Victor Edneral Moscow State University Russia

Normal forms, computer algebra and a problem of integrability of nonlinear ODEs
Victor Edneral Moscow State University Russia Joint work with Alexander Bruno

CADE-2007, February 20-23, Turku, Finland
Introduction Normal Form of a Nonlinear System Euler – Poisson Equations Normal Form of the Euler – Poisson Equations in Resonance Structure of Integrals of the System Necessary Conditions for Existence of Additional Integrals Calculation of the Normal Form of the Euler – Poisson Equations Conclusions CADE-2007, February 20-23, Turku, Finland

Introduction Here, we study the connection between coefficients of normal forms and integrability of the system. For this purpose, we compute normal forms of the Euler – Poisson equations, which describe the motion of a rigid body with a fixed point. This is an autonomous sixth-order system. A lot of books and papers are devoted to integrable systems and to methods for searching for such systems. A.D. Bruno noted that all normal forms of integrable systems are degenerated, so it is interesting to search domains with a such degeneration [Bruno, 2005]. The first attempt to calculate the normal form of the Euler – Poisson system was made in [Starzhinsky, 1977]. However, without computer algebra tools, he was unable to calculate a sufficient number of terms. We use a program for analytical computation of the normal form [V. Edneral, R. Khanin, 2003]. This is a modification of the LISP based package NORT for the MATHEMATICA system. The NORT package [Edneral, 1998] has been designed for the REDUCE system. CADE-2007, February 20-23, Turku, Finland

Normal Form of a Nonlinear System
Consider the system of order n (1.1) in a neighborhood of the stationary point X = 0 under the assumption that the vector function Φ(X) is analytical at the point X = 0 and its Taylor expansion contains no constant and linear terms. CADE-2007, February 20-23, Turku, Finland

Euler – Poisson Equations
CADE-2007, February 20-23, Turku, Finland

Has the system additional local integrals? CADE-2007, February 20-23, Turku, Finland

Local Integrals [Lunkevich, Sibirskii, 1982]. Let us see a system It has two stationary points At S1 it has a center and at S2 – a focus. It is integrable at S1 with the integral You can see that is an invariant line. The system is integrable in semi plane x > -1/2 and not integrable at x < -1/2. CADE-2007, February 20-23, Turku, Finland

[Lunkevich, Sibirskii, 1982]. CADE-2007, February 20-23, Turku, Finland

(x0,y0) D1 D5 D3 Fig. 1 CADE-2007, February 20-23, Turku, Finland

Normal Form of the Euler – Poisson Equations in Resonance
Let the normal form be (6.1) and vector of eigenvalues of the matrix Λ be Let also introduce so called resonance variables. After z1 and z2 we have CADE-2007, February 20-23, Turku, Finland

Lemm 1 [Bruno, 2005]. At the resonance in the normal form where are power series in At this start from free terms but – from linear terms. CADE-2007, February 20-23, Turku, Finland

In resonance variables we will have the system in the form for odd values of (6.2) and for even values of . G0, Hi,k and Fi,k above are linear combination of gr,m, hr.m and fr.m. CADE-2007, February 20-23, Turku, Finland

Structure of Integrals of the System
As it was shown in [Bruno, 1995] an expansion of the first integral of normal form contains only resonance variables with the property Thus the first integral can be written as power series where a0, am and bm are power series in z1, z2, ρ1, ρ2. CADE-2007, February 20-23, Turku, Finland

For the resonance variables, the automorphism can be rewritten as even, if odd. CADE-2007, February 20-23, Turku, Finland

Then we have am = bm and the integral is (7.1) If is odd then the integral A will be CADE-2007, February 20-23, Turku, Finland

Necessary Conditions for Existence of Additional Integrals
From the definition, the derivation in time of any first integral along the system should be zero, i.e. CADE-2007, February 20-23, Turku, Finland

The identity above should be discussed at odd and even values of separately. Corresponding coefficients in formulae below will be slight different. The lowest non vanished coefficients will be different also. It is very important for an estimation of order up to which we a need to calculate the normal form. If we parameterize the identity above up to the common order in z1,2, ρ1,2 variables smaller than 2( ) for the odd value and up to for the even one, we will see that the identity should be right for free and linear in the common order (7.2) CADE-2007, February 20-23, Turku, Finland

So, if we write o(z1,2,ρ1,2), + o(z1,2,ρ1,2), (7.3) o(z1,2,ρ1,2), O(z1,2,ρ1,2). CADE-2007, February 20-23, Turku, Finland

then the equation for the free term has the form (A) Here the vector Ξ ≡ {ξi} can be calculated from the normal form. It will be a function of parameters of the system. The vector α ≡ {αi} defines a0. If you know the first integrals of the system, you can calculate the corresponding α ≡ {αi} for each integral separately. If Ξ ≠ 0 then equation (A) has three dimensional set of solutions α, so only three integrals can be independent. CADE-2007, February 20-23, Turku, Finland

