Recent Progress in the Integration of Poisson Systems via the Mid Point Rule and Runge Kutta Algorithm

Transcription

1 Recent Progress in te Integration of Poisson Systems via te Mid Point Rule and Runge Kutta Algoritm Klaus Bucner, Mircea Craioveanu and Mircea Puta Abstract Some recent progress in te integration of Poisson systems via te mid point rule and Runge Kutta algoritm are discussed and some of teir properties are pointed out Matematics Subject Classification: 70H05, 70K5 Key words: Poisson manifold, mid point rule, Runge Kutta approximation 1 Introduction In te last time many dynamical systems ave been found to be Hamilton Poisson systems Tese include te Euler equations for te free rigid body, te Maxwell Vlasov equations from plasma pysics, te Maxwell Bloc equations from laser matter dynamics and oters, see for details [4] and [6] It is an interesting and very tempting problem to try to numerically integrate tese Hamilton Poisson systems suc tat te corresponding algoritms to preserve as muc as possible from teir Poisson pictures Te goal of our paper is to present some recent progress in te integration of Poisson systems via te mid point rule and Runge Kutta algoritm and to point out some of teir properties Hamilton Poisson systems Let P be a smoot n dimensional manifold and C (P, R) te space of smoot (= C ) real valued functions defined on P Consider a given bracket operation denoted {, }: C (P, R) C (P, R) C (P, R) Definition 1 Te pair (P, {, }) is called a Poisson manifold if {, } satisfies: (PB1) bilinearity, ie {, } is R bilinear (PB) anticommutativity, ie {f, g} = {g, f} for every f, g C (P, R) (PB3) Jacobi s identity, ie {f, {g, }} + {, {f, g}} + {g, {, f}} = 0 for every f, g, C (P, R) Balkan Journal of Geometry and Its Applications, Vol1, No, 1996, pp 9-19 c Balkan Society of Geometers, Geometry Balkan Press

2 10 KBucner, MCraioveanu and MPuta (PB4) Leibniz rule, ie {fg, } = f{g, } + g{f, } for every f, g, C (P, R) Conditions (PB1) (PB3) make (C (P, R), {, }) into a Lie algebra and moreover it is not ard to see tat every Poisson manifold is essentially a union of symplectic manifolds wic fit togeter in a smoot way If (P, {, }) is a Poisson manifold, ten because of (PB1) and (PB4), tere is a tensor field B on P, assigning to eac x P a linear map suc tat B(x): T x P T x P {f, g}(x) = B(x) df(x), dg(x) Here, denotes te natural pairing between vectors and covectors Because of (PB), B(x) is antisymmetric Letting x i, i = 1,,, n, denote local coordinates on P we ave: ij f {f, g} = B x i g x j Definition Let (P 1, {, } 1 ) and (P, {, } ) be Poisson manifolds A mapping φ: P 1 P is called Poisson if for all f, g C (P, R) we ave or equivalently {f, g} φ = {f φ, g φ} 1, JB 1 J T = B, were J is te Jacobian matrix associated to φ Locally te above relation can be written in te following form: B ij 1 (x) yk x i yl x j = Bkl (x) Definition 3 An Hamilton Poisson system is a triple (P, {, }, H), were (P, {, }) is a Poisson manifold and H is a smoot real valued function defined on P called te Hamiltonian or te energy Its dynamics is described by te integral curves of te Hamiltonian vector field X H defined by or locally X H (f) = {H, f}, ẋ i = {x i, H}, i = 1,,, n Moreover a Casimir of our configuration is a smoot function C C (P, R) suc tat {C, f} = 0 for eac f C (P, R) Example 1 (te free rigid body) Te Euler angular momentum equations of te free rigid body are written in te following form (1) ṁ 1 = a 1 m m 3 ṁ = a m 1 m 3 ṁ 3 = a 3 m 1 m,

3 Recent Progress in te Integration of Poisson Systems 11 were a 1 = 1 I 3 1 I, a = 1 I 1 1 I 3, a 3 = 1 I 1 I 1, I 1, I, I 3 being te components of te inertia tensor and we suppose as usually tat I 1 > I > I 3 Te free rigid body is a Hamilton Poisson system wit te pase space P = R 3, te Poisson bracket given by () {f, g} RB (m) = m ( f g) and te Hamiltonian H defined by (3) H(m 1, m, m 3 ) = 1 ( ) m 1 + m + m 3 I 1 I I 3 Moreover, a Casimir of our configuration (R 3, {, } RB ) is given by te function (4) C(m 1, m, m 3 ) = 1 (m 1 + m + m 3) It follows tat te trajectories of motion are intersections of te ellipsoids wit te speres H = constant C = constant Tere is also a very nice interpretation of te rigid body bracket () namely, te rigid body bracket () is te minus Lie Poisson structure on so(3) Indeed, let SO(3) be te Lie group of all linear orientation preserving ortogonal transformations of R 3 to itself Its Lie algebra so(3) is te set of all 3 3 skew symmetric matrices It can be canonically identified wit R 3 via te map : v = p q r R 3 v = r 0 p 0 r q q p 0 so(3) Ten te Lie bracket on so(3) corresponds to te cross product on R 3 in te sense tat [ v, ŵ] = v w Te dual of so(3), ie, so(3) can be also identified wit R 3 and ten te minus Lie Poisson structure on so(3) is given by te matrix Π RB = 0 m 3 m m 3 0 m 1, m m 1 0 wic is noting else tan te Rigid Body bracket () as easily can be verified Example (te Maxwell Bloc equations) Te 3 dimensional real valued Maxwell Bloc equations from laser matter dynamics are usually written as

