ARCHIVŮM MATHEMATICUM (BRNO) Tomus 50 (2014), 297-316 HIDDEN SYMMETRIES OF THE GRAVITATIONAL CONTACT STRUCTURE OF THE CLASSICAL PHASE SPACE OF GENERAL RELATIVISTIC TEST PARTICLE Josef Janyska Abstract. The phase space of general relativistic test particle is defined as the 1-jet space of motions. A Lorentzian metric defines the canonical contact structure on the odd-dimensional phase space. In the paper we study infinitesimal symmetries of the gravitational contact phase structure which are not generated by spacetime infinitesimal symmetries, i.e. they are hidden symmetries. We prove that Killing multivector fields admit hidden symmetries of the gravitational contact phase structure and we give the explicit description of such hidden symmetries. 1. Introduction A classical spacetime is assumed to be an oriented and time oriented 4-dimensional manifold equipped with a scaled Lorentzian metric. In classical general relativity the phase space is usually defined either as the cotangent bundle with the canonical symplectic structure or as a part of the unit pseudosphere bundle given by time-like future oriented vectors. In the second case the phase space is also called the observer space and the metric defines the canonical contact phase structure. Starting from the papers [7, 6, 8, 14, 21] the phase space can be defined as the 1-jet space of motions. In this case we can use the general theory of jets of submanifolds [20] which allows as to define geometrical structures given naturally by the metric and an electromagnetic fields. Namely, the metric field admits the gravitational contact phase structure and the metric and the electromagnetic fields admit the almost-cosymplectic-contact structure [9]. In the case of the cotangent bundle a phase infinitesimal symmetry is assumed to be an infinitesimal symmetry of the kinetic energy function. It is very well known, [1, 18], that such symmetries are given as the Hamiltonian lift (with respect to the canonical symplectic 2-form) of functions constant of motions. Functions constant of motions which are polynomial on fibres of the cotangent bundle are given by 2010 Mathematics Subject Classification: primary 70H40; secondary 70H45, 70H33, 70G45, 58A20. Key words and phrases: phase space, gravitational contact phase structure, gravitational Jacobi phase structure, infinitesimal symmetry, hidden symmetry, Killing multivector field. Supported by the grant GA ČR 14-02476S. DOL 10.5817/AM2014-5-297 298 j. janyska Killing fc-vector fields, k > 1. For k = 1 the corresponding infinitesimal symmetries are the flow lifts of Killing vector fields and they are projectable on infinitesimal symmetries of the spacetime. For k > 2 the corresponding infinitesimal symmetries are not projectable and they are called hidden symmetries. In the case of odd-dimensional phase space (the observer space or the 1—jet space of motions) the metric defines on the phase space a contact structure and a phase infinitesimal symmetry is assumed to be an infinitesimal symmetry of the contact 1-form. Phase infinitesimal symmetries which are projectable on infinitesimal symmetries of the spacetime were studied on the observer space by Iwai [4] and on 1-jet space of motions by Janyska and Vitolo [11]. In both situations projectable symmetries are given by flow lifts of Killing vector fields. In this paper we describe hidden (nonprojectable) infinitesimal symmetries for the phase space given as the 1-jet space of motions. It is proved that hidden symmetries are given by the Hamilton-Jacobi lifts of (special) conserved phase functions and we give explicit construction of hidden symmetries generated by Killing multivector fields. Our theory is explicitly independent of scales, so we introduce the spaces of scales in the sense of [10]. Any tensor field carries explicit information on its scale dimension. We assume the following basic spaces of scales: the space of time intervals T, the space of lengths L and the space of mass M. We assume the speed of light c € T* ® L and the Planck constant e T* ® L2 ® M as the universal scales. 