On the stability and integration of Hamilton-Poisson systems on so(3)

Transcription

1 Rendiconti di Matematica, Serie VII Volume 37, Roma (016), 1 4 On the stability and integration of Hamilton-Poisson systems on so(3) R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING Abstract: We consider inhomogeneous uadratic Hamilton-Poisson systems on the Lie-Poisson sace so (3). There are nine such systems u to affine euivalence. We investigate the stability nature of the euilibria for each of these systems. For a subclass of systems, we find exlicit exressions for the integral curves in terms of Jacobi ellitic functions. 1 Introduction Poisson structures aear in very di erent forms and mathematical contexts such as symlectic manifolds, Lie algebras, singularity theory, and r-matrices. Together with symlectic manifolds, Lie algebras rovide the first examles of Poisson manifolds. Namely, the dual of a finite dimensional Lie algebra admits a canonical Poisson structure, called its Lie-Poisson structure (see, e.g., [7, 30]). Lie-Poisson structures arise naturally in a variety of fields of mathematical hysics and engineering such as classical dynamical systems, robotics, fluid dynamics, and suerconductivity, to name but a few. On Lie-Poisson saces, uadratic Hamilton-Poisson systems have been considered by several authors (e.g., [3, 11,, 4, 30, 33, 34]), most notably in the context of invariant otimal control and geometric mechanics. Rigid body dynamics aear in many areas of engineering such as underwater vehicles, unmanned air vehicles, robotics, and sacecrafts (e.g., [16, 1, 36]). A systematic treatment of stability and integration of homogeneous systems on so (3) was carried out in [17]. Key Words and Phrases: Hamilton-Poisson system Lie-Poisson structure Lyaunov stability Energy-Casimir method Jacobi ellitic function A.M.S. Classification: 34D0, 33E05, E60, 53D17.

2 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [] In this aer we consider inhomogeneous uadratic Hamilton-Poisson systems on the Lie-Poisson sace so (3). There are nine (families of) such systems, under affine euivalence. A system will be referred to as a system of tye I if its set of euilibria is a union of lines and lanes; otherwise, it will be referred to as a system of tye II. For the sake of comleteness, a brief treatment of the homogeneous systems is included. For each system we investigate the (Lyaunov) stability nature of the euilibria. A generalization of the energy-casimir method and the continuous energy-casimir method (see [31]) are used to rove stability. Note that for any system on so(3) the origin is a stable euilibrium state. (Indeed, the Casimir function C() = is a weak Lyaunov function for any Hamiltonian vector field on so(3).) On the other hand, instability usually follows from sectral instability; however, a direct aroach is reuired in some cases. We obtain exlicit exressions for the integral curves of systems of tye I (but not of tye II). In each case we artition the set of initial conditions so as to distinguish between integral curves with di erent ualitative behaviour. The euations of motion are reduced (using the constants of motion) to a single searable di erential euation, which is then transformed into a standard form. An aroriate ellitic integral is used to obtain (after some maniulation) an exlicit exression for the integral curve in terms of Jacobi ellitic functions. Mathematica is used to facilitate most of these calculations. We distinguish between integral curves with di erent ualitative behaviour by determining when the level surfaces of the Hamiltonian and Casimir are tangent to one another. These surfaces are tangent exactly at euilibria. Hence we get a set of critical values (corresonding to euilibria) for the energy states (h 0,c 0 ) of the Hamiltonian and Casimir. This set artitions the sace of energy states into a number of regions. (Within each region, the corresonding nonconstant integral curves can be continuously deformed into one another.) Each region usually corresonds to di erent exlicit exressions for the integral curves. For each system we grah the critical energy states. We select some tyical values for (h 0,c 0 ) from each region (as well as some tyical critical values) for which we then grah the corresonding level surfaces of the Hamiltonian and Casimir. For convenience, we shall refer to this as a tyical configuration. The intersection of these surfaces (i.e., the traces of the corresonding integral curves) and the euilibria are also grahed. The main motivation for our investigation of the uadratic Hamilton-Poisson systems on so (3) comes from our ongoing interest in geometric (otimal) control, articularly on lower-dimensional Lie grous. Similar treatments of uadratic Hamilton-Poisson systems on se() and se(1, 1) can be found in [3, 4] and [7], resectively. After the comletion of this work we learned of several substantial contributions to the (generalized) rigid body dynamics literature from the geometric mechanics ersective. It was Volterra [35] who first found exressions of

3 [3] On the stability and integration of Hamilton-Poisson systems on so(3) 3 integral curves (in terms of sigma-functions and exonents). More recently, an exlicit integration of Zhukovsky-Volterra gyrostat was obtained by Basak [8] (based on an algebraic arametrization of the invariant curves). The stability nature of the euilibria, as well as bifurcations, have been investigated by several authors ([9, 14, 15, 18, 5]). Frauendiener [0] classified uadratic Hamiltonian systems on the unit shere under symlectic transformations; Elie and Lanchares [19] showed that each euivalence class obtained by Fraudiener corresonds to a di erent tye of gyrostat. Nonetheless, we are of the oinion that our alternative investigation, from a Poisson geometry oint of view, is more elementary and direct and as such lends a fresh ersective to the toic. We conclude the aer with some comments and remarks concerning the relationshi between invariant otimal control roblems and uadratic Hamilton- Poisson systems. Quadratic Hamilton-Poisson systems Let g be a real Lie algebra. The (minus) Lie-Poisson structure on g is given by {F, G} () = ([d F (), d G()]). Here g, F, G C 1 (g ), and d F (), d G() g are identified with elements of g. The Lie-Poisson sace (g, {, }) is denoted by g. To each function H C 1 (g ) we associate a Hamiltonian vector field! H on g secified by! H [F ]={F, H}. A function C C 1 (g ) is a Casimir function rovided {C, F} = 0 for all F C 1 (g ). A linear isomorhism : g! g is called a linear Poisson automorhism if {F, G} = {F,G } for all F, G C 1 (g ). Linear Poisson automorhisms are exactly the dual mas of Lie algebra automorhisms. A uadratic Hamilton-Poisson system (g,h A,Q ) is secified by H A,Q : g! R, 7! (A)+Q(). Here A g and Q is a uadratic form on g. If A = 0, then the system is called homogeneous; otherwise, it is called inhomogeneous. (When g is fixed, a system (g,h A,Q ) is identified with its Hamiltonian H A,Q.) We say that two uadratic Hamilton-Poisson systems (g,g) and (h,h) are affinely euivalent if the associated vector fields! G and! H are comatible with an affine isomorhism. That is, two systems are euivalent if there exists an affine isomorhism : g! h such that T! G =! H. (Here T denotes that tangent ma of.) Lemma.1. The following Hamilton-Poisson systems (on g ) are euivalent to H A,Q : (E1) H A,Q, where is a linear Poisson automorhism; (E) H A,rQ, where r 6= 0; (E3) H A,Q + C, where C is a Casimir function.