4 1 KBucner, MCraioveanu and MPuta (5) ẋ 1 = x ẋ = x 1 x 3 ẋ 3 = x 1 x Tey ave a Hamilton Poisson realization wit te pase space P = (R 3 [, ] ), were R 3 [, ] is te Lie algebra R3 wit te bracket operation given by [e 1, e ] = e 3, [e 1, e 3 ] = e, [e, e 3 ] = 0, {e 1, e, e 3 } being te canonical basis of R 3, te minus Lie Poisson structure given by te matrix (6) Π MB = 0 x 3 x x x 0 0 and te Hamiltonian H defined by: (7) H(x 1, x, x 3 ) = 1 x 1 + x 3 Moreover, a Casimir of our configuration is given by (8) C(x 1, x, x 3 ) = 1 (x + x 3) Teorem 1 ([4], [6]) Let (P, {, }, H) be a Hamilton Poisson system and φ t te flow for X H Ten we ave: (i) Eac φ t is a Poisson map (ii) φ t preserves H for eac t R (iii) Eac φ t preserves te symplectic leaves of te Poisson manifold (P, {, }) 3 Mid point rule Let us start wit a Hamilton Poisson system wit te pase space P = R n, te Poisson structure given by te matrix Π and te Hamiltonian H Its dynamics can be written in te following form (31) ẋ = Π(x) H(x) We are interested in te numerical integration of tis system via te mid point rule It is an implicit recurrence given in our case by (3) x k+1 x k ( x k + x k+1 ) = Π H ( x k + x k+1 were is te size step (or time step) If is small enoug, ten (3) defines a diffeomorpism φ H via (33) x k+1 = φ H(x k ) ),

5 Recent Progress in te Integration of Poisson Systems 13 We compute te Frécet derivative Dφ H (x) as follows By definition, y = φ H (x) is te unique solution of te implicit equation (34) F (x, y) def = y x Π Differentiating F (x, φ H (x)) = 0 gives: ( x + y ) H (35) D 1 F + D F Dφ H = 0, ( ) x + y = 0 were D i F, i = 1,, denote te partial Frécet derivatives For small enoug, D F as an inverse, and (35) may be written as (36) Dφ H = (D F ) 1 (D 1 F ) and For te particular case Π(x) = Π = constant it is easy to see tat D 1 F = I n ( ) x + φ ΠH H (x) xx Letting D F = I n ΠH xx ( x + φ H (x) ( ) Q(x) def x + φ = H H (x) xx denote te symmetric Hessian matrix, we obtain (37) Dφ H(x) = [I n ] 1 [I ΠQ(x) n + ] ΠQ(x) Now, following Austin, Krisnaprasad and Wang [1], we can prove: Teorem 31 (Wang [11]) If Π(x) = Π = constant, ten te mid point integrator (31) is a Poisson one Proof We need to sow tat Dφ H(x)Π[Dφ H(x)] T = Π ) Based on te above calculations, tis reduces to sowing tat [(I n ΠQ(x)) 1 (I n + ΠQ(x))]Π[(I n ΠQ(x)) 1 (I n + ΠQ(x))]T = Π or equivalently [I n + ΠQ(x)]Π[I n + ΠQ(x)]T = [I n ΠQ(x)]Π[I n ΠQ(x)]T Tis follows from te fact tat Π T = Π and Q(x) T = Q(x) Indeed, [I n + ΠQ(x)]Π[I n + ΠQ(x)]T = [I n + ΠQ(x)][I n ΠQ(x)] = Π ΠQ(x)Π+

