can be applied in approximately solving the transormed equation. Further, smoothness o the transormation map ensures the correct order o the numerical

Transcription

1 On the construction o geometric integrators in the RKK class Kenth Eng Department o Inormatics University o Bergen N-5020 Bergen Norway October 2, 1998 Abstract We consider the construction o geometric integrators in the class o RKK methods. Any dierential equation in the orm o an innitesimal generator on a homogeneous space is shown to be locally equivalent to a dierential equation on the Lie algebra corresponding to the Lie group acting on the homogenous space. This way we obtain a distinction between the coordinate-ree phrasing o the dierential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group maniold. By choosing the coordinates to be canonical coordinates o the rst kind we obtain the original method o unthe-kaas [14]. ethods similar to the RKK method are developed based on the dierent coordinatizations o the Lie group maniold, given by the Cayley transorm, diagonal Pade approximants o the exponential map, canonical coordinates o the second kind, etc. Some numerical experiments are also given. 1 Introduction In the past ew years there has been a substantial development in the area o geometric integration o dierential equations evolving on Lie groups and more generally homogeneous spaces. The present investigation was stimulated by a paper o unthe-kaas [14], where Runge-Kutta methods o arbitrary high order on homogeneous spaces are introduced. The main idea is to transorm the dierential equation evolving on the homogeneous space to an equivalent dierential equation evolving on a Lie algebra. An appealing eature o Lie algebras is that they are vector spaces. Hence, a classical Runge-Kutta method Kenth.Engo@ii.uib.no, WWW: 1

2 can be applied in approximately solving the transormed equation. Further, smoothness o the transormation map ensures the correct order o the numerical approximation on the homogenous space. The exponetial map rom the Lie algebra to the Lie group is o crucial importance in many o the geometric integration techniques developed. For a moment let us emphasize the maniold structure o Lie groups. Every smooth maniold is modeled by the use o smooth coordinate maps rom an open subset o the maniold to some Banach space. For a Lie group this model space is the Lie algebra and the exponential map is called the canonical coordinates o the rst kind. A point to make is that a maniold can have dierent types o coordinates. Another example o canonical coordinates are coordinates o the second kind. From general Lie theory it is known that every Lie group can be coordinatized by the canonical coordinates o the rst and second kind [19]. Thus, to develop methods appropriate or all Lie groups, coordinatizations based on these two types o coordinates are among the natural choices. For some Lie groups, however, there do exist coordinates other than the canonical coordinates o the rst and second kind. An example is the Cayley transorm. Cayley coordinatizations exist, or instance, or the matricial Lie groups O(n), SO(n), and Sp(n), the orthogonal, the special orthogonal, and the symplectic group, respectively. To develop integration techniques based on the Cayley transorm would entail savings with respect to computational costs because o the lesser expense in computing the Cayley transorm as compared to the exponential map. Introducing local geometric integration techniques, similar to the RKK method, based on dierent coordinatizations o the Lie group is the main goal o this paper. In [4], Celledoni and Iserles establishes the necessary conditions or a map analytic in a neighborhood around 0 o the Lie algebra, with (0) = 1 and 0 (0) = 1, to be a map rom the Lie algebra to the Lie group in the case o the quadratic Lie groups. Diagonal Pade approximants o the exponential unction are analytic unctions that meet these conditions, and hence, are natural candidates as coordinates on the Lie group maniold. Thus, in the case o the quadratic Lie groups, geometric integration methods based on diagonal Pade approximants as coordinates should be considered. There has also been done some work on developing sotware based on these new classes o integration techniques. It turns out that the new methods are well suited or the use o object oriented programming. All the geometric integrators are stated in a coordinate ree language, that is; the methods are stated independently o the actual representation o the elements o the maniold. The algorithm is stated once and or all, in an abstract ashion, regardless o i the algorithm is to handle scalars, vectors, matrices, or even more abstract geometric objects. Traditionally, object orientation is used in all levels o the algorithm itsel, but the new algorithms allow or the object orientation to be employed on the geometry instead. Developing mathematical sotware this way mimics the mathematics in a truly beautiul way, and the mathematics itsel is the guiding star or how the sotware is developed. Dian isaatlab toolbox designed according to this philosophy, which implements many o the methods developed 2

