The Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria On an Extension of the 3{Dimensional Toda Lattice Mircea Puta Vienna, Preprint ESI 165 (1994) November 2, 1994 Supported by Federal Ministry of Science and Research, Austria Available via

2 ON AN EXTENSION OF THE 3-DIMENSIONAL TODA LATTICE Mircea Puta International Erwin Schrodinger Institute for Mathematical Physics December 5, 1994 Abstract. An extension of the 3-dimensional nonperiodic Toda lattice is proposed and some of its geometrical and dynamical properties are pointed out. 1. Introduction. After the Flaschka transformation the dynamics of the 3-dimensional nonperiodic Toda lattice is given by the following system of dierential equations: _a 1 = 2b 2 1 _a 2 = 2(b 2 2? b2 1 ) (1.1) _a >< 3 =?2b 2 2 _b 1 = b 1 (a 2? a 1 ) _b 2 = b 2 (a 3? a 2 ) b 1 > 0 >: b 2 > 0 If we drop the last two conditions we obtain a new system of dierential equations which is a slight extension of the 3-dimensional non-periodic Toda lattice, namely: _a 1 = 2b 2 1 (1.2) _a 2 = 2(b >< 2 2? b2 1) _a 3 =?2b 2 2 _b 1 = b 1 (a 2? a 1 ) >: _b 2 = b 2 (a 3? a 2 ) The goal of this paper is to study some of its geometrical and dynamical properties. Supported by the Federal Ministry of Science and Research, Austria. This work was nalized during the author's stay at the Erwin Schrodinger International Institute for Mathematics and Physics in Vienna. I want to express here all my gratitude to the Institute and to Professor Peter Michor in particular, for their support and hospitality Typeset by AMS-TEX

3 2 2. Lax representations. Following Flaschka [1] rewrite (1.2) in the form (2.1) where and Then we can prove: dl = [L; B] = LB? BL; 2 L = 4 a 1 b b 1 a 2 b b 2 a 3 2 0?b B = 4 b 1 0?b b 2 0 Theorem 2.1. (Flaschka) The ow (1.2) is isospectral, i.e. the eigenvalues of L are independent of t. Proof. Let V (t) be the solution of the matrix equation < dv (t) =?B(t) V (t) (2.2) : V (0) = I 3 : Then one can easily check that Indeed, we have successively Hence L(t) = V (t)l(0)v?1 (t): d [V?1 (t)l(0)v (t)] = dv?1 (t) L(t)V (t) + V?1 (t) d [L(t)V (t)] = V?1 (t)b(t)l(t)v (t) + V?1 (t) dl(t) V (t) or the equivalent + V?1 dv (t) (t)l(t) = V?1 (t)b(t)l(t)v (t) + V?1 (t)[l(t); B(t)]V (t) + V?1 (t)l(t)[?b(t)v (t)] = V?1 (t)b(t)l(t)v (t) + V?1 (t)l(t)b(t)v (t)? V?1 (t)b(t)l(t)v (t)? V?1 (t)l(t)b(t)v (t) = 0 V?1 (t)l(t)v (t) = L(0) L(t) = V (t)l(0)v?1 (t) as required. Now it is clear that the eigenvalues of L are independent on t, hence the ow (1.2) is isospectral.

4 3 3. The Lie-Poisson structure. Let ^e = fe 1 = (1; 0; 0; 0; 0); e 2 = (0; 1; 0; 0; 0); e 3 = (0; 0; 1; 0; 0); e 4 = (0; 0; 0; 1; 0); e 5 = (0; 0; 0; 0; 1)g be the canonical basis of R 5. If we dene: (3.1) [e 1 ; e 4 ] =?e 4 ; [e 2 ; e 4 ] = e 4 [e 2 ; e 5 ] =?e 5 ; [e 3 ; e 5 ] = e 5 and all the other brackets are zero, then (R 5 ; [; ]) is a Lie-algebra. It follows that the minus - Lie-Poisson structure on (R 5 ) ' R 5 is given by the matrix: (3.2) u = or equivalently: b ?b 1 b ?b 2?b 1 b 1 b ff; gg? LP ; u ; ; ; ; = 1? ? 2? 1? 1? 2? 2 3 t for each f; g 2 C 1 (R 5 ; R). It follows that (R 5 ; f; g? ) is a Poisson manifold. Now LP we can prove: Theorem 3.1. The system (1.2) may be realized as a Hamilton-Poisson mechanical system with the phase space P = R 5, the Hamiltonian H given by (3.3) H = 1 2 [a2 1 + a a 3 3] + b b 2 2 and the Poisson structure (3.2). Proof. Indeed, u rh = u [a 1 ; a 2 ; a 3 ; 2b 1 ; 2b 2 ] t = [2b 2 1 ; 2(b2 2? b2 1);?2b 2 2 ; b 1(a 2? a 1 ); b 2 (a 3? a 2 )] t = [_a 1 ; _a 2 ; _a 3 ; b _ 1 ; b _ 2 ] t ; as desired.