6 14 KBucner, MCraioveanu and MPuta + ΠQ(x)Π ΠQ(x)ΠQ(x)Π = Π 4 4 ΠQ(x)ΠQ(x)ΠQ(x)Π, and similarly [I n ΠQ(x)]Π[I n ΠQ(x)]T = [I n ΠQ(x)][Π + ΠQ(x)Π] = = Π + ΠQ(x)Π as required In te particular case wen ΠQ(x)Π ΠQ(x)ΠQ(x)Π = Π 4 4 ΠQ(x)ΠQ(x)Π, Π = [ ] 0n I n I n 0 n we again come upon Feng s teorem, namely Teorem 3 (Feng []) Te mid point integrator is a symplectic one Wen Π(x) is not a constant te mid-point rule is not in general a Poisson integrator Example 31 (Puta and Birtea [8]) Let us take te case of symmetric rigid body, ie I = I 3 Its dynamics is described by te equations: (38) ṁ 1 = 0 ṁ = a m 1 m 3 ṁ 3 = a m 1 m Ten te mid point rule takes te following form 1 = m k 1 (39) = 4mk + 4a m k 1m k 3 a (m k 1) m k 4 + a (mk 1 ) 3 = 4mk 3 4a m k 1m k a (m k 1) m k a (mk 1 ) A straigtforward computation sows us tat te algoritm (39) is not of Poisson type More precisely, it is of Poisson type if and only if = 0 Let F be a first integral of (31), ie or equivalently F = 0, (310) ( F ) T Π H = 0 Assuming F is also tree times differentiable, ten by Taylor s formula we can expand F around x k as: F (x k+1 ) = F (x k ) + ( F (x k )) T (x k+1 x k ) + 1 D F (x k )(x k+1 x k )(x k+1 x k )

7 Recent Progress in te Integration of Poisson Systems 15 (311) D3 F (x k )(x k+1 x k )(x k+1 x k )(x k+1 x k ) + O( x k+1 x k 4 ) It can be cecked tat wen te mid point rule (3) is plugged into (311) we get (31) F (x k+1 F (x k ) = 1 4 D3 F (x k )u u u + O( 4 ), were ( x k+1 + x k ) u = Π H ( x k+1 + x k Equation (31) is an error formula dues to Austin, Krisnaprasad and Wang [1] for conserved quantities of (3), wic contains only tird or iger order terms It follows tat te mid point rule (3) preserves exactly and conserved quantity aving only linear and quadratic terms, including Casimir functions and te Hamiltonian of (31) So we ave proved: Teorem 33 (Austin, Krisnaprasad and Wang [1]) Te mid point integrator (3) conserves all Casimir functions and te Hamiltonian H of (31) if tey contain only linear and quadratic terms Example 3 In te case of te free rigid body (see Example 1), te mid point rule can be written in te following form (313) 1 m k 1 m k 3 m k 3 = a 1 mk+1 + m k = a mk m k 1 = a 3 mk m k 1 ) mk m k 3 mk m k 3 + m k mk+1 Given m k 1, m k, m k 3, equations (313) are solved for 1,, 3 It follows via te above teorem tat tis integrator preserves bot H and C given respectively by (3) and (4), but doesn t preserve te Poisson structure () Example 33 In te case of 3 dimensional real valued Maxwell Bloc equations (see Example ), te mid point rule can be written in te following form (314) x k+1 1 x k 1 x k+1 x k x k+1 3 x k 3 = xk+1 + x k = xk x k 1 = xk x k 1 xk x k 3 + x k xk+1 Given x k 1, x k, x k 3, equations (314) are solved for x k+1 1, x k+1, x k+1 3 It follows via Teorem 33 tat tis integrator preserves bot H and C, given respectively by (7) and (8)

8 16 KBucner, MCraioveanu and MPuta 4 Runge Kutta algoritm Let us start wit te system of differential equations (41) ẋ = f(x), x R n Ten te s stage Runge Kutta algoritm can be written in te following form x k+1 = x k + b i f(y i ) i=1 (4) y i = x k + a ij f(y j ), were 1 i s If our system (41) is Hamiltonian, ie, it can be put in te equivalent form j=1 (43) { qi = H p i ṗ i = H q i, i = 1,,, n, ten we ave te following result proved independently by Lasagni [3], Sanz Serna [9] and Suris [10] Teorem 41 (Lasagni, Sanz Serna and Suris) Assume tat te coefficients of te s stage Runge Kutta algoritm satisfy te relations b i a ij + b j a ji b i b j = 0, 1 i, j s Ten te integrator (41) is a symplectic (so a Poisson) one Proof We sall sketc te proof following Sanz Serna [9] For beginning let us write te relations (4) for our particular system (43) We get p k+1 = p k + b i f(p i, Q i ), (44) i=1 q k+1 = q k + b i g(p i, Q i ), i=1 (45) P i = p k + Q i = q k + a ij f(p j, Q j ), j=1 a ij g(p j, Q j ), j=1 were f and g respectively denote te vectors wit components H/ q i and H/ p i We employ also te notation r i = f(p i, Q i ) and l i = g(p i, Q i ) for te slope of te stages Differentiate (44) and form te exterior product to arrive at dp k+1 dq k+1 = dp k dq k + b i dr i dq k + b j dp k dl j + b i b j dr i dl j i=1 j=1 i,j=1

