summary:Let $X$ a smooth finite-dimensional manifold and $W_\Gamma(X)$ the manifold of geodesic arcs of a symmetric linear connection $\Gamma$ on $X$. In a previous paper [Differential Geometry and Applications (Brno, 1995) 603-610 (1996; Zbl 0859.58011)] the author introduces and studies the Poisson manifolds of geodesic arcs, i.e. manifolds of geodesic arcs equipped with certain Poisson structure. In this paper the author obtains necessary and sufficient conditions for that a given Lagrange function generates a Poisson manifold of geodesic arcs. These conditions are formulated in terms of tangent Frobenius algebras

Klapka, Lubomír

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Lagrange functions generating Poisson manifolds of geodesic arcs

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WSGP 19Lubomír KlapkaLagrange functions generating Poisson manifolds of geodesic arcsIn: Jan Slovák and Martin Čadek (eds.): Proceedings of the 19th Winter School "Geometry andPhysics". Circolo Matematico di Palermo, Palermo, 2000. Rendiconti del Circolo Matematico diPalermo, Serie II, Supplemento No. 63. pp. 113--119.Persistent URL: http://dml.cz/dmlcz/701654Terms of use:© Circolo Matematico di Palermo, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czRENDICONTI DELCIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 63 (2000) pp. 113-119 LAGRANGE FUNCTIONS GENERATING POISSON MANIFOLDS OF GEODESIC ARCS LUBOMIR KLAPKA ABSTRACT. Necessary and sufficient conditions are found under which a given La-grange function generates a Poisson manifold of geodesic arcs. These conditions are framed in terms of tangent Frobenius algebras. 1. BASIC NOTIONS In this paper notions of geodesies, Lagrangian mechanics, linear connections, Poisson manifolds, Frobenius algebras and homogeneous functions are used in the usual sense (see, e.g. [1], [2], [4], [5] and [6]). In all local expressions we use the standard summation convention. Let us consider the closed interval [0,1] C R, a smooth finite-dimensional manifold X, the tangent bundle TX, the canonical projection n : TX —> X, and a smooth symmetric linear connection F on TX. A geodesic [0,1] -> X of the connection T is called a geodesic arc. Let Wr{X) be the set of all geodesic arcs. It is well known that there exists a bijective mapping /3r : Wr{X) 3 7 -> 7(0) E codom/3r, where 7 is the prolongation of the geodesic arc 7 on tangent bundle TX. The subset codom/3r C TX is open and contains the zero section. The set IVr(X) equipped with a structure of smooth fibered manifold such that (3r is an isomorphism of smooth fibered manifolds is called a manifold of geodesic arcs. Let M be the set of all polynomial mappings [0,1] -> [0,1] of degree < 1. Then it is known that for any \x € M there exists the smooth mapping R^ : Wr{X) B 7 —> 7°/^ € Wr{X). A Poisson manifold of geodesic arcs is a manifold of geodesic arcs JVr(K") equipped with a Poisson structure such that all mappings i?M, where /i e M, are endomorphisms of Wr{X). General Poisson manifolds of geodesic arcs are the subject of the paper [3]. 1991 Mathematics Subject Classification. 16L60, 16P10, 53B05, 53C15, 53C22, 58D15, 58F05, 70H99. Key words and phrases. Frobenius algebras, fibrations of algebras, linear connections, Poisson manifolds, geodesies, geodesic arcs, manifolds of geodesic arcs, symplectic geometry, Lagrangian mechanics. The paper is in final form and no version of it will be submitted elsewhere. 114 L. KLAPKA A mapping whose codomain is R will be called a function. Let us consider a smooth regular Lagrange function L, where dom L C TX is an open submanifold equipped with the canonical symplectic structure. Any mapping [0,1] —> X satisfying the corre-sponding Euler-Lagrange equations is called an extremal arc of the Lagrange function L. Let WL(X) be the set of all extremal arcs of L. The set WL(X) equipped with a symplectic structure such that the bijective mapping (3L : WL(X) 3 7 - ) 7(0) G codom f3L C dom L is an isomorphism of symplectic manifolds is called a symplectic manifold of extremal arcs. We say that the Lagrange function L generates a Poisson manifold of geodesic arcs Wr(X) if and only if (1) WL(X) C Wr(X) is a symplectic submanifold, (2) WL(X) 9 7 ^ 7(0) G X is a surjective mapping. Let us remark that using local expressions (see [3]) we get the following two assertions: No Poisson manifold of geodesic arcs is symplectic, so WL(X) ^ Wr(X). If L is a Lagrange function satisfying (2), then there exists at most one Poisson manifold of geodesic arcs satisfying (1). Let Z —> X be a fibered manifold, / be a function such that dom / C Z. We say that / is X-projectable if there exists a function / such that / = / o #, where n : Z —> X is the canonical projection. 