Hierarchy of integrable geodesic flows

Publicacions Matem`atiques, Vol. 44 (2000), 257–276 HIERARCHY OF INTEGRABLE GEODESIC FLOWS Peter Topalov Abstract A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics. 1. Introduction In papers [1], [2], [3] a simple approach was suggested for obtaining first integrals of Hamiltonian systems if a trajectorial diffeomorphism is given. This approach is closely related to the ideas presented in [4]. Recall briefly the main construction (for details see [1], [2]). Let v and ¯vbe Hamiltonian systems on the symplectic manifolds (M2n,ω) and ( ¯ M2n,¯ω) with Hamiltonians Hand ¯ Hrespectively. Consider the isoenergy surfaces Qdef ={x∈M2n|H(x)=h},¯ Qdef ={x∈¯ M2n|¯ H(x)=¯ h}, where hand ¯ hare regular values of the functions Hand ¯ H. Definition 1. A diffeomorphism φ:Q→¯ Qis said to be trajectorial,if it takes the trajectories of the system vto the trajectories of the system ¯v. Let φ:Q→¯ Qbe a trajectorial diffeomorphism. Let us denote the restrictions ω|Qand ¯ω|¯ Qalso by the letters ωand ¯ω. Then the pullback vanish (of course, also L¯vω= 0). It is obvious that the kernels of the forms ωand φ∗¯ωcoincide with the linear span of the vector v. Therefore, these forms induce two non-degenerate tensor fields on the quotient bundle TQ/ v. Thus, the characteristic polynomial of the operator ω−1◦φ∗¯ωis preserved by the flow v. In papers [2], [3] the construction was applied to a classical example where such a diffeomorphism exists —the geodesic flows corresponding to a pair of geodesically equivalent metrics. Let gand ¯gbe Riemannian metrics on the manifold Mn. 258 P. Topalov Definition 2. The pseudo-Riemannian metrics gand ¯gare called geodesically equivalent iff they have the same geodesics (considered as unparameterized curves). For geodesically equivalent Riemannian metrics, a trajectorial diffeomorphism Φ is given by the formula Φ : (x, ξ)→ x, ||ξ||g ||ξ||¯gξ, where ξ∈T xMn. Denote by Mq(g, ¯g)(1≤q≤n) the set of points y∈Mnsuch that the Riemannian metrics gand ¯ghave exactly qdistinct eigenvalues in an open neighborhood of the point y. If the set Mnis everywhere dense in Mnwe say that the Riemannian metrics gand ¯gare strictly non-proportional on Mn. In the papers [2], [3] the following was proved. Theorem 1. Suppose the Riemannian metrics gand ¯gare geodesically equivalent; then a) the geodesic flow of the metric gadmits nintegrals (k≥0) Ik def =(−1)kdet(g) det(¯g)k+2 n+1 ¯g(Skv,v),(1) where Sk def =¯ Gk−σ1(¯ G)¯ Gk−1+···+(−1)kσk(¯ G)(2) and ¯ Gdef =(gij ¯gjk),σi(¯ G)are the elementary symmetric polynomials of degree i; b) the integrals Ik(k≥0) are in involution1; c) if Mq=∅, then the rank of the differentials dI0,... ,dI n−1is equal to qalmost everywhere in TMq. Moreover, on TMqwe have rk(dI0,... ,dI n−1) = rk(dI0,... ,dI q−1)≤q.(3) This theorem is closely related to some results proved by U. Dini, P. Painlev´e, T. Levi-Civita and R. Liouville (see [5], [6]). In the present paper we assign to any pair of geodesically equivalent metrics a hierarchy of integrable geodesic flows (Theorem 5). The paper is organized as follows. In Section 2 we present some important facts from the theory of geodesic mappings needed for the sequel. In Section 3 we assign to any pair of strictly non-proportional geodesically equivalent metrics gand ¯ga family of completely integrable 1I.e., making the Legendre transformation we obtain nfunctions commuting with respect to the canonical symplectic structure on T∗M. Hierarchy of integrable geodesic flows 259 Riemannian metrics S(g, ¯g) (see Theorem 5). Some simple properties of these families are obtained. In Section 4 the results obtained in Section 3 are applied to the metrics dg2def = n  i=1 dxi2 (4) and d¯g2def =1 n i=1 xi ai2 n  i=1 dxi2 ai (5) which are geodesically equivalent on the standard ellipsoid Eh def = n i=1 xi2 ai=h,h>0. We give explicit