4 4 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [4] The three-dimensional orthogonal Lie algebra so(3) = { A R 3 3 : A > + A = 0 has standard (ordered) basis E 1 = , E = , E 3 = The commutator relations are given by [E,E 3 ]=E 1,[E 3,E 1 ]=E, and [E 1,E ]= E 3. Let (E 1,E,E 3) denote the dual of the standard basis. We shall write an element = 1 E 1 + E + 3 E 3 so (3) as 1 3. The grou of linear Poisson automorhisms takes the form { 7! : R 3 3, > = 1, det =1} = SO(3). Note that C() = is a Casimir function. Remark.. The Hamiltonian vector fields on so (3) are comlete as their integral curves evolve on the comact subsets C 1 (c 0 ), c 0 0(cf. [1]). Remark.3. The Hamiltonian vector field associated to a function H C 1 (so (3) ) can be exressed as! H = 1 rc rh. Hence the (regular) level sets of H and C are tangent exactly at euilibria. A classification of the uadratic Hamilton-Poisson systems on so (3) was obtained in []. We shall base our investigation of uadratic systems on this classification. For the sake of comleteness, we rovide a sketch of the roof. Theorem.4. Let H be a uadratic Hamilton-Poisson system on so (3). If H is homogeneous, then it is euivalent to exactly one of the systems: H 0 () = 0 (tye I) H 1 () = 1 1 (tye I) H () = (tye I)

5 [5] On the stability and integration of Hamilton-Poisson systems on so(3) 5 If H is inhomogeneous, then it is euivalent to exactly one of the systems: H 0 1, () = 1 > 0 (tye I) H0 1 () = 1 1 (tye I) H1 1 () = (tye I) H 1, () = > 0 (tye II) H 1, () = > 0 (tye I) H, () = > 0 (tye I) H 3, () = , > 0 (tye II) H 4, () = > 0 (tye II) H 5, () = > 0, 1 > 3 > 0 or > 0, 1 = 3 > 0. (tye II) Here, 1,, 3 arametrize families of class reresentatives, each di erent value corresonding to a distinct (non-euivalent) reresentative. Remark.5. A stronger form of euivalence, namely orthogonal euivalence, has been considered in [33]. Proof (Sketch). We note that the euivalences (E1)-(E3) are not always sufficient to reduce a system to its normal form. In such cases, we find an exlicit affine isomorhism with resect to which the vector fields are comatible. Let H() =A + Q >, where Q is a symmetric 3 3 matrix. Here A = a 1 E 1 + a E + a 3 E 3 so(3) is identified with >. a 1 a a 3 We may assume that Q is ositive definite. Given a linear Poisson automorhism : 7!, we have (H )() = A+ Q > >. As any symmetric matrix can be diagonalized by an orthogonal matrix (see, e.g., [3]), it follows that there exists a linear Poisson automorhism such that (H )() = A + diag(λ 1, λ, λ 3 ) > with λ 1 λ λ 3 > 0. Thus (H )() λ 3 C() = A + diag(λ 1 λ 3, λ λ 3, 0) > with λ 1 λ 3 λ λ 3 0. If λ 1 λ 3 = 0, then (by (E1) and (E3)) H is euivalent to an intermediate system G 0 B () =B, where B = A. On the other hand, if λ 1 λ 3 > 0, then (H )() λ 3 C() = A +(λ 1 λ 3 ) diag 1, λ λ3 λ 1 λ 3, 0 >. Thus H is euivalent to H 0 () = A + 1 +, = λ λ3 λ 1 λ 3. If = 0, then H 0 () = A + 1 and so H 0 is euivalent to (an intermediate system) G 1 B () = B with B = A. (A similar argument holds when = 1.) On the other hand, suose 0 < < 1. Then the vector fields associated to H 0 () =a a 0 + a and G B() =b b + b

6 6 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [6] (here A 0 = A) are comatible with the affine isomorhism 3 (1 ) 0 0 7! 4 0 (1 ) ale + 1 a 0 1 (1 ) 1 (1 ) a0 1 a 0 3 rovided b 1 = (1 ) 1 a 0 1 1, b = (1 ) a0, and b 3 = (1 ) a 0 3. Suose that H is homogeneous, i.e., A = 0. Then, by the above argument, H is euivalent to G 0 0 = H 0, G 1 0 = H 1, or G 0 = H. The systems H 1 and H are not euivalent as the set of euilibria for H 1 is the union of a lane and a line whereas the set of euilibria for H is the union of three lines. The remainder of the roof involves considering each of the intermediate inhomogeneous systems G 0 B, G1 B, and G B and using a combination of linear Poisson automorhisms and affine isomorhisms to reduce these systems as much as ossible. One then verifies that each reresentative obtained is distinct and noneuivalent. It turns out that any homogeneous system on so (3) is euivalent to a system on se (), see [1]. The Euclidean Lie algebra < = se () = 4x 1 0 x 3 5 = x 1 Ẽ 1 + x Ẽ + x 3 Ẽ 3 : x 1,x,x 3 R : ; x x 3 0 has nonzero commutators [Ẽ, Ẽ3] =Ẽ1 and [Ẽ3, Ẽ1] =Ẽ. The Lie-Poisson sace se () has Casimir function C( ) e = 1 +. The systems ( ) so (3), 1 1 : ṗ1 =0, ṗ = 1 3, ṗ 3 = 1, ( ) se (), 1 3 : 1 = 3, = 1 3, 3 =0 ] are comatible with the linear isomorhism : so (3)! se (), = On the other hand, the systems ( so (3), ) : ṗ1 = 3, ṗ = 1 3, ṗ 3 = 1, ( se (), + ) 3 : 1 = 3, = 1 3, 3 = 1

7 [7] On the stability and integration of Hamilton-Poisson systems on so(3) 7 are comatible with the linear isomorhism : so (3)! se (), = We shall make use of such an euivalence in the investigation of the system H 1, to relate to some results reviously obtained. For uadratic Hamilton-Poisson systems on so(3), it turns out that the integral curves are often exressible in terms of Jacobi ellitic functions. Given the modulus k [0, 1], the basic Jacobi ellitic functions sn(,k), cn(,k), and dn(,k) can be defined as (see, e.g., [6, 8]) where am(,k)=f (,k) 1 sn(x, k) = sin am(x, k), cn(x, k) = cos am(x, k), dn(x, k) = 1 k sin am(x, k) is the amlitude and F (',k)= R ' 0 dt 1 k sin t The number K is given by K = F (,k). (The functions sn(,k) and cn(,k)are4k eriodic, whereas dn(,k) is K eriodic.) Nine other ellitic functions are defined by taking recirocals and uotients; in articular, we have nd(,k)= 1 sd(,k)= sn(,k) dn(,k) cn(,k) and cd(,k)= dn(,k) dn(,k), 3 Homogeneous systems We consider the two homogeneous systems H 1 and H (see Theorem.4). The integral curves of the system H 1 can easily be found in terms of elementary functions; it is then a simle matter to determine the stability nature of the euilibria. On the other hand, the integral curves of the system H can be found in terms of basic Jacobi ellitic functions. The stability nature of the euilibria can be determined via the energy-casimir methods (and the investigation of sectral stability). Proofs will be omitted; somewhat less refined versions of these results were obtained elsewhere (cf. [17], see also [3]). Throughout, we shall arametrize the euilibrium states by µ,, R, 6= System H 1 The system H 1 () = 1 1 has euations of motion ṗ 1 =0, ṗ = 1 3, ṗ 3 = 1.

8 8 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [8] 4 c0 3 ( ) isolated oint E 0 3 E 0 3 intersection 1 emty intersection E 0 E h0 0 E 1 0 E 1 Figure 1: Critical energy states for H 1 and a corresonding tyical configuration. The euilibria are e µ, 1 = (0, µ, ) and e µ =(µ, 0, 0). The states eµ, 1 6= 0 are unstable whereas the states e µ are stable. In Figure 1 we grah the critical energy states (c 0,h 0 ) and a corresonding tyical configuration. The integral curves of the system are given by 1 (t) = 1 (0) (t) = (0) cos( 1 (0) t)+ 3 (0) sin( 1 (0) t) 3 (t) = 3 (0) cos( 1 (0) t) (0) sin( 1 (0) t). 3. System H The system H () = has euations of motion ṗ 1 = 3, ṗ = 1 3, ṗ 3 = 1. The euilibria are e µ 1 =(µ, 0, 0), e = (0,, 0), and e 3 = (0, 0, ). There are three ualitatively di erent cases for the intersection of a ellitic cylinder (H ) 1 (h 0 ) and a shere C 1 (c 0 ), corresonding to (a) c 0 < h 0, (b) c 0 =h 0, and (c) c 0 > h 0. In Figure we grah the critical energy states (h 0,c 0 ); in Figure 3 we grah the corresonding tyical configurations. Figure : Critical energy states for H.