3 in this area o numerical ODE solvers. The interested reader is recommended to visit: the home page o Dian. The paper is organized as ollows. Section 2 reviews all the necessary theory about actions and equivariant maps between homogeneous spaces. In Section 3 we consider the innitesimal description o actions, relatedness o vector elds and develop the main theorem rom which we iner the numerical algorithms. Section 4 presents dierent numerical algorithms based on dierent coordinates and some numerical examples. Finally, some concluding remarks. 2 Actions on maniolds and equivariant maps We assume the reader to be amiliar with the basic concepts o maniold, Lie group and Lie algebra. Two good reerences on this material - among a wealth o others - are Section 4.1 o [1] and Chapter 9 in [13]. Throughout this paper we will tacitly assume that every maniold is a dierentiable maniold. Actions o Lie groups on maniolds are o crucial importance to us. The action o the Lie group G is the tool or moving around on the maniold. We start with the standard denition o a Lie group action on a maniold. Denition 2.1 (Lie group action) Let be a maniold and let G be a Lie group. A (let) action o a Lie group G on is a smooth mapping :G! such that: i) (e; x) =xor all x 2; and ii) (g; (h; x)) = (gh; x) or all g; h 2 G and x 2. For every g 2 G let g :!denote the smooth map given by x 7! (g; x), and or every m 2 let m : G! denote the smooth map g 7! (g; m). This notation might cause some conusion at rst. Remember that the map is dened by the properties o the subindex element, so keep in mind the type o this element. With this notation Denition 2.1 is equivalent to the ollowing denition: Denition 2.2 (Lie group action { alternative denition) Let, G and be as in Denition 2.1. A (let) action o G on is a map such that g 7! g is a group homomorphism o G into Di(). Di() being the group o dieomorphisms o. In other words; i) e = Id; and ii) gh = g h or all g; h 2 G. Further, one can classiy actions on maniolds as transitive, i or every pair x; y 2there exists at least one element g 2 G such that g (x) =y. 3

4 Denition 2.3 (Homogeneous space) A maniold with a transitive Lie group action is called a homogeneous space. That is; the triple (;G;) constitutes the homogeneous space. In this paper we will exclusively consider ordinary dierential equations evolving on homogeneous spaces. Considering only this type o domain should by no means be considered as a restriction, as most domains used in mathematical modeling are included in the category o homogeneous spaces. We now turn our attention to a particular class o smooth maps between homogeneous spaces. Denition 2.4 (Equivariant map) Let and N be maniolds and let G be a Lie group which acts on by g :!and on N by g : N!N. A smooth map :!N is called equivariant with respect to these actions i, or all g 2 G, that is, the ollowing diagram commutes: g = g; (1),,,,! N g g,,,,! Loosely speaking, the eects o the two actions on and N are related through the equivariant map. To study the action o G on N we can instead study the action o G on. As we will see, the concept o an equivariant map is very important inthedevelopment onumerical algorithms. The idea is to choose the maniold tobeavector space which supports the classical Runge-Kutta method. Instead o studying the action o G on some non-linear homogeneous space N,we nd an appropriate equivariant map that `transorms' us to an equivalent action o G on our vector space. As the st step to accomplish this we establish the equivariance o p or p 2. Remember that p is a map rom G to. Thus, any action on a homogeneous space can be equally well studied as an action o the Lie group on itsel. Dene L g to be the let translation map on G, that is L g (h) =gh, 8h 2 G. Lemma 2.5 (Equivariance o p ) Given a let action : G! Di() o G on. Then or all p 2 the smooth map p : G!is equivariant with respect to the action L g o G on itsel, and the action g o G on, g 2 G: N p L g = g p 4