A single possibility to have an additional integral in this case is that the tree known integrals are dependent each from other. Mathematically it can be written as the vector equality Because for checking this condition we need calculate only the lowest orders of the normal form it is possible to calculate it in analytical form in variables of the system. CADE-2007, February 20-23, Turku, Finland

The dimension of solutions (α, β) of the system above is
If Ξ ≡ 0 then from (5.2), (5.3) we will have the condition (B) η and ζ are coefficients which can be calculated from coefficients of the normal form as functions of parameters. The dimension of solutions (α, β) of the system above is where M is a matrix (4 x 5) which consists from vectors CADE-2007, February 20-23, Turku, Finland

Let us say that formal integral (7.1) is local independent on known integrals if its linear approximation in z, ρ, ω is linear independent from first approximations of known integrals. Main theorem [Bruno, 2005]. For existence of an additional formal integral at the family of the stationary point Fδ, it is necessary a satisfaction of one of two sets of conditions: Ξ ≠ 0 and V = 0, Ξ = 0 and rank(M) < 2. Note, that if the original system has five first integrals, then right hand side of normalized equation is linear, and rank(M) = 0. CADE-2007, February 20-23, Turku, Finland

Calculation of the Normal Form of the Euler – Poisson Equations
Near stationary points of families Sσ we computed normal forms of the System up to terms of some order m For that, we used the program [Edneral, Khanin, 2003]. All calculations were lead in rational arithmetic and float point numbers in this paper are approximations of exact results. CADE-2007, February 20-23, Turku, Finland

Case of Resonance (1:2) We will use the uniformization then domain corresponds the interval and at δ = 1 we have CADE-2007, February 20-23, Turku, Finland

Components of external products will be The system will have only two solutions So at δ = 1 in the interval above CADE-2007, February 20-23, Turku, Finland

At δ = -1 Solutions CADE-2007, February 20-23, Turku, Finland

So only solution h4 lies in the mechanical semi-interval But h4 is a special point with all zero eigenvalues and we can conclude that With respect of the Main theorem of existence of an additional integral Ξ ≠ 0 and V = 0, Ξ = 0 and rank(M) < 2, we should look for points where Ξ = 0. CADE-2007, February 20-23, Turku, Finland

Due to automorphism (5.1) and to Property 1, the normal form has corresponding automorphism and the sum k ≡ q3 + q4 + q5 + q6 is even for all its terms. We considered sums For the normal form, it occurs that, for m = q1 + q2 + k = 4 we will have CADE-2007, February 20-23, Turku, Finland

and all lower terms cancel. Here, the quantities with a hat ĝk, k = 1, …, 6, denote the normal forms calculated up to order four. It can be demonstrated that the vector Ξ has a components So we can calculate Ξ now. Coefficients a and b depend on δ2 and c. For δ2 = 1, both coefficients a and b are pure imaginary. For δ2 = –1, they are pure imaginary if c (0, c2) and are real if c (c2, 2]. CADE-2007, February 20-23, Turku, Finland

Finally we opened that ξ3 = ξ4 = 0 at for for We found also that at all points above ξ1 = ξ2 = 0. Thus we satisfy case 2 of the main theorem about the necessary condition of local integrability, if the rank of matrix M is smaller then 2. CADE-2007, February 20-23, Turku, Finland

We calculated the matrix M and its rank at the all points where Ξ = 0. Particularly at c = ½ the matrix M is for for CADE-2007, February 20-23, Turku, Finland

Case of Resonance (1:3) CADE-2007, February 20-23, Turku, Finland

We calculated rank(M) at the points above and opened, that it is equal to 2 in all cases except point c = ½, where this rank is equal 1 and point c = 1, where it has a zero value. CADE-2007, February 20-23, Turku, Finland

Conclusions The normalized system of Euler – Poisson has no additional formal integrals at the mechanical values of the parameter c in resonances (1:2) and (1:3), i.e., it is nonintegrable, except known cases c = ½ and c = 1. We have a new workable approach for searching additional formal integrals. We have a new method for a proof of nonintegrability in some domains. CADE-2007, February 20-23, Turku, Finland

References Bruno, A.D.: Theory of normal forms of the Euler – Poisson equations. Preprint of the Keldysh Institute of Applied Mathematics of RAS No 100. Moscow (2005). Lunkevich, V.A,, Sibirskii, K.S., Integrals of General Differential System at the Case of Center. Differential Equation, v. 18, No 5 (1982) 786–792, in Russian. CADE-2007, February 20-23, Turku, Finland

Absence of formal integral
Absence of local integral Nonintegrability of shortened system Nonintegrability of original system CADE-2007, February 20-23, Turku, Finland

Victor Edneral Moscow State University Russia

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