9 [9] On the stability and integration of Hamilton-Poisson systems on so(3) 9 (a) c 0 < h 0 (b) c 0 =h 0 (c) c 0 > h 0 Figure 3: Tyical configurations for H. Theorem 3.1. The euilibrium states have the following behaviour: (i) The states e µ 1 (ii) The states e (iii) The states e 3 are stable. are (sectrally) unstable. are stable. Theorem 3.. Let ( ) be an integral curve of the system H Let h 0 = H ((0)) and c 0 = C((0)). through (0). (a) If 0 <c 0 < h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where Here = h 0 and k = 1 (t) =σ h 0 dn ( t, k) (t) = c 0 h 0 sn ( t, k) 3 (t) =σ c 0 h 0 cn ( t, k). c 0 h 0 h 0.

10 10 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [10] (b) If c 0 =h 0 > 0, then there exist t 0 R and σ 1, σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) =σ 1 h0 sech h0 t (t) =σ 1 σ h0 tanh h0 t 3 (t) =σ h0 sech h0 t. (c) If c 0 > h 0 > 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) = h 0 cn ( t, k) (t) =σ h 0 sn ( t, k) 3 (t) =σ dn ( t, k). Here = c 0 h 0 and k = 4 Inhomogeneous systems of tye I h 0 c 0 h 0. In this section we consider those inhomogeneous systems whose euilibria are unions of lines and lanes (tye I). There are five such systems (in fact two systems and three one-arameter families of systems, see Theorem.4). Note that the systems which are euivalent to H0 1 are homogeneous systems in disguise. For each system we obtain exlicit exressions for the integral curves: for H1, 0 in terms of elementary functions and for the remaining systems in terms of rational functions of (ossibly suare roots of) Jacobi ellitic functions. We rovide a detailed roof for obtaining the integral curves for one sub-case of the system H,. The integral curves for the remaining systems are obtained in a similar fashion and hence the roofs are omitted. For each system the stability nature of all euilibria is determined. We rovide a detailed roof for the system H1 1. Similar arguments hold for determining the stability nature of the euilibria of the remaining systems and thus the roofs are omitted, excet where instability does not follow from sectral instability. We note that the system H1, is euivalent to a system on se () which has been considered reviously in [4]. Again, the euilibria are arametrized by µ,, R, 6= System H 0 1, The system H 0 1, () = 1, > 0 has euations of motion ṗ 1 =0, ṗ = 3, ṗ 3 =.

11 [11] On the stability and integration of Hamilton-Poisson systems on so(3) 11 The euilibria are e µ 1 =(µ, 0, 0); all euilibria are stable. In Figure 4 we grah the critical energy states (h 0,c 0 ) and a corresonding tyical configuration. (The value = 1 was used in Figure 4.) Figure 4: Critical energy states for H 0 1, and a corresonding tyical configuration. The integral curves of this system are given by 4. System H 1 1 The system H 1 1 () = (t) = 1 (0) (t) = (0) cos( t)+ 3 (0) sin( t) 3 (t) = 3 (0) cos( t) (0) sin( t). has euations of motion ṗ 1 = 3, ṗ = 1 3, ṗ 3 = 1 1. The euilibria are e µ 1 = (0, µ, 0) and e =(, 1, 0). There are three ualitatively di erent cases for the intersection of a arabolic cylinder (H1 1 ) 1 (h 0 ) and a shere C 1 (c 0 ), corresonding to (a) c 0 <h 0, (b) c 0 = h 0, and (c) c 0 >h 0. In Figure 5 we grah the critical energy states (h 0,c 0 ); in Figure 6 we grah the corresonding tyical configurations. Figure 5: Critical energy states for H 1 1.

12 1 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [1] Theorem 4.1. The euilibrium states have the following behaviour: (i) The states e µ 1, µ ale 1 are stable. (ii) The states e µ 1, µ>1 are (sectrally) unstable. (iii) The states e are stable. (a) c 0 <h 0 (b) c 0 = h 0 (c) c 0 >h 0 Figure 6: Tyical configurations for H 1 1. Proof. Let H λ () =λ 1 H1 1 () +λ C(). (i) Suose µ<1, µ 6= 0, and let λ 1 = 1 and λ = 1 µ. We have d H λ(µ, 0, 0) = 0 and that the Hessian d H λ (µ, 0, 0) = diag( µ 1 µ, 1 µ, 1 µ ) is definite. Thus, by the generalized energy- Casimir method, the states e µ 1, µ<1, µ 6= 0 are stable. Suose µ = 1. Then H(e 1 1)=C(e 1 1) = 1. It is a simle matter to show that (H1 1 ) 1 (1) \ C 1 (1) = {e 1 1}. Thus, by the continuous energy-casimir method, the state e 1 1 is stable. Likewise, the origin is stable. (ii) The linearization of the system at e µ 1 has eigenvalues λ 1 = 0, λ,3 = ± µ 1. Thus the states e µ 1, µ>1 are sectrally unstable. (iii) Let λ 1 = 1 and λ = 1. We have d H λ(, 1, 0) = 0 and d H λ (, 1, 0) = diag(0, 1, 1) is definite when restricted to W =san {(1,, 0), (0, 0, 1)}. Hence, by the generalized energy-casimir method, the states e are stable.

13 [13] On the stability and integration of Hamilton-Poisson systems on so(3) 13 Theorem 4. ([]). Let ( ) be an integral curve of the system H1 1 (0). Let h 0 = H1 1 ((0)) and c 0 = C((0)). through (a) If c 0 <h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t+t 0 ), where 1 (t) =σ 1+k sn ( t, k) δ dn ( t, k) δ (t) =h 0 + δ 1 k sn ( t, k) 3 (t) = σk cn ( t, k) δ 1 k sn ( t, k). Here = h h 0 1+δ, k = 0 1 δ h, and δ = 0 1+δ h 0 c 0. (b) If c 0 = h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t+t 0 ), where 1 (t) =σ h 0 1 sech h0 1 t (t) =h 0 (h 0 1) sech h0 1 t 3 (t) =σ(h 0 1) sech h0 1 t tanh h0 1 t. (c) If c 0 >h 0, then there exists t 0 R such that (t) = (t + t 0 ), where Here = δ, k = 1 (t) = (h 0 + δ 1) cn ( t, k) (t) =h 0 (h 0 + δ 1) cn ( t, k) 3 (t) = δ(h 0 + δ 1) dn ( t, k) sn ( t, k). h 0+δ 1 δ, and δ = 1+c 0 h System H 1, The system H 1, () = , > 0 has euations of motion ṗ 1 = 3, ṗ =( + 1 ) 3, ṗ 3 = ( + 1 ). The euilibria are e µ 1 =(µ, 0, 0), e =(,, 0), and e 3 =(, 0, ). The system (so(3),h 1, ) is euivalent to the system (se(), e H ), where eh ( ) = Stability and integration of e H were treated in [4].