5 That is; the ollowing diagram commutes: G L g,,,,! p G,,,,! p The proo o this is a mere consequence o the denition o an action, see Denition 2.2. In the case o a right action, L g must be substituted with R g, the right translation map on G. To be able to apply classical Runge-Kutta methods, the domain space ought to be a vector space. The Lie algebra o a Lie group is a vector space with the additional structure o a commutator. Hence, the Lie algebra o the Lie group acting on the homogeneous space is the natural choice o space or our Runge-Kutta method. Let : g! G be a local coordinate map on G, hence is smooth and invertible. Our goal is to nd an action o G on g such that will be an equivariant map with respect to this action o G on g and the let action o G on itsel. g Lemma 2.6 (The action B:G!Di(g)) For g 2 G denote the let action o G on itsel by L. B dened as B g =,1 L g (2) with respect to the coordinate map : g! G, is a local action o G on g by construction. In the case where is the exponential map, B is nothing else than the well-known Baker-Campbell-Hausdor (BCH) ormula. For x; y 2 g, BCH(x; y) is dened to be exp(bch(x; y)) = exp(x)exp(y). What we get is B g (u) = log(g exp(u)), which implies exp(bch(log(g);u)) = g exp(u). Since the composition o two equivariant maps is an equivariant map, we can construct an equivariant map rom g to with respect to the actions B on g and on. Just choose the composition p : g!and the ollowing diagram commutes: g,,,,! G,,,,! p B g L g g g,,,,! G,,,,! p The commutative diagram (3) sums up all the important concepts introduced in this section. We have constructed an action o the Lie group G on the Lie algebra g and an equivariant map p rom g to. We are now in a position were studying the action o B on g is equivalent to study the action on. (3) 5

6 3 Innitesimal description o actions and relatedness We now turn to the innitesimal description o actions and equivariant maps. Let us rst dene an innitesimal generator o an action: Denition 3.1 (Innitesimal generator) Suppose :G!Di() dened by g 7! g is an action o the Lie group G with Lie algebra g on the maniold. The innitesimal generator o the action corresponding to 2 g is (x) = d dt exp(t) (x); 8x 2: (4) t=0 The innitesimal generator is a vector eld on. What the innitesimal generator describes is the direction o the motion on the maniold, subject to the action o the Lie group element exp(t). This is the tangent o the ow and the direction o where to proceed. The next Lemma answers what the ow o an innitesimal generator is. Lemma 3.2 (Flow o innitesimal generator) Suppose :G!Di() is an action o the Lie group G on the maniold with innitesimal generator. Then exp(t) :!is the ow o. A ow is an R-action on. Proo: We simply take the time derivative o the ow and apply a simple trick: d dt exp(t)(m) = d ds exp(s) (m) s=t = d ds exp((s+t)) (m) (5) s=0 = d ds exp(s) exp(t) (m) (6) s=0 = exp(t) (m) The equality o (5) and (6) is justied by noticing that [t; s] = 0 and remembering that is an action. This completes the proo. Reerring to Denition 2.2, an action o a Lie group G on a maniold can be regarded as a homomorphism rom G to Di(). Similarly, we can dene a Lie algebra action. Denition 3.3 (Lie algebra action) A Lie algebra action o g on is a map : g 7! X() such that 7! is a Lie algebra homomorphism o g into X(). X() is the let Lie algebra 6

7 o Di() with the negative Lie-Jacobi bracket as Lie algebra bracket. Lie-Jacobi bracket is given as: The Z i =[X; Y ] i = X j X j, j ; where X; Y; Z are vector elds over. Both a Lie group action and a Lie algebra action can be viewed as homomorphisms, and the latter can be interpreted as the tangent lit o the ormer. The precise requirements to establish this connection are given by a theorem o Palais (See [1] page 270, or [18]). Theorem 3.4 (Palais) Let G be a simply connected Lie group, a compact maniold, and : g! X() a Lie algebra homomorphism. Then there exists a unique action : G! Di() such that T =. Thus, in the case o a simply connected Lie group we have an bijection between the Lie group actions on a compact maniold and the Lie algebra actions on. In cases where the group is not simply connected consider instead the action o the universal covering group which is always simply connected. In our numerical calculations we are always working locally, hence we can always nd a local domain on the maniold being compact. Next, we turn our attention to vector elds. Denition 3.5 (Relatedness o vector elds) Let :!N be a smooth map between maniolds. The vector elds X 2 X() and Y 2 X(N ) are called -related, denoted X Y,iTX=Y. The denition means that the ollowing diagram commutes: T X x? T,,,,! TN,,,,! In the case o being a dieomorphism, X Y is equivalent tosaying that X is the pull-back oy. Where is the numerics in all o this abstract theory? The guiding principle is the ollowing simple observation. Let exp(t) and exp(t) be two ows on and N, respectively, generated by the element 2 g and the action on and on N o the Lie group G. Recall that a ow can be regarded as an action o R on the maniold. Our goal is to solve or a ow on a simpler maniold so that the eect is the same as i one solved or the ow onn. `Simpler' means in our context that the maniold is a vector space. We now consider the innitesimal consequences o equivariance. Suppose that the map :!N is equivariant with respect to the two ows exp(t) x??y N 7