2. FlBRATIONS OF ALGEBRAS Throughout this paper an algebra A is a finite-dimensional R-module A together with a bilinear multiplication A x A -> A which makes A into an associative ring with the unity element. A structure tensor of A is the tensor of the type (2,1) associated with this multiplication. An algebra A is called commutative if A is a commutative ring. Any algebra A is a left .4-module. The dual R-module A* equipped with the multiplication A x A* ^ (a, a) -> (A ^ 6 -> a(ba) £ R) G A* is a left ,4-module as well. An algebra A is a Frobenius algebra if and only if the left A-modules A and A* are isomorphic. An algebra A* is a dual Frobenius algebra of A if and only if the following conditions hold: (i) A is a Frobenius algebra; (ii) there exists an isomorphism of algebras A -> A*; (iii) the isomorphism of algebras A —> A* is an isomorphism of left _4-modules. The unity element in the dual Frobenius algebra A* will be denoted by (-) :A^a-> (a) eR. Let A be an algebra. Denote by exp the mapping that takes each point a € A to y(l) € A, where y : R -> A is the solution of the differential equation dy/dr = ay under the condition y(0) — 1. The mapping exp exists and the solution y is given by ! / : 1 B T - 4 exp(ra) G A. Moreover, the mapping exp is a local diffeomorphism. This means that for any a0 G A there is a neighborhood [ / 3 c 0 such that the mapping U 3 a —> exp a G expU is a diffeomorphism. Therefore, we can locally define a smooth mapping In : exp U -> U by the formula In o exp|U = idfj. Let A be an commutative algebra. By the above for each pair a, b G A it follows that exp(a + 6) = exp(a) exp(6) and so dexp(a-f-òr) dт = &exp(a) T = 0 LAGRANGE FUNCTIONS GENERATING POISSON MANIFOLDS OF GEODESIC ARCS 1 1 5 A vector bundle Z —> X is called a fibration of algebras if the following conditions hold: (i) any fiber of Z is an algebra; (ii) the structure tensor field is smooth. If Z -> X is a fibration of algebras, then the mapping Z 3 z -» exp(z) € Z is a local diffeomorphism. Over the manifold X we shall consider partly a fibration of tangent algebras TX, partly a fibration of cotangent algebras T*X. If gk are components of a cotangent algebra structure tensor field, then the com-mutativity gives (4) íľ = íř and the assoeiativity gives (5) 9Ґ9*=9ÍГ im9m-There exists a differential invariant of a structure tensor field. This invariant is a tensor field of the type (3, 2). Its components are (6) rilm _ nüд9Г , nimд9f , siH"1 , nsiЧm , ЉГ , sЖ Jjk -9s-^k~+9s ЭxT+Зk-g^Ţ+Әj-g^ + Эj^+gj Q^ J'___ _ „ - M _ „«___ - „«__! _ nsl___ _ nsm_9l 9s dxi ys dxi yj dxk 9k dx° yk dx* 9k dx* It is easy to prove that (4), (5), (6) imply Jf? = j j f = J}f = -Jfҷ. 3. LOCAL EXPRESSIONS On TX we shall use standard local fiber coordinates xl,v%. The canonical symplectic structure on codom /?£ C dom L is defined by the relations m {̂ }=o, H £ H {&&}-"• The Hamilton function is defined by the relation (8) H = vi_^i-L. Lemma 1. A given smooth Lagrange function L, where domL C TX, 7r(codom/?i) = X, generates a Poisson manifold of geodesic arcs if and only if on a neighborhood of every point vQ G codom @L there exist X-projectable functions g\k = gkl, T3kl = TJlk such that (91 Jk J __________ -Xi [J) 9l V dvkdv3-6>> (^c^^ d2L , k , d2L , dL n (10) WW^M-MM***?-0' 116 L. KLAPKA Proof. Let us suppose that L generates a Poisson manifold of geodesic arcs Wr(X). Then from (1) we get codom/?L C codom/3r- On codom/?