14 14 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [14] Exlicitly, the systems (se(), H e ) and (so(3),h1, ) are comatible with the affine isomorhism : se()! so(3) given by ! (4.1) Hence, any integral curve of (so (3),H 1, ) is just the image under of an integral curve of (se (), H e ). The exressions for the integral curves slit into a number of cases. (Some divisions are based on ualitative grounds, whereas others where retrosectively made to facilitate integration.) An index of the conditions defining these cases aears in Table 1. In Figure 7 we grah the critical energy states (h 0,c 0 ); in Figure 8 we grah the corresonding tyical configurations. (The value = 3 was used in both these figures.) Conditions (! ± =h 0 + ( ± +4h 0 )) Index h 0 ale 0 h 0 > 0 + h 0 >c 0 + c 0 + h 0 c 0 ( +h 0 ) c 0 >! + + h 0 = c 0 + c 0 + h 0 c 0 ( +h 0 ) + h 0 <c 0 + c 0 + h 0 c 0 ( +h 0 ) 1a(i) 1a(ii) 1a(iii) c 0 =! + c 0 < + h 0 1b(i) c 0 = + h 0! < c 0 <! + 1c c 0 > +h 0 c 0 = +h 0 1b(ii) c 0 < +h 0 c 0 =! + c(ii) c 0 >! + c(i) a b! < c 0 <! + c(iii) Table 1: Index of cases for integral curves of H 1,. We give a roof detailing how the exressions for the integral curves on so (3) are obtained from those on se () only for case 1a(i). (The remaining cases follow a similar argument and thus the roofs are omitted.)

15 [15] On the stability and integration of Hamilton-Poisson systems on so(3) 15 Figure 7: Critical energy states for H 1,. Theorem 4.3. The euilibrium states have the following behaviour: (i) The states e µ 1, µ ( 1, ) [ [, 1) are stable. (ii) The states e µ 1, <µ< are (sectrally) unstable. (iii) The state e 1 is unstable. (iv) The states e are (sectrally) unstable. (v) The states e 3 are stable. Proof. (iii) Consider the euilibrium state e 1. We have that (t) = 3 t +t, t +t, +t is an integral curve of the system H1, such that lim t! 1 (t) =e 1. Let B " be the oen ball of radius " = centred at the oint e 1. For any neighbourhood V B " of e 1 there exists t 0 < 0 such that (t 0 ) V. Furthermore k(0) e 1 k = > ", i.e., (0) 6 B ". Hence the state e 1 is unstable. Note 4.4. In Theorems we shall find it convenient to use 0 = +4h 0 instead of h 0 ; also, we shall make use of the following notation δ = 1 4 ( 4c 0 ) ( +4c 0 ) and ± = 1 4c 0 0 ± 4δ.

16 16 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [16] Case 1a(iii) Case 1b(i) Case 1c Case a Case b Case c(i) Figure 8: Tyical configurations for H 1,. Theorem 4.5 (case 1a(i)). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case 1a(i) are satisfied, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) = 0 + sn ( t, k) + sn ( t, k) (t) =σ 0 δ cn ( t, k) + sn ( t, k) 3 (t) = σδ k 0 dn ( t, k) + sn ( t, k).

17 [17] On the stability and integration of Hamilton-Poisson systems on so(3) 17 Here k = s = 1 4c δ, s +4δ 4c δ 4c , and k 0 = δ 4c δ. Proof. Any integral curve of H1, is the image under of an integral curve of H. In [4], exlicit exressions for all integral curves of H are determined; there are a number of cases (corresonding to di erent exlicit exressions). The exression for the integral curve ( ) of H through a oint (0) se () involves the constants h 0 = H e ( (0)) and c 0 = C( (0)). e The various cases are exressed in terms of ineualities in h 0 and c 0. We wish to find the image ( ( )) of each such integral curve and to exress c 0 and h 0 in terms of the constants h 0 = H1, ( ( (0))) and c 0 = C( ( (0))). Moreover, we wish to find the corresonding conditions for the various cases on so(3) in terms of ineualities in h 0 and c 0. Let se() and let h 0 = e H ( ) = and c 0 = e C( ) = 1 +. Corresondingly, let h 0 = H 1, ( ( )) and c 0 = C( ( )); we have h 0 = 1 ( 1 + ) 1 4 and c 0 = ( 1 + ) Hence h 0 = 1 c and c 0 = h c 0. (4.) We can invert these relations to get c 0 = h and h 0 = c 0 h 0 1. (4.3) Therefore, ( H e ) 1 ( h 0 ) \ C e 1 ( c 0 ) if and only if () (H1, ) 1 (h 0 ) \ C 1 (c 0 ) whenever (4.) or (4.3) holds. We consider the first case for the integral curves of H treated in [4]. Let ( ) be an integral curve of H and let h 0 = H ( (0)) and c 0 = C( (0)). If the conditions c ale 0, h0 > c 1 0, h 0 > h 0 c 0 (4.4)

18 18 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [18] hold, then there exist σ { 1, 1} and t 0 R such that (t) = (t + t 0 ), where 8 >< >: 1 (t) = c 0 h0 δ h0 + δ sn( t, k) h0 + δ h0 δ sn( t, k) (t) = σ c 0 δ 3 (t) = σ δ k 0 cn( t, k) h0 + δ h0 δ sn( t, k) dn( t, k) h0 + δ h0 δ sn( t, k). s Here δ = h 0 c 0, = ( h 0 + δ)( 1 h 0 + δ), ( h k 1 0 δ)( = h 0 δ) ( h 1 0+ δ)( h 0+ δ) and k 0 =. We now find the corresonding integral curves of δ ( + δ h 0)( δ+ h 0) H 1,. Let ( ) be an integral curve of H 1, and let h 0 = H 1, ((0)) and c 0 = C((0)). We have that 1 (( )) is an integral curve of H. By (4.3) we have that 1 (( )) satisfies the reuisite conditions (4.4) of the above result if and only if the conditions h 0 ale 0, c 0 > h 0 + ( + +4h 0 ), +h 0 >c 0 + c 0 + h 0 c 0 ( +h 0 ) hold. Suosing these conditions hold, there exist σ { 1, 1} and t 0 R such that 1 ((t)) = (t + t 0 ), i.e., (t) = ( (t + t 0 )). Finally, we let (t) = ( (t)) and relace e h 0 and ec 0 with exressions in h 0 and c 0 (using (4.3)) and simlify to obtain the result. Theorem 4.6 (case 1a(ii)). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case 1a(ii) are satisfied, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where Here = 3 4c (t) = 0 + sin( t) + sin( t) cos( t) (t) =σ 0 δ + sin( t) σδ 3 (t) = + sin( t).