8 and exp(t). The next theorem states that i is an equivariant map, the innitesimal generators o the actions with respect to the same element 2 g are -related vector elds. Theorem 3.6 Let :! N be a smooth map between maniolds. Further, let exp(t) and exp(t) be the ows o the innitesimal generators and N. Then is an equivariant map with respect to the ows exp(t) and exp(t), that is; exp(t) = exp(t), iand N are -related, that is; N. Proo: (From [2]). Take the time derivative o the relation exp(t) (m) = exp(t) (m), m 2, and use the ow/vector eld property toget T d dt exp(t)(m) = d exp(t) (m) dt T exp(t) (m) =N exp(t) (m) ) T = N = N exp(t) (m) Starting with this relation, let c(t) = exp(t) (m) denote the integral curve o through m 2. Then d dt ( c)(t) =Tdc(t) dt =Tc(t)=Nc(t): (7) This says that c is the integral curve o N through the point (c(0)) = (m). By uniqueness o integral curves, we get exp(t) (m) = c(t) = exp(t)((m)). This completes the proo. In the case o the equivariant map p : g!,wehave the ollowing commutative diagram relating the ows on the maniold and the Lie algebra g: g,,,,! G,,,,! p B exp(t) L exp(t) exp(t) g,,,,! G,,,,! p From Theorem 3.6 we know that the innitesimal generators o the ows B exp(t) and exp(t) on g and are p -related. That is; p =T p T g : (8) It is implicit, and an assumption that we know that the dierential equation evolving on the maniold can be written as an innitesimal generator o the action. Having established equation (8), the next question is: What is g? 8

9 Theorem 3.7 Let : g! G be a coordinate map on G and p equivariant with respect to the ows B exp(t) and exp(t). The innitesimal generator o B satisying equation (8), is g (u) =d,1 u (). d : g! g is the trivialization o T dened as d u = TR (u),1 T u. Proo: Since every Lie group action on a maniold by equivariance o p is equivalent to an action o G on itsel, we only need to consider innitesimal generators over g and G. T denotes the tangent lit o a mapping between maniolds to a mapping between tangent bundles. Denote by d the part o such a lited mapping that maps g to g. Hence takes u 7! (u) and the lit is T u :Tgu!TG (u). Tgutgand TG (u) can be identied with g through a translation TR, R being the right translation map on G. Because o this, T u can be represented by a map d u : g! g, that is: T u =TR (u) d u : According to Theorem 3.6 and B exp(t) = L exp(t) the ollowing equality must be satised: T u g (u) = G (u)=tr (u) () TR (u) d u g (u) =TR (u) () (9) Choosing g (u) =d,1 u () in (9) completes the proo. The ollowing commutative diagram sums up what we have accomplished: Tg T,,,,! TG Tp d,1 u () TR p() g,,,,!,,,,! T G,,,,! p x??(p) (10) As a concluding remark o this section we emphasize the two-old role o the Lie algebra g. Given a dierential equation on wewant to solve an equivalent dierential equation on g. This is because g has the structure o a vector space. To do this we construct an equivariant map rom g to. On the other hand, g is also the Lie algebra o the Lie group that acts on the maniold. Because o this we also nd elements 2 g as parameters in the innitesimal generators. 4 Applications o the theory We want to numerically solve dierential equations evolving on a homogeneous space. Our strategy is to transorm the vector eld on to an equivalent vector eld on the Lie algebra g. The reason or choosing the Lie algebra is because this maniold is a vector space, and or instance Runge-Kutta methods can be used to solve or the dierential equation on g. The solution on is then advanced through the equivariant map. 9