5 4 Remark 3.1. If we write the above Poisson structure in the following more convenient form: ?b 1?b 1 0?b 2?b 2 0 ff; gg? det 4 for each f; g 2 C 1 (R 5 ; R) then it is easy to see that the function C given by (3.4) C = a 1 + a 2 + a 3 is a Casimir of our conguration, i.e. for each f 2 C 1 (R 5 ; R). fc; fg? LP = 0; 4. Stability and stabilization. It is clear that equilibrium states of the system (1.2) are: (4.1) (; ; ; 0; 0); ; ; 2 R: Consider rst the system (1.2) linearized about the equilibrium (4.1). Its eigenvalues are given by the solutions of (4.2) 3 ( +? )( +? ) = 0: If ; ; 2 R; < or ; ; 2 R; <, or ; ; 2 R; < < at least one root of the equation (4.2) is positive and so the corresponding equilibrium states are unstable. If ; ; 2 R; then the equilibrium states (4.1) are spectrally stable. Are they nonlinear stable? Using the energy-casimir method we can prove the following statement: Theorem 4.1. The equilibrium states (4.3) (; ; ; 0; 0); 2 R are nonlinear stable. Proof. Recall that the energy-casimir method (see e.g. Holm, Marsden, Ratiu and Weinstein [2] or [3]) requires nding a constant of motion for the system, usually the energy H, and a family C of constants of motion such that for some C 2 C; H+C has a critical point at the equilibrium of interest. Often the C's are taken to be Casimir - functions that commute with all other functions under the Poisson bracket. Then in the nite dimensional case, the deniteness of 2 (H + C) at the critical point is sucient to prove stability. Now let us take the modied energy-casimir function: (4.4) H + '(C) = 1 2 [a2 1 + a2 2 + a2 3 ] + b2 1 + b2 2 + '(a 1 + a 2 + a 3 );

6 where ' is a smooth arbitrary real valued function, i.e. ' 2 C 1 (R; R). Then we have: (H + '(C)) = a 1 a 1 + a 2 a 2 + a 3 a 3 + 2b 1 b 1 + 2b 2 b 2 + _'(a 1 + a 2 + a 3 ): At the equilibrium of interest (4.3) the rst variation vanishes if and only if (4.5) _'(3) =? Then 2 (H + '(C)) = (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 + 2(b 1 ) 2 + 2(b 2 ) 2 + '(a 1 + a 2 + a 3 ) 2 : and at the equilibrium of interest (4.3) we have 2 (H + '(C))(; ; ; 0; 0) = (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 + 2(b 1 ) 2 + 2(b 2 ) 2 If we choose ' such that + '(3)(a 1 + a 2 + a 3 ) 2 : '(3) 0; then the second variation at the equilibrium of interest is positive denite and we have nonlinear stability. Remark 4.1. It is an open problem to decide the stability or the instability of the equilibrium states which are not mentioned in our previous considerations. 5. Stabilization by one control. In this section we shall prove that the equilibrium states: (5.1) (; ; 0; 0; 0); 2 R; > 0 (5.2) (0; 0; ; 0; 0); 2 R; < 0 (5.3) (0; ; ; 0; 0); 2 R; < 0 of the system (1.2) may be nonlinear stabilized by a particular linear control applied to the axis 0b 2 [resp. 0b 2, resp. 0b 1 ]. The system (1.2) with one control about the axis 0b 2 can be written in the following form: (5.4) _a 1 = 2b 2 1 _a 2 = 2(b >< 2 2? b2 1) _a 3 =?2b 2 2 _b 1 = b 1 (a 2? a 1 ) >: _b 2 = b 2 (a 3? a 2 ) + u where u 2 C 1 (R 5 ; R). In all that follows we shall employ the feedback: (5.5) u = kb 2 ; where k 2 R is the control parameter. 5

7 6 Theorem 5.1. The controlled system (5.4), (5.5) is a Hamilton-Poisson mechanical system with the phase space P = R 5, the Poisson structure (3.2) and the Hamiltonian H given by: (5.6) H = 1 2 [a2 1 + a2 2 + a2 3 ] + b2 1 + b2 2 + ka 3 Proof. One can easily check that: _a i = fa i ; Hg? ; i = 1; 2; 3 LP _b i = fb i ; Hg? ; i = 1; 2; LP which prove the assertion. Using now the energy-casimir method we can prove: Theorem 5.2. The controlled system (5.4), (5.5) may be nonlinear stabilized about the equilibrium states (5.1) for k =. Proof. Let us take the modied energy-casimir function (5.7) H + '(C) = 1 2 [a2 1 + a a 2 3] + b b ka 3 + '(a 1 + a 2 + a 3 ) where ' 2 C 1 (R; R). Then we have: (H + '(C)) = a 1 a 1 + a 2 a 2 + a 3 a 3 + 2b 1 b 1 + 2b 2 b 2 + ka 3 + _'(a 1 + a 2 + a 3 ) At the equilibrium of interest (5.1) the rst variation vanishes if and only if (5.) _'(2) =? k = Then 2 (H + '(C)) = (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 + 2(b 1 ) 2 + 2(b 2 ) 2 + '(a 1 + a 2 + a 3 ) 2 ; and at the equilibrium of interest (5.1) it is positive denite if we choose ' such that: '(2) 0: Hence the equilibrium states (5.1) are nonlinear stable. If we employ now the feedback: (5.9) u =?kb 2 then we can prove in a similar manner the following results:

8 7 Theorem 5.3. The controlled system (5.4), (5.) is a Hamilton-Poisson mechanical system with the phase space P = R 5, the Poisson structures (3.2) and the Hamiltonian H given by: (5.10) H = 1 2 [a2 1 + a2 2 + a2 3 ] + b2 1 + b2 2 + ka 1 + ka 2 Theorem 5.4. The controlled system (5.4), (5.) may be nonlinear stabilized about the equilibrium states (5.3) for k =. The system (1.2) with one control about the axis 0b 1 can be written in the following form: (5.11) _a 1 = 2b 2 1 Let us employ now the feedback: _a 2 = 2(b >< 2 2? b2 1 ) _a 3 =?2b 2 2 _b 1 = b 1 (a 2? a 1 ) + v >: _b 2 = b 2 (a 3? a 2 ) (5.12) v =?kb 1 where k 2 R is the control parameter. Then we can prove: Theorem 5.5. The controlled system (5.11), (5.12) is a Hamilton-Poisson mechanical system with the phase space P = R 5, the Poisson structure (3.2) and the Hamiltonian H given by: (5.13) H = 1 2 [a2 1 + a2 2 + a2 3 ] + b2 1 + b2 2 + ka 1 + ka 1 : Theorem 5.6. The controlled system (5.11), (5.12) may be nonlinear stabilized about the equilibrium states (5.2) for k =. Remark 5.1. It is easy to see that the equilibrium states: (5.14) (; 0; ; 0; 0); 2 R; 6= 0 are all unstable. 6. Stabilization by two controls. In this section we shall prove that the equilibrium states (6.1) (; 0; 0; 0; 0); 2 R; > 0 of the system (1.2) may be nonlinear stabilized by two controls about the axes 0b 1 and 0b 2.

9 The system (1.2) with two controls about the axes 0b 1 and 0b 2 can be written in the following form: (6.2) Let us employ now the feedback: (6.3) _a 1 = 2b 2 1 _a 2 = 2(b >< 2 2? b2 1) _a 3 =?2b 2 2 where k 2 R is the control parameter. _b 1 = b 1 (a 2? a 1 ) + u >: _b 2 = b 2 (a 3? a 2 ) + v u = kb 1 v? kb 2 Theorem 6.1. The controlled system (6.2), (6.3) is a Hamilton-Poisson mechanical system with the phase space P = R 5, the Poisson structure (3.2) and the Hamiltonian H given by: (6.4) H = 1 2 [a2 1 + a a 2 3] + b b ka 2 + ka 3 : Proof. Indeed, we easily check that which give the result. _a i = fa i ; Hg? ; i = 1; 2; 3 LP _b i = fb i ; Hg? LP ; i = 1; 2 Using now the energy-casimir method we can prove: Theorem 6.2. The controlled system (6.2), (6.3) may be nonlinear stabilized about the equilibrium states (6.1) for k =. Proof. Let us take the modied energy-casimir function (6.5) H + '(C) = 1 2 [a2 1 + a a 2 3] + b b ka 2 + ka 3 + '(a 1 + a 2 + a 3 ); where ' 2 C 1 (R; R). Then we have: (H + '(C)) = a 1 a 1 + a 2 a 2 + a 3 a 3 + 2b 1 b 1 + 2b 2 b 2 + ka 2 + ka 3 + _'(a 1 + a 2 + a 3 ) At the equilibrium of interest (6.1) the rst variation vanishes if and only if (6.6) Then _'() =? k = 2 (H + '(C)) = (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 + 2(b 1 ) 2 + 2(b 2 ) 2 + '(a 1 + a 2 + a 3 ) 2 and at the equilibrium of interest (6.1) it is positive denite if we choose ' such that: '() 0: Therefore the equilibrium states (6.1) are nonlinear stable.

10 9 Remark 6.1. It is easy to see that the equilibrium states are all unstable. (0; ; 0; 0; 0); 2 R; 6= 0 References 1. H. Flaschka, The Toda lattice, I Phys. Rev. B (1974), D. Holm, J. Marsden, T. Ratiu and A. Weinstein, Nonlinear stablility of uid and plasma equilibria, Phys. Rep. 123 (195), M. Puta, Hamiltonian mechanics and geometric quantization, Kluwer Academic Publishers 270 (1993). Seminarul de Geometrie-Topologie, University of Timisoara, Bvd. v. P^arvan 4, Timisoara 1900, Romania

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