19 [19] On the stability and integration of Hamilton-Poisson systems on so(3) 19 Theorem 4.7 (cases 1a(iii) & a). Let ( ) be an integral curve of the system H1, through (0). Let h 0 = H1, ((0)) and c 0 = C((0)). If the conditions of case 1a(iii) or a are satisfied, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where Here = δ and k = 1 (t) = 0 + cn ( t, k) + cn ( t, k) sn ( t, k) (t) = σ 0 δ + cn ( t, k) dn ( t, k) 3 (t) = σδ + cn ( t, k). (3 4δ 4c 0+ 0)( +4δ 4c 0+ 0) δ. Theorem 4.8 (case 1b(i)). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case 1b(i) are satisfied, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where Here = ( 0) 0. ( 1 (t) = ) 0 0 cosh( t) sinh( t) (t) =σ 0 ( 0 ) 0 cosh( t) 3 (t) =σ ( 0 ) 0 cosh( t) 0 cosh( t). Theorem 4.9 (case 1b(ii)). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case 1b(ii) are satisfied, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) = 3 t + t, (t) = σ t + t, 3(t) = σ + t. Theorem 4.10 (cases 1c & c(iii)). Let ( ) be an integral curve of the system H1, through (0). Let h 0 = H1, ((0)) and c 0 = C((0)). If the conditions of case 1c or c(iii) are satisfied, then there exists t 0 R such that (t) = (t + t 0 ),

20 0 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [0] where 1 (t) = " 1 Here = s k = cd ( t, k) cd ( t, k) (t) =" sd ( 1 t, k) 1+k cd ( t, k) 1 + nd ( t, k) cd ( t, k) cn ( 1 3 (t) =" t, k) 1 k cd ( t, k) 1 + nd ( t, k) cd ( t, k) = c 0 ( = ( + 0 ) 0 ( +4 0 ) + ) s k 0 = = 1 0 ( ( + 0 ) ) ( ( + 0 ) ) ( 0 ) and " 1 = ( + 0 ) " = k0 0 (4 k ) s ( 1 )( " 3 = ( + 0 ( + 0 )) ( ) ) k 3 0 ( + 0 ). Theorem 4.11 (case b). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case b are satisfied, then there exist t 0 R and σ 1, σ { 1, 1} such that (t) = (t+t 0 ), where Here = 0. 1 (t) = σ 1 0 cosh( t) sinh( t) (t) = σ 1 σ 0 σ 1 0 cosh( t) σ 3 (t) = σ 1 0 cosh( t).

21 [1] On the stability and integration of Hamilton-Poisson systems on so(3) 1 Theorem 4.1 (case c(i)). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case c(i) are satisfied, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) = k σ dn ( t, k) k 0 σ + dn ( t, k) (t) = δ 0 cn ( t, k) k 0 σ + dn ( t, k) 3 (t) = σδk 0 sn ( t, k) k 0 σ + dn ( t, k). Here k = s = 1 4c δ δ 4c δ and k 0 = s 4c δ 4c δ. Theorem 4.13 (case c(ii)). Let ( ) be an integral curve of the system H 1, through (0). Let h 0 = H 1, ((0)) and c 0 = C((0)). If the conditions of case c(ii) are satisfied, then there exists t 0 R such that (t) = (t + t 0 ), where Here = 0( 0 ). 1 (t) = + 0 ( 0) 0 0 cos( t) (t) = 0 ( 0 ) sin( t) 0 cos( t) 3 (t) =( 0 ) 0 cos( t) 0 cos( t). 4.4 System H, The system H, () = , > 0 has euations of motion ṗ 1 = ( + ) 3, ṗ = 1 3, ṗ 3 = 1 ( ). The euilibria are e 1 =(,, 0), e µ = (0, µ, 0), and e 3 = (0,, ).

22 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [] Theorem The euilibrium states have the following behaviour: (i) The states e 1 are stable. (ii) The states e µ, µ ( 1, ) [ (, 1) are (sectrally) unstable. (iii) The states e µ, ale µ ale are stable. (iv) The states e 3 are stable. There are five cases for the intersection of a ellitic cylinder (H, ) 1 (h 0 ) and a shere C 1 (c 0 ). (We note that if the intersection is nonemty, then h 0 ale c ) We further subdivide one of these cases into two subcases to facilitate integration. An index of the conditions defining these cases aears in Table. In Figure 9 we grah the critical energy states (h 0,c 0 ); in Figure 10 we grah the corresonding tyical configurations. (The value = 1 was used in both these figures.) Conditions Index c 0 + c 0 <h 0 a c 0 + c 0 = h 0 b c 0 c 0 <h 0 < c0 + c 0 =h 0 c(i) c 0 c 0 6=h 0 c(ii) c 0 c 0 = h 0 d c 0 c 0 >h 0 e Table : Index of cases for integral curves of H,. Figure 9: Critical energy states for H,. We will find the following lemma useful when verifying that a given curve is an integral curve.

23 [3] On the stability and integration of Hamilton-Poisson systems on so(3) 3 (a) Case a (b) Case b (c) Case c(ii) (d) Case d (e) Case e Figure 10: Tyical configurations for H,. Lemma If ( ) is a curve such that H, ((t)) = h 0, C((t)) = c 0, and ṗ (t) = 1 (t) 3 (t) for t R, then ( ) is an integral curve of the system H,. d Proof. As dt C((t))= d 1ṗ 1 + ṗ + 3 ṗ 3 = 0, dt H, ((t)) = ṗ + 1 ṗ 1 + ṗ = 0, and ṗ = 1 3, we have 1 ṗ 1 = ( + ) 1 3 and 3 ṗ 3 = ( ) 1 3. It follows that ( ) is an integral curve of the system H,. We now resent the exressions for the integral curves in the first case. Theorem 4.16 (case a). Let ( ) be an integral curve of the system H, through (0). Let h 0 = H, ((0)) c and c 0 = C((0)). If 0 + c 0 <h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) =σ δ c 0 + δ dn( t, k) + δ δ sn( t, k) (t) = 1 (δ + c 0 h 0 ) + δ +(δ c 0 +h 0 ) δ sn( t, k) + δ δ sn( t, k) 3 (t) = σ δ cn( t, k) +c 0 h 0. + δ δ sn( t, k) Here δ = c 0 +4h 0 4c 0 ( + h 0 ), = c 0 + δ, k = =h 0 c 0. +δ c 0 δ c 0, and Remark If we take the limit of the exression for (t), as tends to 0, then we obtain integral curves for the first case of the system H.

24 4 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [4] One might consider limiting h 0 to c0 + c 0 in case (a) in order to roduce integral curves for case (b). However, this limit degenerates and so a more direct aroach is reuired. Theorem 4.18 (case b). Let ( ) be an integral curve of the system H, through (0). Let h 0 = H, ((0)) c and c 0 = C((0)). If 0 + c 0 = h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where Here = c 0. 1 (t) = σ(c0 ) c0 (t) = c 0 3 (t) = σ(c0 ) c0+ cosh( 1 t) c0 + cosh( t) (c 0 ) c0 + cosh( t) sinh( 1 t) c0 + cosh( t). We rovide a detailed roof for case c(i) to show how the integral curves may be obtained. For this case, in the reduction to standard form, the roots of the two uadratics need to be deinterlaced. Conseuently, the exressions for the corresonding integral curves are more involved. Theorem 4.19 (case c(i)). Let ( ) be an integral curve of the system H, through (0). Let h 0 = H, ((0)) and c 0 = C((0)). If c 0 =h 0, then there exists t 0 R such that (t) = (t + t 0 ), where 1 (t) = k 4 c 0 cn( t, k) + c 0 c 0 + dn( t, k) s 1 + dn( t, k) k 0 + dn( t, k) (t) = + c 0 c 0 dn( t, k) c 0 + c 0 c 0 + dn( t, k) s 3 (t) = k 4 sn( t, k) c 0 + c 0 k 0 + dn( t, k) c 0 + dn( t, k) 1 + dn( t, k). Here = +c, k = 4 c 4 0 +c 0 0 c 0 +c, and k 0 +c = 0 c 0 0+ c 0 +c. 0+ c 0 Proof. We start by exlaining how the exression for ( ) was found. Suose ( ) is an integral curve of H, such that c 0 =h 0, where h 0 = H, ( (0)) and c 0 = C( (0)). Note that c0 c 0 <h 0 < c0 + c 0 is trivially satisfied when c 0 > 0. As ( ) satisfies ( d dt ) = 4 1 3, H, ( ( )) = c0, and C( ( )) = c 0, we have d dt = (c 0 )(c 0 + ).