10 4.1 Algorithmic overview Formalizing the situation, we want to solve the dierential equation y 0 = F (y); y 0 = y0 (0) 2 where F is assumed to be written as the innitesimal generator F (y) =(y)= d dt (exp(t (s; y));y) t=0 or some : R!gthat describes the dierential equation. What we obtained in the previous section is the ollowing commutative diagram (a compressed version o (10)): Tg TyT,,,,,! T d,1 u ((s;y)) F (y)=(y) g,,,,! y Observe that once the map describing the dierential equation on has been ound, the transormation to an equivalent system on g is simply done by choosing as parameter input to the vector eld d,1 u, the output o the unction. Hence, we are able to solve all those dierential equations evolving on the homogeneous space that can be described as an innitesimal generator with respect to the action o G on or some map into the Lie algebra. A general time-dependent dierential equation in this amily is described by a map : R! g. Assuming that dierential equations on can be written as an innitesimal generator is by no means a limitation. All Lie type equations t into this ramework, as well as isospectral ows, rigid rames, and o course all classical equations evolving on R n. For more details on this, see [16]. The ollowing algorithm incorporates all the above ideas. Algorithm 4.1 Assume that the vector eld F over the homogeneous space can be written as an innitesimal generator o the action o the Lie group G on the maniold. That is; F (y) =(y)or some map : R! g which describes the dierential equation. The ollowing is a q'th order Runge-Kutta based numerical method or the solution y(t) 2: - Choose the initial point u 0 =0in g. For any initial point y 0 2 we have y 0 = y0 (u 0 ). - Solve u 0 =d,1 u ((s; y)), u(0) = u 0 by taking one step with any q'th order Runge-Kutta method. Denote the solution by u 1. - Advance the solution on rom y 0 to y 1 by the equivariant map y0. That is; y 1 = y0 (u 1 ). 10

11 This is a q'th order method on since y is smooth, and a q'th order modication on g will result in a corresponding modication on with at least order q. 4.2 Dierent coordinate implementations We will now see how the unction d,1 u ((s; y)) maniests itsel or dierent choices o the local coordinate map Canonical coordinates o the rst kind Choosing = exp : g! G in Algorithm 4.1 we obtain the original method o unthe-kaas [14]. Since every Lie group can be coordinatized by the exponential map there is no restriction on the applicability othemethod. In the case o the exponential map, the vector eld on g, dexp,1 u, is represented by the innite sum dexp,1 u (v) =v, 1 1X 2 [u; v]+ k=2 B k k! k z } { [u; [u; [ ;[u; v] ]]]: (11) [; ] denotes the commutator in the Lie algebra g, and B k is the kth Bernoulli number. In an implementation o the RKK method this innite sum must be truncated. For a q'th order classical Runge-Kutta method we introduce the ollowing q'th order truncation o (11): dexpinv(u; v; q) = q,1 X k=0 B k k! k z } { [u; [u; [ ;[u; v] ]]]: This truncated version o dexp,1 u is used as the vector eld on g. Channell and Scovel reormulate the Ge-arsden Lie-Poisson integration algorithm by coordinatizing the Lie group through the exponential map [5]. Their approach is very similar to ours, albeit not as general. They also nd the rst terms in the expansion (11) or dexp, Canonical coordinates o the second kind Let x 1 ;::: ;x n g denote a basis o the n-dimensional Lie algebra g. Every element in the Lie algebra can be written as a linear combination o elements in this basis. Let the n-tuple o real numbers = ( 1 ;::: ; n ) represent the element ( 1 x n x n ) o the Lie algebra. Canonical coordinates o the second kind are then dened as the ollowing map depending on the basis o the Lie algebra: : =( 1 ;::: ; n )7! exp( 1 x 1 ) exp( 2 x 2 ) exp( n x n ) (12) 11