£,, relations (7) imply (11) {x\xj} = 0. Since xl are X-projectable functions, (11) can be extended to codom/?p. Let us consider a Poisson structure on codom PY such that PY is an isomorphism of Poisson manifolds. If k E [0,1], v e codom /3r? * : [0,1] 3 r -» fer € [0,1], then ProRKop~l(v) = kv. Let x%, vk be standard fibef coordinates on a neighborhood of a point v0 G codom PY-Since /3YORKO/3~1 is an endomorphism of the Poisson manifold codom PY, {xl,vk} are homogeneous functions of degree 1 in vx. Because codom /Jr contains the zero section of TX, these homogeneous functions are polynomials (see, e.g. [5]). Then there exist X-projectable functions g\k such that (12) {x\v>} = <fivk. Relations (1), (7), (12) imply (9), relation (9) implies g% = gft. Denoting by T)k = Tlkj components of the connection F, from (1) we get (10). Conversely, let g\k = gf1, TJkt = Fjk be X-projectable functions on a neighborhood of a point vQ G codom pi such that (9) and (10) hold. Then from (7), (9), (10) we have (11), (12) and (iз) {v\vj} = (9ҐrĹ-ámň Imì Consider the Hamilton vector field (H generated by the Hamilton function (8) on codom Pi. Its components are given, according to (7) and (10), by the relations (14) {x\H} = v\ {v\H} = -rjkxPvk. Since xl, g]k, Tkl are X-projectable functions, vl are globally defined on any fiber, and 7r(codom/?/,) = X, we can extend the Poisson structure defined by the relations (11), (12), (13), the linear symmetric connection F defined by the components Tljk, and the Hamilton vector field £H defined by the relations (14) from the symplectic manifold codom pi to the whole manifold TX. The set WY(X) of all geodesic arcs of the con-nection F can be equipped with a Poisson structure such that /5Y is an isomorphism of Poisson manifolds. Hence, for all li € M we have R(, = PY1 ° 6»(i)-»(o) ° exp(ji(0)f//) o /3r, where fl^ij-^o) is the mapping TX 3 v -» (/x(l) - fi(0))v e TX and exp(/i(0)£H) is the flow of the vector field /i(0)£H. Since R^ : WY(X) -> WY(X) is the composition of four homomorphisms, it is an endomorphism of the Poisson manifold and so Wr(X) is the Poisson manifold of geodesic arcs. From (9), (10) we get (1) and from 7r(codom/3L) = X we get (2). Thus, the Lagrange function L generates the Poisson manifold of geodesic arcs VVr(K). This completes the proof. LAGRANGE FUNCTIONS GENERATING POISSON MANIFOLDS OF GEODESIC ARCS 1 1 7 Lemma 2. Let us suppose that a Lagrange function L, domL C TX, 7r(codom/?_.) = X, satisfies condition (9) of Lemma 1. Then condition (10) of Lemma 1 is satisfied if and only if on a neighborhood of every point VQ € codom/3/, there exist X-projectable functions g\ such that (15) Jjlm = 0, (16) H — QiV1 - const, (17) 9?9k = 8). Proof. Let (9), (10) hold. According to (9), d2L/dvk dvj are homogeneous functions of degree —1 in v%. Hence, i Q*L d2L = ^ ' dvl dvk dvi + dvk dv* ~ From (8), (18) we obtain d2H/dvk dvj = 0. Therefore the Hamilton function is a polynomial of degree less than 2 in vl. Differentiating (10) with respect to vm we see that d2L/dxl dvm — d2L/dxm dvl are homogeneous function of degree 0 in vl. Thus, according to (8), (10), dH/dx1 must be a homogeneous function of degree 1 in v%. We have proved (16). From (9), (10) we get " 9 » ^{WV'-TS)-*' where (20) r = T)kv>vk. Differentiating (19) with respect to t/J, vk, vl, according to (6), (9) we obtain (21) W-*±- J*- *±- J" v' = ^ v ' y" dvi dvP dvk dvi dv' dvr ms dvJ dvk dvl ' Differentiating (8) with respect to u-7, according to (16) we obtain (22» sSV-*-Combining (9), (20), (21), (22) we obtain (15), (17). Conversely, suppose that (9), (15), (16), (17) hold and P is given by (19). Rela-tions (8), (16), (19) imply (23) ^_2rW'(^-^W-; dvi Uk \dxl dxi) From (6), (17) we get дgj дgk _ 1 rilm, According to (15), (23), (24), F are homogeneous functions of degree 2 in vl. Further, according to (15), (21), P are polynomials in v%. Hence, there exist K-projectable functions rj f c = F fc/ such that (20) holds. Finally from (9), (19), (20) we get (10). This completes the proof. 118 L. KLAPKA 4. LAGRANGE FUNCTIONS Theorem. A given smooth Lagrange function L, where dornL C TX, codornL = E, generates a Poisson manifold of geodesic arcs if and only if the three following conditions hold: 1. there exists a fibration of tangent commutative Frobenius algebras TX such that for every v € codom fa L(v) — (v (In v — 1)) 4- const, 2. there exists a fibration of dual Frobenius algebras T*X such that the differential invariant (6) is zero, 3. 7r (codom/?/,) = X. Proof. Let us suppose that L generates a Poisson manifold of geodesic arcs Wr(X). Then from (2) we get 7r(codom/?