25 [5] On the stability and integration of Hamilton-Poisson systems on so(3) 5 After deinterlacing the roots of the two uadratics we get d dt = (c c 0 + )(c 0 + c 0 + ). We transform this euation into standard form. Making the change of variables s = r1 r yields 1 t = (r 1 r ) A 1 A Z r B1 A 1 r 1 r ds r B1 A 1 s s B A. Here A 1 = c 0 < 0 A = c 0 > 0 B 1 = c 0 > 0 B = c 0 < 0 r 1 = c 0 r = c 0. By alying the ellitic integral formula (see [6, 8]) Z a x dt (a t )(t b ) = 1 a dn 1 1 a x, a b a, b ale x ale a we obtain (t) = c 0 + c 0 c 0 dn ( t, k) + c 0 c 0 + dn ( t, k) where = +c 0 c 0 and k = 4 c 4 0 +c 0 +c. As 0+ c 1 (t) = c0 0 (t) 1 (t), we have c 0 (1 + dn ( t, k)) c 0 ( + c 0 ) c 0 + dn ( t, k) 1 (t) = ( + c 0 c 0 + dn ( t, k) ) = 3 c 0 (1 + dn ( t, k)) ( dn ( t, k) k 0 ) ( + c 0 c 0 + dn ( t, k) ) where k 0 = +c 0 c 0 +c 0+ c 0. We now multily this euation by cn ( t, k) cn ( t, k) = k cn ( t, k) dn ( t, k) (k 0 ) = k cn ( t, k) (dn ( t, k) k 0 )(dn ( t, k)+k 0 )

26 6 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [6] and take the suare root to obtain k s c 0 cn( t, k) 1 (t) =σ 1 + c dn( t, k) c 0 + dn( t, k) k 0 + dn( t, k) for some σ 1 { 1, 1}. Similarly, using c 0 = 1 (t) + (t) + 3 (t) and multilying by sn ( t, k) sn ( t, k) = k sn ( t, k) (1 dn ( t, k))(1 + dn ( t, k)) yields k s c 0 sn( t, k) 3 (t) =σ + c 0 k 0 + dn( t, k) c 0 + dn( t, k) 1 + dn( t, k) for some σ { 1, 1}. We show that ( ) is an integral curve for certain values of σ 1 and σ. We have d dt (t) 1 (t) 3 (t) = k 3 c0 (1 σ 1 σ ) cn( t, k) sn( t, k) ( + c 0 c 0 + dn( t, k) ). Therefore d dt (t) = 1 (t) 3 (t) whenever σ 1 = σ = 1. We have by construction that, H, ( (t)) = h 0 and C( (t)) = c 0. Conseuently, by Lemma 4.15, it follows that ( ) (as stated in the theorem) is an integral curve; it is not difficult to show that 0 <k<1 and that (t) is defined for all t R. Let ( ) be an integral curve through (0), let h 0 = H, ((0)), c 0 = C((0)), and suose that c 0 =h 0. We claim that (t) = (t + t 0 ) for some t 0 R. We have (0) + 1 (0) + 1 (0) = c0 and 1(0) + (0) + 3 (0) = c 0. Therefore (0) + 1 (0) ale c0 and so + c 0 ale (0) ale + + c 0. We also have 1 (0) + (0) ale c 0, which imlies that + c 0 ale (0) ale + + c 0. Thus + c 0 ale (0) ale + + c 0. Now (0) = + c 0 and ( K ) = + + c 0. Thus there exists t [0, K ] such that (0) = (t ). As 1 (0) = c0 (0) 1 (0) = c0 (t ) 1 (t ) = 1 (t ) it follows that 1 (0) = ± 1 (t ). Furthermore 1 (t+ K )= 1(t) and (t+ K )= (t). Thus there exists t 1 R ( t 1 = t or t 1 = t + K ) such that 1(0) = 1 (t 1 ) and (0) = (t 1 ). On the other hand 3 (0) = c 0 1 (0) (0) = c 0 1 (t 1 ) (t 1 ) = 3 (t 1 )

27 [7] On the stability and integration of Hamilton-Poisson systems on so(3) 7 and so 3 (0) = ± 3 (t 1 ). Furthermore 1 ( t) = 1 (t), ( t) = (t), and 3 ( t) = 3 (t). Thus there exists t 0 R ( t 0 = t 1 or t 0 = t 1 ) such that (0) = (t 0 ). Conseuently, the integral curves t 7! (t) and t 7! (t + t 0 ) solve the same Cauchy roblem, and therefore are identical. Case c(ii) is very similar to case c(i), although the comutations are more involved. The identity cn ( 1 t + 1 K, k) = k 0 (1 sn( t,k)) k 0 +dn( t,k) roved to be useful in deriving the below exression for 1 (t). Theorem 4.0 (case c(ii)). Let ( ) be an integral curve of the system H, through (0). Let h 0 = H, ((0)) c and c 0 = C((0)). If 0 c 0 <h 0 < c 0 + c 0 and c 0 6=h 0, then there exists t 0 R such that (t) = (t + t 0 ), where Here and 1 (t) =&" 1 cn( 1 t + 1 K, k) 1+k sn( t, k) k 0 + dn( t, k)! + &! sn( t, k) (t) =" (δ+ 0) + (δ+ 0)! + + &! sn( t, k)! + &! sn( t, k) 3 (t) =&" 3 cn( 1 t 1 K, k) 1 k sn( t, k) k 0 + dn( t, k)! + &! sn( t, k). = 1 0 = +h 0 r + k = = δ 4 6δ r 4 k 0 = = δ 0 ( + δ 0 ) ( δ + 0 ) δ = ( + c 0 ) 0! = (δ + 0 ) (δ 0 ) & = sgn(δ 0 ) " 1 = 1 (δ 0) k 0 (! + ) ( 0 + ( δ) 0 1 )( 0 ( + δ) ) " = + (δ + 0) (δ 0 ) " 3 = 1 (δ 0) k 0 (! + ) ( δ + ( 0 ) δ 1 )( δ ( + 0 ) δ + 1 ). One might consider limiting h 0 to c0 c 0 in case (e) in order to roduce integral curves for case (d). However, like for case (b), this limit degenerates and again a more direct aroach is reuired.

28 8 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [8] Theorem 4.1 (case d). Let ( ) be an integral curve of the system H, through (0). Let h 0 = H, ((0)) c and c 0 = C((0)). If 0 c 0 = h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) = σ(c0 ) c0+ (t) = c (t) = σ(c0 ) c0 sinh( 1 t) c0 + cosh( t) (c 0 ) c0 + cosh( t) cosh( 1 t) c0 + cosh( t). Here = c 0. We resent the exressions for the integral curves of case (e). Theorem 4. (case e). Let ( ) be an integral curve of the system H, through (0). Let h 0 = H, ((0)) c and c 0 = C((0)). If 0 c 0 >h 0, then there exist t 0 R and σ { 1, 1} such that (t) = (t + t 0 ), where 1 (t) =σ δ cn( t, k) +h 0 + δ δ sn( t, k) (t) = 1 (δ c 0 +h 0 ) + δ +(δ + c 0 h 0 ) δ sn( t, k) + δ δ sn( t, k) 3 (t) =σ δ δ + c 0 dn( t, k). + δ δ sn( t, k) Here δ = c 0 +4h 0 4c 0 ( + h 0 ), = δ + c 0, k = = c 0 h 0. +δ c 0 δ c 0, and Remark 4.3. If we take the limit of the exression for (t), as tends to 0, then we obtain integral curves for the third case of the system H. 5 Inhomogeneous systems of tye II Among the inhomogeneous systems on so (3), there are four kinds of systems whose euilibria cannot be exressed as unions of lines and lanes (tye II). In fact, there is one one-arameter family of systems, two two-arameter families of systems, and one three-arameter family of systems (see Theorem.4). The stability nature of all euilibria is determined for the system H 1,. On the other hand, for each of the remaining systems (i.e., those with homogeneous art H ) we determine the stability nature of all but one or two euilibrium oints. Again, we omit