12 A straight orward calculation will produce the desired dierential o this mapping, namely d: d u (v) = nx k=1 Ad Q k,1 i=1 exp(uixi)v kx k (13) Inverting to nd d,1 is not a trivial task, and has very recently been studied by Owren and arthinsen in [17]. Wei and Norman, in [20, 21], studied the global properties o canonical coordinates o the second kind. Their main theorem states that the coordinates o the second kind are global whenever the Lie algebra g is o nite dimension and solvable Cayley coordinates o the rst kind The Cayley transorm, cay : g!gis dened as: cay(u) = 1+ u 2 1, u 2 As mentioned in the introduction, the Cayley transorm exists or several o the matricial Lie groups. The appealing eature o this map is the lesser expense in computing it, as compared to the exponential map. Thus using a method based on the Cayley transorm should be an appealing option whenever possible. By a simple calculation the corresponding vector eld on the Lie algebra dcay,1 u is calculated to be dcay,1 u (v) =v, 1 2 [u; v], 1 uvu; (14) 4 where [; ] denotes the matrix commutator and `' denotes matrix multiplication. Notice the resemblance o this expression to that o dexp,1 u in (11). The rst two terms are identical. The Cayley transorm is the rst diagonal Pade approximant to the exponential, and it is thereore natural that dcay,1 u behaves in a similar manner to dexp,1 u. dcay,1 u is also exactly computable, since no truncation o the series is necessary. The Cayley transorm has been used in the numerical solution o unitary dierential systems and isospectral ows, see the work o Diele et.al. [6, 7] and Zanna [23]. There is also an interesting application o the Cayley transorm in the energy-momentum method in the solution o Hamiltonian systems on Lie groups, see Lewis and Simo [11] Cayley coordinates o the second kind Conorming with the use o the term `canonical coordinates o the second kind' as synonymous to products o exponentials, we choose to name products o Cayley transorms as `Cayley coordinates o the second kind'. 12

13 Similarly to the situation in Subsection 4.2.2, we have a Lie algebra with basis x 1 ;::: ;x n g and the n-tuple o real numbers =( 1 ;::: ; n ) represents an element o the Lie algebra. Cayley coordinates o the second kind are then dened as the ollowing product o Cayley transorms, depending on the basis: : =( 1 ;::: ; n )7! cay( 1 x 1 )cay( 2 x 2 ) cay( n x n ) The calculation o the dierential o the mapping ollows along the same lines as or in (13), except that some care most be taken in the `elimination' o the dcays. Our calculation assumes that the basis elements o the Lie algebra satises the relation x 3 =,x, which isvalid or instance or so(n). Hence, the result is: d u (v) = nx k=1 v Ad Q k k,1 x i=1 cay(uixi) k (15) 1+ u2 k 4 Inversion o (15) will ollow along similar lines as in the inversion o (13). For details relating to this inversion, see [17] Higher order diagonal Pade coordinates Things are very simple or the Cayley transorm. I we want to use higher order Pade approximations to the exponential map [3, 8], things get more complicated. The main reason or this is the non-commutativity o matrix multiplication. The (p; p)-diagonal Pade approximant is dened as R pp (u) = N pp(u) N pp (,u) ; where N pp (u) = px j=0 c (p) j u j = px j=0 (2p, j)!p! (2p)!j!(p, j)! uj : From this, dr pp;u (v) is readily being ound to equal dr pp;u (v) =N pp (,u),1 N pp (,u) d dt N pp (u + tv) t=0 Substituting or N pp into (16), we get N pp (,u)dr pp;u (v)n pp (u) = p,1 px X j=0 m=0 k=0, N pp (u) d dt N pp (,u, tv) t=0 mx N pp (u),1 : (16) c (p) j c (p) m+1 ((,1)j +(,1) m )u j+k vu m,k : (17) 13