/J = X. Hence, (4), (9), (10) follow from Lemma 1 and (15), (16), (17) follow from Lemma 2. The functions g\k are components of a tensor field. Its type is (2,1). Since 7r(codom/3L) = X, this field is defined on the whole man-ifold X. Consider the associated bilinear multiplication T*X x T*X -> T*X for any x € X. According to (4), this multiplication is commutative. Differentiating (9) with respect to vm, and multiplying by (gpmg^3' -g?mgpj) vr v\ we obtain (5). Therefore, the multiplication T*XxT*X -» T*X is associative. Since by (17) there exists the unity el-ement, the cotangent space T*X is a commutative algebra. Put g^ = dexp{ o \(v)/dv3\ where A : TXX D codom fa -» T*X is the Legendre transformation \{(v) = dL(v)/dv\ Since (3) implies (25) ^ ) = f f f e x p f c ( p ) , we obtain According to (5), (9), (25), all second derivatives of the mapping expo A are zeros. Whence, g^'s are independent of vJ 's. Multiplying (26) by v\ according to (9) we obtain (27) expjo\(v) = gijV%. Therefore, there exists a linear mapping <D : TXX ~» T*X such that (28) ^kxncodom/?L = exp o A. Because A, exp are local diffeomorphisms, in is a linear isomorphism. Since (9), (26), (27) imply g{jg*J = gtjg-3, we have (tp(a)(p(b))(c) = (<p(c)<p(a))(b) for all a,b,c £ TXX. If we consider the structure of algebra on TXX such that ip is an isomorphism of algebras, we get tp(ab)(c) = ip(ca)(b). If a = 1, then <p(b)(c) = <p(c)(b). Hence, ^(a6)(c) = <£>(&)(ca), and so <p(ab) = atp(b). Therefore, ip is an isomorphism of left FcK-modules, TXX is a .Frobenius algebra, and T*X is a dual Frobenius algebra of TXX. Since x £ X is arbitrary, we get a fibration of tangent commutative Frobenius algebras TX and a fibration of dual Frobenius algebras T*X. From (15) it follows that the corresponding differential invariant (6) is zero. Suppose that v E codom fa. Then v e TXXflcodom /3L, where x = ir(v). Since tp is an isomorphism of Frobenius algebras LAGRANGE FUNCTIONS GENERATING POISSON MANIFOLDS OF GEODESIC ARCS 1 1 9 and isomorphism of left modules, from (28) we obtain v%dL(v)/dvl = \(v)(v) = \n((p(v))(v) = (p(lnv)(v) = (lnv(p(l))(v) = tp(l)(vlnv) = (vlnu). Since (16), (17) imply H(v) = (v) — const, from (8) we get condition 1 of the Theorem. Conversely, suppose that the conditions 1-3 of the Theorem are satisfied. Denoting by g% and gi components of the cotangent algebra structure tensor field and the cotangent algebra unity element field, we have (15) and (17). From (8) and condition 1 of the Theorem by a straightforward computation we get (9), (16). Thus, from Lemmas 1 and 2 it follows that the Lagrange function L generates a Poisson manifold of geodesic arcs. This completes the proof. Acknowledgments. The author is grateful for the support by Grant No. 201/98/0853 of the Czech Grant Agency, and Project No. VS 96003 "Global Analysis" of the Min­istry of Education, Youth and Sports of the Czech Republic. He would like to express his thanks to Jarolim Bures for critical remarks, Peter Michor for bibliographic rec­ommendations, and Jan Slovak for terminology consultations. His very special thank is due to Jifi Vanzura who has recognized that the tangent algebras used in this work were the Frobenius ones. REFERENCES [1] R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd Ed., Benjamin-Cummings. Reading, 1978. [2] C W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, In-terscience Publishers, New York, 1962. [3] L. Klapka, Poisson manifolds of geodesic arcs, in: Differential Geometry and Applications, Proc. Conf. Brno, Czech Republic, 1995 (Masaryk University, Brno, 1996) 603-610. Electronic version in ELibEMS at http://www.emis.de/proceedings/6ICDGA/. [4] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Volume I, Interscience Pub­lishers, New York, 1963. [5] I. Kolaf, P. W. Michor, J. Slovak, Natural Operations in Differential Geometry, Springer, Berlin 1993. [6] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983) 523-577. SILESIAN UNIVERSITY AT OPAVA MATHEMATICAL INSTITUTE BEZRUČOVO NÁM. 13 746 01 OPAVA 1 CZECH REPUBLIC E-mail: LUBOMIR.KLAPKA@MATH.SLU.CZ 

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Lagrange functions generating Poisson manifolds of geodesic arcs

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Institute of Mathematics AS CR, v. v. i.