formulae for the family S(dg2|Eh,d¯g2|Eh) in terms of some metrics on the whole Rn(Theorem 6). It is interesting that this family contains the metric on the Poisson sphere, well-known in classical mechanics. It immediately permits us to prove that the geodesic flow on the standard ellipsoid and the geodesic flow on the Poisson sphere have the same Liouville foliation (Corollary 3). This theorem is a multidimensional generalization of the well-known one in the two-dimensional case. In addition, we give an explicit formula for a metric geodesically equivalent to the metric on the Poisson sphere (Corollary 2). In Section 5 we obtain a family of completely integrable Hamiltonians with respect to the standard Lie-Poisson bracket on e(3)∗(the dual space to the Lie algebra e(3)). In particular, we obtain an algebraic Hamiltonian H(1) such that the corresponding Hamiltonian system sgrad H(1) is orbitally equivalent to the Euler case of the free motion of the rigid body (see Corollary 6). Throughout the paper the standard agreement holds, i.e. Riemannian metric means a positive-definite symmetric form and pseudo-Riemannian metric means a non-degenerate symmetric form. We consider mainly Riemannian metrics; nevertheless many results hold in the pseudoRiemannian case. The author is grateful to A. V. Bolsinov, A. T. Fomenko, V. V. Kozlov, V. S. Matveev, S. Tabachnikov and I. A. Taimanov for useful discussions. This paper was written during my stay at the Max-Planck-Institut f¨ur Mathematik in Bonn. It is pleasure to thank the Institut for its hospitality and financial support. The author is partially supported by MESC grant MM-810/98. 260 P. Topalov 2. Geodesically equivalent metrics and the corresponding 1-parameter family The following lemma is needed for the sequel. Lemma 1 (see [8]). 1) Suppose the pseudo-Riemannian metrics gand ¯gare geodesically equivalent; then the tensors aij and λi aij def =Aα igαj,(6) λi def =−Aα iψα,(7) where (n+1)ψi=1 2∂iln ¯g g,g= det(gij), and the operator Ais given by formula Ai j(g, ¯g)def = ¯g g 1 n+1 ¯giαgαj,(8) satisfy the equation aij,k =λigjk +λjgik.(9) Here aij,k denotes the covariant derivative ∇kaij, where ∇is the Levi-Civita connection corresponding to the metric g. 2) Conversely, if a non-degenerate symmetric tensor field aij and an 1-form λisatisfy equation (9), then the metric ¯gij def =ˆg gˆgij,(10) where ˆgij def =giαaαβgβj, is geodesically equivalent to g. It is obvious that Ais self-adjoint with respect the both metrics. Now, using Lemma 1 we can prove Proposition 1. Suppose the pseudo-Riemannian metrics gand ¯gare geodesically equivalent; then if for some parameters αand βthe operator (αA+β)is invertible on Mn, then the pseudo-Riemannian metrics g and ¯gα,β(X,Y )def =1 det(αA +β)g((αA +β)−1X,Y ),(11) are geodesically equivalent. Remark 1.Remark here that ¯g0,1=gand ¯g1,0=¯g. Hierarchy of integrable geodesic flows 261 Remark 2.If the manifold Mnis compact, then formula (11) gives a 1-parameter family of geodesically equivalent metrics. Moreover, in a small neighborhood of each point of Mnwe also obtain 1-parameter family. Remark 3.Let us take two representatives ¯gα,β and ¯gλ,µ. Now, we are able to apply Proposition 1 again. The corresponding family of geodesically equivalent metrics is ¯ga,b =1 det((aλ +bα)A+(aµ +bβ))g((aλ +bα)A+(aµ +bβ))−1.(12) Therefore, if (α:β)=(λ:µ), then we obtain the same family. Remark here also that A(¯gα,β,¯gλ,µ)= λA +µ αA +β.(13) Proof of Proposition 1: Let gand ¯gbe geodesically equivalent metrics. Using (6), we obtain that the symmetric form a=gA satisfies equation (9) for some λ. Hence, the form ac def =gA +cg (c= const) also satisfied this equation. We use here that ∇is the Levi-Civita connection corresponding to the metric g. Assume that acis non-degenerate on Mn. We get ˆgdef =g(gA +cg)−1g=g(A+c)−1and using the converse formula (10) obtain that the pseudo-Riemannian metric ¯gc=1 det(A+c)g(A+c)−1 (14) is geodesically equivalent to g. Now, using Proposition 1 we are able to present another ‘non-symplectic’ proof of the first item of Theorem 1. Corollary 1. If the Riemannian metrics gand ¯gare geodesically equivalent, then the geodesic flow of the metric gadmits a 1-parameter family of first integrals Iα,β(g, ¯g)def = det(αA +β)g(αA +β)−1.