29 [9] On the stability and integration of Hamilton-Poisson systems on so(3) 9 roofs for stability results, excet where instability does not follow from sectral instability. However, a full roof is rovided for the system H 5,, as the argument and comutations are more involved. We found it unfeasible to comute exressions for the integral curves, due to comutational comlexity. Some indication of this comlexity can be inferred from the grahs of the critical energy states. 5.1 System H, 1 The system H, () 1 = , > 0 has euations of motion ṗ 1 = 3, ṗ = (1 + 1 ) 3, ṗ 3 = 1 (1 + 1 ). The euilibria are e µ 1 = (eµ 1, (1 e µ ), 0) and e µ =( eµ 1, (1 + e µ ), 0). In Figure 11 we grah the critical energy states (h 0,c 0 ); in Figure 1 we grah the corresonding tyical configurations. (The value = 1 was used for both these figures.) Figure 11: Critical energy states for H 1,. Theorem 5.1. The euilibrium states have the following behaviour: (i) The states e µ 1 are stable. (ii) The states e µ, µ< 3 ln are (sectrally) unstable. (iii) The state e µ, µ = 3 ln is unstable. (iv) The states e µ, µ> 3 ln are stable.

30 30 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [30] (a) (b) (c) (d) (e) Figure 1: Tyical configurations for H 1,. Proof. (iii) Let µ = 3 ln ; we consider the euilibrium state eµ =( 1 3, 1 3 +, 0). We have that! 4 3 (t) = t t 3, 3 + ( t ), 8 t ( t ) is an integral curve of the system H, 1 such that lim t! 1 (t) =e µ. Let B " be the oen ball of radius " = 1 3 centred at the oint e µ. For any neighbourhood V B " of e µ there exists t 0 < 0 such that (t 0 ) V. Furthermore k(0) e µ k = > ", i.e., (0) 6 B ". Thus the state e µ, µ = 3 ln is unstable. 5. System H 3, The system H 3, () = , 1, > 0 has euations of motion ṗ 1 = ( + ) 3, ṗ =( ) 3 ṗ 3 = 1 ( ). The euilibria of this system are given by e µ 1 = ( e µ 1, (1 1 e µ ), 0 ), e µ = ( e µ 1, (1 + 1 e µ ), 0 ) e 3 = ( 1,, ). In Figure 13 we grah the critical energy states (h 0,c 0 ); in Figure 14 we grah the corresonding tyical configurations. (The values 1 = 1 and = 1 were used in both the figures.)

31 [31] On the stability and integration of Hamilton-Poisson systems on so(3) 31 Figure 13: Critical energy states for H 3,. Theorem 5.. The euilibrium states have the following behaviour: (i) The states e µ 1 1, µ<ln are (sectrally) unstable. (ii) The states e µ 1 1, ln ale µ are stable. (iii) The states e µ, µ< 1 3 ln 1 are (sectrally) unstable. (iv) The states e µ, µ> 1 3 ln 1 are stable. (v) The states e 3 are stable. Remark 5.3. The euilibrium state e µ, µ = 1 3 ln 1 is sectrally stable. However, we were unable to determine its Lyaunov stability nature. We susect that this state is unstable (see Figure 14f). 5.3 System H4, The system H4, () = , 1 3 > 0 has euations of motion ṗ 1 =( 3 3 ), ṗ = 3 1 +( ) 3, ṗ 3 = ( ). The euilibria of this system are given by e µ 1 = ( 1 (eµ 1 ), 0, 3 (1 1 e µ ) ), e µ = ( 1 (eµ + 1 ), 0, 3 (1 + 1 e µ ) ) e 3 =( 1,, 3 ).

32 3 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [3] (a) (b) (c) (d) (e) (f) Figure 14: Tyical configurations for H 3,. When 1 = 3 the set of unstable euilibria degenerates (see Figure 15); we treat this case searately. In Figures 15iii and 16 we grah the critical energy states (h 0,c 0 ) and the corresonding tyical configurations. (We used the values 1 = 1, 3 = 1 5 for Figures. 15i, 15iii, and 16 and the values 1 = 3 = 1 for Figure 15ii.) Theorem 5.4. If 1 > 3 > 0, then the euilibrium states have the following behaviour: (i) The states e µ 1 are stable. (ii) The states e µ, 1 3 ln 1 3 <µ<ln 1 are (sectrally) unstable. (iii) The state e µ, µ = ln 1 is unstable. (iv) The states e µ, µ ( 1, 1 3 ln 1 3) [ (ln 1, 1) are stable. (v) The states e 3 are (sectrally) unstable.

33 [33] On the stability and integration of Hamilton-Poisson systems on so(3) 33 (i) 1 > 3 (ii) 1 = 3 (iii) Figure 15: Euilibria and critical energy states for H 4,. If 1 = 3 > 0, then the euilibrium states have the following behaviour: (vi) The states e µ 1 are stable. (vii) The state e µ, µ = ln 1 is unstable. (viii) The states e µ, µ 6= ln 1 are stable. (ix) The states e 3 are (sectrally) unstable. Remark 5.5. The euilibrium state e µ, µ = 1 3 ln 1 3, 1 6= 3 is sectrally stable. However, we were unable to determine its Lyaunov stability nature. We susect it is unstable (see Figure 16e). ln 1 Proof. (iii) Consider the euilibrium state e =( 1, 0, 3 ). We have that (t) = 4( ) 4 + ( ) t 1, ( ) t +( ) t, 3 4( ) 4 + ( ) t is an integral curve of the system H4, ln 1 such that lim t! 1 (t) =e. Let B " ln 1 be the oen ball of radius " = centred at the oint e. For any neighbourhood V B " of e there exists t 0 < 0 such that (t 0 ) V. Furthermore ln 1 ln 1 k(0) e k = ( ) > ", i.e., (0) 6 B ". Hence the state e ln 1 is unstable.

34 34 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [34] (a) (b) (c) (d) (e) (f) (g) Figure 16: Tyical configurations for H 4,. 5.4 System H 5, The system H 5, () = , with > 0, 1 > 3 > 0 or > 0, 1 = 3 > 0, has euations of motion ṗ 1 = 3 ( + ) 3, ṗ = 3 1 +( ) 3, ṗ 3 = 1 ( ). The euilibria are x, x, 3x 1+x 1+x, x 6= 1, x 6= 1 1. These oints are the union of three curves which have resective arametrizations e µ 1 = e µ 1, + 1 e µ, 3( 1 +e µ ) 1 +e µ e µ = tanh(µ) , 1 + tanh(µ), 1 3( + e µ ) e µ 3 = e µ 1 1, 1 1 +e µ, 1 4 3( 1 e µ ).

