Normal forms, computer algebra and a problem of integrability of nonlinear ODEs

Transcription

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

2 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 2

3 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 sixthorder 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. 3

4 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. 4

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10 Euler Poisson Equations 10

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14 Has the system additional local integrals? 14

15 Local Integrals [Lunkevich, Sibirskii, 1982]. Let us see a system It has two stationary points At S 1 it has a center and at S 2 a focus. It is integrable at S 1 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. 15

16 [Lunkevich, Sibirskii, 1982]. 16

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21 C2 R + D2 D4 R C1 C3 (x 0,y 0 ) D1 D5 D3 Fig. 1 21

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28 Normal Form of the Euler Poisson Equations in Resonance Let the normal form be and vector of eigenvalues of the matrix Λ be (6.1) Let also introduce so called resonance variables. After z 1 and z 2 we have 28

29 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. 29

30 In resonance variables we will have the system in the form for odd values of and (6.2) for even values of. G 0, H i,k and F i,k above are linear combination of g r,m, h r.m and f r.m. 30

31 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. 31

32 For the resonance variables, the automorphism can be rewritten as if even, odd. 32

33 Then we have am = bm and the integral is (7.1) If is odd then the integral A will be 33

34 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. 34

35 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) 35

36 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). 36

37 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. 37

38 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 V det i a a a (1) 1 (2) 1 (3) 1 j a a a (1) 2 (2) 2 (3) 2 a 11 ( ) k a a a (1) 3 (2) 3 (3) 3 l a a a (1) 4 (2) 4 (3) 4 = 0 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. 38

39 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 5 rank(m), where M is a matrix (4 x 5) which consists from vectors 39

40 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: 1. Ξ 0 and V = 0, 2. 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. 40

41 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. 41

42 Case of Resonance (1:2) We will use the uniformization then domain corresponds the interval and at δ = 1 we have 42

43 Components of external products will be The system will have only two solutions So at δ = 1 in the interval above 43

44 At δ = -1 Solutions 44

45 So only solution h 4 lies in the mechanical semi-interval But h 4 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 1. Ξ 0 and V = 0, 2. Ξ = 0 and rank(m) < 2, we should look for points where Ξ = 0. 45

46 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 46

47 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 ^ 1 z 4 z 2 z =, =, g ξ 1 ^ g 2 ξ ξ = a, =b, 3 ξ 4 z 4 z 2 2 z2 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]. 47

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51 Finally we opened that ξ 3 = ξ 4 = 0 at for for We found also that at all points above ξ 1 = ξ 2 = 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. 53

52 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 54

53 Case of Resonance (1:3) 55

54 c7=1/9 56

55 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. 57

56 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. 58

57 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) , in Russian. 59

58 Absence of formal integral Absence of local integral Nonintegrability of shortened system Nonintegrability of original system 60