(15) Proof of Corollary 1: The corollary easily follows from the next theorem (see [6]). 262 P. Topalov Theorem 2 (Painlev´e).If the metrics gand ¯gare geodesically equivalent, then the function I0(g)def =det(g) det(¯g)2 n+1 ¯g(16) is an integral of the the geodesic flow of the metric g. In a small neighborhood of each point of Mnwe can find a 1-parameter family ¯g1,β of geodesically equivalent Riemannian metrics. Therefore, locally the Riemannian metrics gand ¯g1,β are geodesically equivalent, where β∈(a, b), a<b. It permits us to apply Theorem 2. We have I1,β(g)def =det(g) det(¯g1,β)2 n+1 ¯g1,β (17) = det(A+β)g(A+β)−1 (18) =I0+I1β+···+In−1βn−1,(19) where the functions Ikare the same as in Theorem 1. The last equality in this chain can easily be proved. Therefore, the functions Ikare integrals in a small neighborhood of each point of Mn. But they are globally defined on Mn. Thus, they are integrals. Remark 4.If we apply Corollary 1 to the geodesically equivalent metrics gand ¯gα,β ((α:β)= (0 : 1)), then we’ll obtain the same family of integrals. Remark 5.Let us consider the map Φ : ξ→ ||ξ||g ||ξ||¯gξ2, Φ:(TM)0→(TM)0.(20) It is obvious that Φ maps each geodesic trajectory of the metric ginto a geodesic trajectory of the metric ¯g. Therefore, the pull-back Φ∗ (Iα,β(¯g,g)) gives a family of integrals for the geodesic flow of the metric g. It can easily be checked that Φ∗(Iα,β(¯g,g)) = g(ξ,ξ) I1,0(g, ¯g)Iβ,α(g, ¯g).(21) Thus, we won’t be able to obtain new family of integrals. 2Sometimes we will denote this mapping by Φ(g, ¯g). Hierarchy of integrable geodesic flows 263 3. Hierarchy of integrable flows As we have seen if the Riemannian metrics gand ¯gare geodesically equivalent, then they are contained in a family of geodesically equivalent metrics ¯gα,β =1 det(αA +β)g(αA +β)−1,(22) where αand βare parameters. Remark 6.In the following sections we consider only Riemannian metrics although the most of the constructions pass in pseudo-Riemannian case. It is interesting that using geodesically equivalent Riemannian metrics gand ¯gwe can produce another families of geodesically equivalent metrics. We need Theorem 3 (see [8]).If the Riemannian metrics gand ¯gare geodesically equivalent, then for each integer kthe Riemannian metrics g(k)def = gAkand ¯g(k)def =¯gAkare also geodesically equivalent. This theorem may be proved by direct calculations. For details see paper [8]. We will slightly generalize this result later using some other arguments. At first we need some notations. Let Bbe a self-adjoint operator on the connected Riemannian manifold (Mn,g). By definition, put r(B)def = inf x∈Mmin{spec B(x)}(23) and R(B)def = sup x∈M max{spec B(x)}.(24) Denote by Ithe set obtained by adding to the interval (r, R) its endpoints iff they are achieved for some x∈Mn. Let us consider the set of all real Laurent series La(x)=kck(x−a)kwhich are convergent on some open neighborhood of I.Denote by ω+(B)the cone of all finite linear combinations of such series which give positive functions on I.Of course, we are able to consider more general set of functions but it will only complicate our construction. 