14 Setting p = 1 in (17), we get the Cayley coordinate situation: N 11 (,u)dr 11;u (v)n 11 (u) =v This expression can be inverted directly and we get dr,1 11;u(v) =N 11 (,u)vn 11 (u); which in expanded orm is identical to (14). For p =2we obtain and or p =3we get N 22 (,u)dr 22;u (v)n 22 (u) =v, 1 uvu; (18) 12 N 33 (,u)dr 33;u (v)n 33 (u) = v, 1 10 uvu (u2 v + uvu + vu 2 ) u2 vu 2 : (19) For the general inversion o (17) to nd dr,1 pp we need to solve a linear matrix system in v. We will now demonstrate one such procedure or dr 22;u (See e.g. [10]). To nd the inverse unction dr,1 22;u we realize, by examining (18), that we need to solve a matrix equation o the type Z, XZX = W (20) or the matrix Z, where X; Z; W 2 R nn. One way to solve equation (20) is to rewrite it as a linear system in Z. Assembling all the columns o a matrix A into a n 2 1vector and denoting it ~a, we obtain the ollowing equivalent linear system (I n I n + X T X)~z = ~w; (21) where denotes the Kronecker product, and I n is the n n identity matrix. This procedure generalizes easily to the dr pp -case. 4.3 Numerical examples We consider the ollowing equation evolving in SO(3). y 0 (t) =a(y(t))y(t) (22) Here, a : SO(3)! so(3) is the ollowing skew-symmetric matrix: a(y) = 1 2 (y,yt ) (23) We solve the equation (22) with initial condition y(0) = q, where q is the special orthogonal matrix obtained by the command [q,r] = qr(magic(3)) 14

15 in ATLAB. The equation is integrated rom t =0to t =1,with step sizes h =[2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 ]. We are interested in the ops impact the choice o dierent coordinate maps have or methods o dierent order. We consider local coordinates o the type exp; cay and R 22. In the numerical calculations we truncate the dexp,1 -series to the appropriate order, whereas or the dcay,1 and dr,1 22 we calculate these expressions exactly. The comparison is done or the ollowing our Runge-Kutta methods: E1 - Eulers method (rst order), E2 - odied Eulers method (second order), RK4 - `The Runge-Kutta method' (ourth order), and butcher6 - Butchers sixth order method. The experiments were done using ATLAB and the Dian Toolbox. In Figure 1 we see our plots, one or each o the methods. Along the horizontal axis is the ops count as measured by ATLAB, and along the vertical axis is the global error in the solution or t = 1. In the gure, exp coordinates are denoted by a continuous line with circles, cay coordinates are denoted by a dotted line, and R 22 coordinates by a dash-dotted line EULERS ETHOD E ODIFIED EULERS ETHOD E2 GLOBAL ERROR GLOBAL ERROR FLOPS FLOPS 10 0 THE RUNGE KUTTA ETHOD RK BUTCHERS ETHOD GLOBAL ERROR GLOBAL ERROR FLOPS FLOPS Figure 1: Global error versus ops count. From the plots we can conclude that the Cayley coordinates does constitute a saving in this particular problem or all the dierent methods. It should be warned, however, that the Cayley coordinates do run into trouble i,1 is an 15

16 eigenvalue o y(t). In case o the R 22 coordinates we have to solve a linear system in each step in order to determine R,1 22. This is expensive and does not compare with the exponential coordinates, unless or very high order methods. 5 Conclusion We have constructed a local group action on the Lie algebra g equivalent in the sense o equivariance to the group action on the maniold. Hence, this gives us the ollowing group action diagram: g,,,,! G,,,,! p B g L g g g,,,,! G,,,,! p Accompanying the above nite picture o the situation, there is also an innitesimal diagram: Tg T,,,,! TG Tp d,1 u () TR p() g,,,,!,,,,! T G,,,,! p x??(p) Every vector eld locally written in the orm o an innitesimal generator on the maniold can be transormed to an equivalent vector eld on the Lie algebra g. These vector elds are p -related. We emphasize that all the above constructions are local in nature, since we are considering local coordinate maps on the Lie group maniold. Addressing the aspects o developing object oriented sotware what we have obtained is a nice separation o `how' a dierential equation is dened on the domain, verses `what', in terms coordinates, is used in the numerical integrator. In other words, a dierential equation can be dened in a coordinate-ree manner, independently o the actual coordinates used to coordinatize the Lie group in the numerical method. Acknowledgements I will thank Hans unthe-kaas or valuable input and many interesting discussions in the writing o this paper, and Per Christian oan or his many comments. I also grateully acknowledge the travel grant awarded by L.eltzers Hyskoleond, giving me the opportunity to present parts o this work at NO- DE98 in Arizona, USA, April Finally, I want to acknowledge the continuous support rom everybody in the SYNODE project. 16

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