35 [35] On the stability and integration of Hamilton-Poisson systems on so(3) 35 Figure 17: Critical energy states for H 5,. The first case corresonds to x< 1, the second to 1 <x< 1 1, and the third to x> 1 1. The araboloid (H5, ) 1 (h 0 ) and shere C 1 (c 0 ) are tangent at so (3) if and only if h 0 =H5, (), c 0 =C(), and = ale 1 3 for some ale R. For 6= 0, this yields = 1 (ale 1), (ale 1 ), 3 ale, ale 6= 0, 1, 1. In other words, aart from at the origin, the level surfaces of H5, and C are tangent at the oints e ale 1 = (ale 1), 3 (ale 1 ),, ale 6= 0, 1 ale, 1. We shall find it more convenient to use this arametrization of the euilibria (covering all euilibrium oints excet the origin) in determining the stability nature of the euilibria. We note that e ale = e µ 1 for 1 < ale = eµ + 1 (e µ < 1, µ R + 1 ) e ale = e µ for 1 0 < ale = +e µ < 1,µR e ale = e µ 3 for ale = eµ e µ < 0, µ<ln 1 1 or ale = eµ e µ 1 > 1, µ>ln 1. In Figure 17 we grah the critical energy states (h 0,c 0 ); in Figure 18 we grah the corresonding tyical configurations. (We used the values 1 = 1, = 5, 3 = 1 for these figures.) The olynomial P (ale) = 3 ( 1 3ale +ale ) 3 + ale 3 ( 8 (ale 1) 3 1(ale 1) 3) will be central to our discussion of the stability nature of the euilibria.

36 36 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [36] (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 18: Tyical configurations for H 5,. Lemma 5.6. The olynomial P (ale) has exactly two real roots ale 1 (0, 1 ) and ale ( 1, 1). Proof. We have that P ( ale) = 3 9 3ale 33 3ale ( ) ale 3 ( ) ale 4 ( ) ale 5 ( ) ale 6. Thus P ( ale) < 0 for ale 0 and so P has no nonositive real roots. Furthermore, P (0) = 3 < 0, P ( 1 )= 8 > 0, and P (1) = 1 < 0. Therefore P has at least one root in (0, 1 ) and at least one root in ( 1, 1). As ( 1 3ale +ale ) 3 > 0, (ale 1) 3 > 0, and (ale 1) 3 > 0 for ale > 1, it follows that P (ale) < 0 for ale > 1. Thus P has no real roots in (1, 1).

37 [37] On the stability and integration of Hamilton-Poisson systems on so(3) 37 It remains to be shown that P has at most one real root in (0, 1 ) and at most one real root in ( 1, 1). Suose ale ( 1, 1). Then we have that 1 6ale +8ale 0 and so d dale P (ale) ale 1 = 48 (ale 1) ale 6 1ale ( 1 6ale +8ale ) 6 3(ale 1) (3 + ale(4ale 5)) < 6 3ale ( 1 6ale +8ale ) 6 3( 1+ale) (3 + ale( 5+4ale)) = 6( 3 3 ale) (3 + 4(ale 1)ale) < 0. Hence P is strictly decreasing on ( 1, 1). Therefore P has at most one real root in ( 1, 1). Similar comutations, although somewhat more involved, show that P is strictly increasing on (0, 1 ); hence P has at most one real root in (0, 1 ). Theorem 5.7. The euilibrium states have the following behaviour: (i) The states e ale, ale ( 1, ale ), or corresondingly e µ 1 (sectrally) unstable. (ii) The states e ale, ale (ale, 1), or corresondingly e µ 1 (iii) The states e ale, ale (0, ale 1 ), or corresondingly e µ, µ> 1, are (sec- (iv) The states e ale, ale (ale 1, 1 ), or corresondingly eµ, µ< 1 trally) unstable. (v) The states e ale, ale ( 1, 0) [ (1, 1), or corresondingly e µ 3 stable. (vi) The origin e µ 1 3, µ = ln is stable. 1(ale 1), µ < ln (1 ale, are ) 1(ale 1), µ>ln (1 ale ), are stable. 1 ale1 ln ale 1, are stable. ln 1 ale1 ale 1, µ 6= ln 1, are Remark 5.8. The states e ale1 and e ale are sectrally stable. However, we were unable to determine their Lyaunov stability nature. We susect that they are unstable (see Figures. 18e and 18i). Proof. The linearization of the system at e ale has eigenvalues ± P (ale) ale (1 3ale+ale ) and 0. Hence, as ale ( 1 3ale +ale ) > 0 for ale 6= 0, ale 6= 1, ale 6= 1, we have a ositive real eigenvalue if and only if P (ale) > 0. We have that P (0) = 3 < 0, P ( 1 )= 8 > 0, and P (1) = 1 < 0. Furthermore, by the foregoing lemma, P has exactly two real roots ale 1 (0, 1 ) and ale ( 1, 1). Therefore P (ale) > 0 for ale (ale 1, ale ) and P (ale) ale 0 for ale ( 1, ale 1 ] [ [ale, 1). Conseuently, the euilibrium states e ale, ale (ale 1, 1 ) and eale, ale ( 1, ale ) are sectrally unstable; all other states are sectrally stable. Consider the energy function H λ = λh5, λalec. We have d H λ (e ale )=0 and d H λ (e ale ) = diag((1 ale)λ, λ(1 ale), aleλ). Suose ale ( 1, 0) [ (1, 1) and

38 38 R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING [38] let λ = ale. Then d H λ = diag((ale 1)ale, ale(ale 1), ale λ) is ositive definite. Therefore the states e ale, ale ( 1, 0) [ (1, 1) are stable. On the other hand, assume that ale (0, 1). It is a simle matter to show that ker d C(e ale ) if and only if 1 = (1 ale)( ale+ 3(ale 1)3) 1ale(ale 1) basis 3 (1 ale) (1 ale) 1 (ale 1), 1, 0, The restriction of d H λ (e ale ) to ker d C(e ale ) is Q = 4 (8 (ale 1)3 + 1 (ale 1)3 )λ 1 (1 ale) 1 ale, 0, (ale 1)3 λ 1 ale(ale 1) 4 3(ale 1)3 λ 1 ale(ale 1) ( 3 (ale 1)3 + 1 ale3 )λ 1 ale, i.e., ker d C(e ale ) has Suose ale (0, ale 1 ) and let λ = 1. Then the first minor 8 (ale 1)3 + 1 (ale 1)3 > 0 1 (1 ale) and det Q = P (ale). Hence, as P 1 (1 ale) ale is negative on (0, ale 1 ), we have det Q>0 and so the states e ale, ale (0, ale 1 ) are stable. Suose ale (ale, 1) and let λ = 1. We have that ( 3 (ale 1)3 + 1 ale3 ) > 0 and det Q = P (ale). Hence, as P 1 ale 1 (1 ale) ale is negative on (ale, 1), we have det Q>0 and so the states e ale, ale (ale, 1) are stable Comments and concluding remarks Quadratic Hamilton-Poisson systems lay a notable role in the context of invariant otimal control. To each invariant otimal control affine roblem on a Lie grou one can associate, via the Pontryagin Maximum Princile, a uadratic Hamilton- Poisson system on the dual sace of the corresonding Lie algebra. The extremal controls are (linearly) related to the integral curves of this Hamiltonian system. (For more details see, e.g., [5, 13, 6, 3].) Accordingly, in order to obtain the extremal trajectories for an otimal control roblem on SO (3), one needs to find the integral curves of the associated uadratic Hamilton-Poisson system on so (3). Any such uadratic Hamilton-Poisson system is euivalent to one of the normal forms enumerated in Theorem.4. Two illustrative examles follow. Examle 6.1. Consider the otimal control roblem on SO (3) secified by 8 ġ = g(e 1 + u 1 E + u E 3 ), g SO (3), u =(u 1,u ) R >< g(0) = g 0 and g(t )=g T >: J = Z T 0 ( c1 u 1 (t) + c u (t) ) dt! min, c 1 c > 0.