264 P. Topalov Theorem 4. Suppose the Riemannian metrics gand ¯gare geodesically equivalent and F(x)∈ω+(A); then the metrics gFdef =gF(A)and ¯gFdef = ¯gF(A)are also geodesically equivalent. These metrics are contained in a family of geodesically equivalent Riemannian metrics ¯gF α,β def =1 det(αA +β)gF(A)(αA +β)−1,(25) where αand βare parameters such that the operator (αA +β)is nondegenerate. Proof of Theorem 4: Let gand ¯gbe geodesically equivalent Riemannian metrics on a manifold Mn. Denote by ρ1,... ,ρ m(1 ≤m≤n) the common eigenvalues of the metrics gand ¯g. Suppose the functions ρ1,... ,ρ m are different at every point of an open domain D⊂Mn. In the paper [6], T. Levi-Civita proved that for every point P∈Dthere is an open neighborhood U(P)⊂Dand a coordinate system ¯x=(¯x1,... ,¯xm) (in U(P)), where ¯xi=(x1 i,... ,x ki i), (1 ≤i≤m), such that the quadratic forms of the metrics gand ¯ghave the following form: g(˙ ¯x, ˙ ¯x)= m  i=1 Πi(¯x)Ai(¯xi,˙ ¯xi),(26) ¯g(˙ ¯x, ˙ ¯x)= m  i=1 ρiΠi(¯x)Ai(¯xi,˙ ¯xi),(27) where Ai(¯xi,˙ ¯xi) are positive-definite quadratic forms in the velocities ˙ ¯xi with coefficients depending on ¯xi, Πi def =(φi−φ1)...(φi−φi−1)(φi+1 −φi)...(φm−φi),(28) ρi=1 φ1...φ m 1 φi (29) and φ1,φ 2,... ,φ m(0 <φ 1<φ 2<···<φ m) are smooth functions such that φi=φi(¯xi),if ki=1 constant,else. Hierarchy of integrable geodesic flows 265 Definition 3. Let metrics gand ¯gbe given by formulae (26) and (27) in a coordinate chart U. Then we say that the metrics gand ¯ghave LeviCivita local form of (type m), and the coordinate chart Uis Levi-Civita coordinate chart (with respect to the metrics). In the paper [6], Levi-Civita proved that the metrics gand ¯ggiven by formulae (26) and (27) are geodesically equivalent. Denote by Mthe set of all point in Mnwhich are contained in some Levi-Civita chart, i.e. x∈Miff there is a Levi-Civita chart Uwhich contains x. By definition, Mis an open subset of Mn.In[2], [3]was proved that Mis everywhere dense in Mn. In every Levi-Civita coordinate chart we have ¯g g1 n+1 =1 φk1−1 1...φ km−1 m1 φ1...φ m = const.1 φ1...φ m ,(30) (¯gikgkj) = diag 1 ρ1,... , 1 ρ1    k1 ;... ;1 ρm,... , 1 ρm    km .(31) Therefore, A(g, ¯g) = const.diag(φ1,... ,φ 1    k1 ;... ;φm,... ,φ m    km ).(32) Hence, g(F(A)˙ ¯x, ˙ ¯x)= m  i=1 Πi(¯x)A i(¯xi,˙ ¯xi),(33) and ¯g(F(A)˙ ¯x, ˙ ¯x)= m  i=1 ρiΠi(¯x)A i(¯xi,˙ ¯xi),(34) where A i=fi(¯xi)Aiand fi(¯xi) is a smooth function depending of the variable ¯xi. Therefore, in any Levi-Civita chart the metrics gF(A) and ¯gF(A) are geodesically equivalent. Let us consider the map Φ : ξ→ ||ξ||g ||ξ||¯gξ, Φ:(TM)0→(TM)0.(35) 272 P. Topalov This construction generalize the well-known metric on the Poisson sphere considered in the classical mechanics (see [10]). Corollary 2. The restriction of the metric dg2 (1) to the ellipsoid Ehhas the same geodesic lines as the metric on the Poisson sphere. Using Proposition 2, we obtain Corollary 3. The geodesic flows on the ellipsoid and the Poisson sphere have identical Liouville foliations. If n= 2 the last result is well-known. Moreover, in [11] A. T. Fomenko and A. V. Bolsinov using the theory of orbital equivalence of the integrable Hamiltonian systems proved that for every 2-dimensional ellipsoid we can take a suitable Poisson sphere such that the corresponding geodesic flows are continuously orbitally equivalent. It is interesting to solve the same problem in multidimensional case. The standard sphere and the Poisson sphere. In the case k=−1 we have dg2 (−1) =A(x)dx, dx (55) =A−1dx, dx−2A−1x, dxA−2x, dx A−1x, A−1x.(56) The last term on the right side vanish on TEh. Therefore, we can think that dg2 (−1) =A−1dx, dx. Changing the variables x=√Ay we see that g(−1) is the standard metric on the sphere. Further, d¯g2 (−1) =!A−1dx, A−1dx−A−1x,A−1dx2 A−1x,A−1x" A−1x, A−1x.