w2z4. Before returning to the variables (q¯,p¯) we denote(59)F1≔p¯12+q¯12+p¯52+q¯52,F2≔p¯22+q¯22+p¯42+q¯42,I3≔p¯32+q¯32,I6≔p¯62+q¯62.Then the truncated normal form H¯=H2+H¯4 becomes(60)H¯=Ω12F1+Ω22F2+Ω32I3+Ω62I6+b83F128Ω12+3F228Ω22+I324Ω32+I624Ω62+F1F22Ω1Ω2+I3I6Ω3Ω6+I3F1Ω1Ω3+I3F2Ω2Ω3+I6F1Ω1Ω6+I6F2Ω2Ω6-12p¯1q¯5-q¯1p¯5Ω1-p¯2q¯4-q¯2p¯4Ω32+1Ω1Ω2p¯12+q¯12p¯22+q¯22+p¯42+q¯42p¯52+q¯52-2q¯1q¯5+p¯1p¯5q¯2q¯4+p¯2p¯4-p¯1q¯5-q¯1p¯5p¯2q¯4-q¯2p¯4.This normal form is integrable: the independent first integrals areF1,F2,I3,I6,G≔(p¯1q¯5-q¯1p¯5)-(p¯2q¯4-q¯2p¯4)  and  H¯4.However, the construction of all action-angle variables is unclear so far.This finishes the proof of Theorem3.


## 4. Concluding Remarks

This paper presents partial results on integrability of normal forms of the periodic KG lattices.We study the normal forms because the original systems are nonintegrable. This is proven rigorously in the case of two degrees of freedom (Theorem2) and that is one of the differences with the FPU chain. One can carry out the nonintegrability proof for n=3 in the same line, but with more efforts. There is a technical difficulty to carry through that proof in the higher-dimensions, however. The variational equation (VE) does not split nicely and one needs either a tool to deal with higher dimensional (NVE) or another particular solution with the VE along it suitable enough. Nevertheless, we claim that the periodic KG lattice is nonintegrable for all n≥2.The calculation of the normal forms goes in the standard way, because we treat only low dimensions. It is also facilitated by the assumption ana. The result in Theorem 3 allows us in most cases to view the periodic KG Hamiltonian (4) as a perturbation of a nondegenerate Liouville integrable Hamiltonian, namely, the truncated up to order four Birkhoff-Gustavson normal form. One can also verify the other KAM condition, known as Arnold-Moser’s condition [6].As a consequence we have found an integrable KAM nondegenerate normal form of 1 : 2 : 2 Hamiltonian resonance (see Remark5).The outcome of Theorem3 suggests that when one considers the integrability of the normal forms in higher-dimensional periodic KG lattices, certain finite set of values of a even in the domain (3) has to be ruled out initially. For the values of this set there probably exist additional fourth-order resonant relations, the corresponding normal forms are probably more complicated than the generic ones, and, hence, more difficult to study.In any case, the results of this paper serve to understand the lattices with many particles. We intend to study them with the approach of Rink [4], based on using the symmetry properties, which we also enjoy here, to construct suitable normal forms.Finally, we do not address here the symmetric invariant manifolds in the KG lattices because they can be retrieved from [14].



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