9 Recent Progress in te Integration of Poisson Systems 17 Our next step is to eliminate dr i dq k and dp k dl j from tis expression Tis is easily acieved by differentiating (45) and taking te exterior product of te result wit dr i, dl j Te outcome of te elimination is dp k+1 dq k+1 dp k dq k = b i [dr i dq i + dp i dl i ] i=1 (b i a ij + b j a ji b i b j )dr i dl j i,j=1 Te second term in te rigt and side vanises by ypotesis To finis te proof is ten sufficient to sow tat, for eac i, dr i dq i + dp i dl i = 0 In fact, dropping te subscript i tat numbers te stages, we can write = n µ,ν=1 dr dq + dp dl = n (dr µ dq µ + dp µ dl µ ) = µ=1 ( fµ dp ν dq ν + f µ dq ν dq µ + g µ dp µ dp ν + g ) µ dp µ dq ν p ν q ν p ν q ν To see tat tis expression vanises, express f µ and g ν as derivatives of H and recall te skew symmetry of te exterior product If te system (41) is of Poisson type, ie it is equivalent to ẋ = Π H, qed ten we ave: Teorem 4 (McLaclan [5]) If Π is constant, ten te s stage Runge Kutta integrator is of Poisson type Proof It is known tat s stage Runge Kutta algoritm is invariant under linear maps, tat is, canging variables in te map or in te vector field results in te same Runge Kutta map s stage Runge Kutta for te Poisson system is, terefore, equivalent to s stage Runge Kutta for te system in canonical form wit Poisson tensor given by te matrix: 0 I n 0 I n n For tis system, s stage Runge Kutta algoritm leaves te last n variables fixed, so it is equivalent to s stage Runge Kutta for a Hamiltonian system in te first m variables, for wic it is a symplectic map and so a Poisson map Tus, s stage Runge Kutta for te original system preserves te symplectic leaves and is symplectic on tem as required

10 18 KBucner, MCraioveanu and MPuta qed If te matrix Π is not constant, ten s stage Runge Kutta algoritm is not in general a Poisson one Example 41 (Puta [7]) Let us consider te Hamilton Poisson system ( R, Π = [ 0 x x 0 ] ), H(x 1, x ) = Ax 1 + Bx + C, A 0 Ten te 1 stage Runge Kutta algoritm is not of Poisson type Indeed, te dynamics of our system is given by { ẋ1 = Bx ẋ = Ax Ten te 1 stage Runge Kutta algoritm wit size step is given by x k+1 1 = x k 1 + bb 1 + aa xk (46) ( x k+1 = 1 ba ) 1 + aa Now, an easy computation sows us tat it doesn t preserve te Poisson tensor Π Let us mention also tat te algoritm (46) is also not energy preserving Moreover te following assertions are equivalent: (i) (46) is a Poisson integrator; (ii) (46) is an energy integrator; (iii) A = 0 References [1] M A Austin, P S Krisnaprasad, L S Wang, Almost Poisson integration of rigid body systems, Journ of Computational Pysics, vol 107 No 1(1993) [] K Feng, Lecture Notes in Numerical Metods in Partial Differential Equations, Springer Verlag, New York/Berlin, 1987 [3] F Lasagni, Canonical Runge Kutta metods, ZAMP 39, 1988, [4] J Marsden, Lectures on Mecanics, London Matematical Society, Lecture Notes Series, 174, Cambridge University Press 199 [5] R McLaclan, Comment on Poisson scemes for Hamiltonian systems on Poisson manifolds, Computers Mat Applic Vol 9 No 3, 1, 1995 [6] M Puta, Hamiltonian systems and geometric quantization, Mat and its Applic vol 60, Kluwer, 1993 [7] M Puta, Poisson integrators, An Univ Timişoara, vol 31, Fasc (1993), 67-73

11 Recent Progress in te Integration of Poisson Systems 19 [8] M Puta, P Birtea, Gauss Legendre algoritm and te symmetric rigid body, Proceedings of 4 t National Conference of Geometry and Topology, Timişoara, Romania, July 5-9, 1994, Part two Communications, [9] J M Sanz Serna, Runge Kutta scemes for Hamiltonian systems, BIT 8(1988) [10] B Y Suris, Integrable mappings of te standard type, Funct Anal Appl 3(1)(1989), [11] D L Wang, Symplectic Difference Scemes for Hamiltonian Systems on Poisson Manifolds, Computing Center, Academia Sinica, Beijing, Cina 1993 Matematisces Institut der Tecniscen Universität Müncen, Arcisstrasse 1, 8090 Müncen, Germany West University of Timişoara, Department of Matematics, B dul VPârvan 4, 1900 Timişoara, Romania