(57) Corollary 4. The restriction of the metric d¯g2 (−1) =A−1dy, dy−A−1y,dy2 A−1y,y A−1y,y (58) to the sphere Sn={y,y=h>0}has the same geodesic lines as the standard metric on the sphere. Hierarchy of integrable geodesic flows 273 Using Proposition 3, we obtain Corollary 5. The metric on the Poisson sphere d¯g2 (1)|Ehand the standard metric on the sphere dg2 (−1)|Ehare completely integrable on T∗M such that the corresponding integrals are the same. 5. A family of integrable Hamiltonians in e(3) ∗ It is well-known that the Euler-Poisson equations which describe the motion of the rigid body can be written as Euler equations on the dual space of the Lie algebra e(3) (see [12], [13]). More precisely, if the coordinates in e(3)∗are denoted by (r1,r 2,r 3,s 1,s 2,s 3), then the LiePoisson bracket can be given by relations {si,s j}=8ijksk,{ri,r j}=0, {si,r j}=8ijkrk. Therefore, if H=H(r, s) is a Hamiltonian function, then the Euler equations take the form of Kirchoff equations ˙s=s×∂H ∂s +r×∂H ∂r ,(59) ˙r=r×∂H ∂s .(60) The Lie-Poisson bracket has two annihilators F1=r2 1+r2 2+r2 3and F2=r1s1+r2s2+r3s3. The Hamiltonian function for the rigid body is Hdef =1 2s2 1 I1 +s2 2 I2 +s2 3 I3+mg l,r,(61) where s=(s1,s 2,s 3) denotes the angular momentum of the body, l=(l1,l 2,l 3) denotes the coordinates of the center of gravity of the body, I= diag(I1,I 1,I 3) is the tensor of inertia, r=(r1,r 2,r 3) —the coordinates of the unit vertical vector in the space, and mg is the weight of the body. The cotangent bundle T∗S2supplied with the canonical symplectic structure dp ∧dq is symplectomorphic to the manifold Oh def ={F1= h, F2=0}⊂e(3)∗(see [14]). Therefore, any integrable Hamiltonian on T∗S2gives an integrable Hamiltonian on Oh. Moreover, if the first Hamiltonian is polynomial in impulses, then the second one is also polynomial in momenta s=(s1,s 2,s 3) and has the same degree. Using the results of [14] and Lemma 2 it is easy to prove 274 P. Topalov Lemma 3. Suppose that the Riemannian metric dg2=gij dxidxj,x= (x1,x 2,x 3)is smooth in a neighborhood of the sphere S2 h={x, x=h} and the restriction dg2|S2 hgives a metric whose geodesic flow is completely integrable; then the Hamiltonian function Hg(r, s)def =det E det G 1 e(ne,n g)  i,j gij(r)sisj,(62) where de2=(dx1)2+(dx2)2+(dx3)2,G=(gij),E=(eij )and neand ngare the unit external normal vectors to the sphere S2 hcalculated with respect to the metrics eand grespectively at the point x=r∈S2 h,is completely integrable on the submanifold Oh. In Section 4 we found a family of metrics on Rnsuch that their restriction on the ellipsoid Ehgive a family of integrable metrics. Now, using these metrics and Lemma 3 we are able to find a family of integrable Hamiltonians in e(3)∗. Theorem 7. Consider the operators A(r)def =A−(√Ar)⊗(√Ar) r, r+(A−1 2r)⊗(A−1 2r)(63) and B(k)(r)def =√AAk(r)√A.(64) For any fixed integer kthe Hamiltonians H(k)def =1 A−1r, rk+1 B(k)s, s (65) and ¯ H(k)def =1 A−1r, rk−1B(k−1)s, s (66) are Liouville integrable on any orbit Oh. Moreover, the corresponding Hamiltonian systems are orbitally equivalent. Example. If k= 1, then we obtain that the Hamiltonians H(1) =1 A−1r, r2A2s, s−Ar, s2 r, r+r, s2# (67) and ¯ H(1) =As, s(68) are orbitally equivalent. Hierarchy of integrable geodesic flows 275 Corollary 6. The Hamiltonian system with Hamiltonian H(1) and the Euler case of the free motion of the rigid body are orbitally equivalent on the orbit Oh,h>0. References [1] P. J.Topalov, Tensor invariants of natural mechanical systems on compact surfaces, and the corresponding integrals, Mat. Sb. 188(2) (1997), 137–157. [2] P. J. Topalov and V. S. Matveev, Geodesic equivalence and integrability, Preprint of MPI-Bonn (1998), MPI 98-74. 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Kozlov, The de Maupertuis principle and geodesic flows on a sphere that arise from integrable cases of the dynamics of a rigid body, Uspekhi Mat. Nauk 50(3) (1995), 3–32 (in Russian); translation in Russian Math. Surveys 50(3) (1995), 473–501. Department of Differential Equations Institut of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev Str., bl. 8 Sofia 1113 Bulgaria E-mail address:[email protected] Primera versi´o rebuda el 30 d’abril de 1999, darrera versi´o rebuda el 23 de setembre de 1999.