2. Preliminaries 2.1. Schouten bracket. In 1940 Schouten [17] introduced the differential invariant (Schouten bracket) of two contravariant tensor fields (multivector fields). We recall basic facts about the Schouten bracket of skew symmetric and symmetric multivector fields. Let M be an n-dimensional diflerentiable manifold and let us denote by (xA) local coordinates on M. The induced fibred coordinates on TM and T*M will be denoted by (xA;iA) and (x^ixx), respectively. Let us recall the expression of the Schouten bracket [P, Q] of a skew symmetric p-vector P and a skew symmetric g-vector Q [12, 19] i[P,Q]/3 = (-1)^+9 lPdlQP + (-1)" lQdlPP , for each closed form (3 of degree p + q — 1. Such bracket satisfy the graded antisymmetry and the graded Jacobi identity, so the sheaf of sections T(/\TM) = ©fcr(AfeTM) is a graded Lie algebra. According to [15, 22] we can define the Schouten bracket for a symmetric fc-vector fe l k l field K and a symmetric I-vector field L as the (k +1 — l)-vector field [K, L] defined fe i for decomposable fields K — X\ V • • • V X]„ and L — Y\ V • • • V YJ as [x1v---vxk,Y1v---vYl] = ^[xJ,yJ]vx1v...xJ---vxfcvyiV...yJ ■■■vYi. i,3 hidden symmetries 299 The Schouten bracket of symmetric multivector fields is antisymmetric and satisfy the Jacobi identity, so T(STM) = Q)kT(SkTM) is a Lie algebra with grading but not a graded Lie algebra. Any symmetric vector field K = KXl---Xk d^® - ■ -®d\k, KXl---Xk = Kx'(iy-X'(k~> for any permutation of indices a, defines a function on T*M polynomial and homogeneous of degree k on fibres. So we have the mapping ir* : T(SkTM) —> C°°(T*M) given in coordinates by 7r*(£) = £Al-AfciAl...iAfc. tt* is a homomorphism of Lie algebras, where on C°°(T*M) we consider the canonical Poisson bracket {, } induced by the canonical symplectic 2-form lu = dx\A k I k I dxx, [15]. I.e. tt*([K, L]) — {tt*(K), tt*(L)} which gives the following coordinate fe i expression, for a symmetric fc-vector field K and a symmetric ^-vector field L, (2.1) \K, L] = -—I—- V (k KpKiiy-Kik-^ dpL^w-^+'-v y ' L 1 (k + l-l)\^y 9 l k _ ilpK(1)...K(1-1) dpKKm-K(k+'-v) dXl ■ ■ ■ (9Afe+;_! , where a runs all permutations of indices 1,... ,k + I — 1. 2.2. Killing tensor fields. We recall basic facts about Killing tensor fields on a Riemannian or a pseudo-Riemannian manifold (M,g). In what follows we shall denote by g the contravariant metric. A Killing (0, k)-tensor field is a symmetric (0, &)-tensor field K = K\1...\k dXl ■ ■ ■ ® dXk such that the function K\1...xk xXl ... xXk is constant on geodesic curves of the Levi Civita connection. This condition is equivalent with V(Al^A2...Afe+1) =0. Equivalently we can define a Killing (k,0)-tensor field (k-vector field) as a fe fe symmetric (k, 0)-tensor field K — KXl",Xk d\1 ® ■ ■ ■ ® d\k such that gP(Xl\7pKX2'"Xk+1^ — Y/(Ai^A2---Afc+i) — o . It is easy to see that K is a Killing (0, &)-tensor field if and only if K$ — (#" ® • • • ® g^)K is a Killing (k, 0)-tensor field. Remark 2.1. We have the canonical Killing (0,2) and (2,0) tensor fields given by g and g, respectively. n Killing multivector fields can be equivalently defined as symmetric multivector fields satisfying the Killing tensor equation (2.2) [K,g] = 0, 300 j. janyška [22]. Then from the Jacobi identity we get that the sheaf of Killing multivector fields fe i is closed with respect to the Schouten bracket, i.e. if K, L are Killing multivector fe i fields then [K, L] is also a Killing multivector field. fe Remark 2.2. In [18] it was proved that for a symmetric fc-vector field K the fe fe function ir*(K) is constant of motion if and only if K is Killing. On the other fe hand it is equivalent with the fact that the Hamiltonian lift of ir*(K) given by the canonical symplectic 2-form is an infinitesimal symmetry of the kinetic energy function T^g^^ixi^- n 2.3. Structures of odd dimensional manifolds. Let M be a (2n + ^-dimensional manifold. A pre cosymplectic (regular) structure (pair) on M is given by a 1-form lu and a 2-form CI such that lu A Cln ^ 0. A contravariant (regular) structure (pair) (E, A) is given by a vector field E and an antisymmetric 2-vector A such that EAAn ^ 0 . We denote by Clb: TM ->• T*M and A*: T*M ->• TM the corresponding "musical" morphisms. By [13] if (lu,CI) is a pre cosymplectic pair then there exists a unique regular pair (E, A) such that (2.3) (ííb|imA0_1 = AVmS> , ÍiSW=l, itfíí = 0 , i^A = 0 . On the other hand for any regular pair (E, A) there exists a unique (regular) pair (lu, CI) satisfying the above identities. The pairs (lu, CT) and (£", A) satisfying the above identities are said to be mutually dual. The vector field E is usually called the Reeb vector field of the pair (lu, CI). In fact geometrical structures given by dual pairs coincide. An almost-co symplectic-contact structure (pair) [9] is given by a pair (lu, CI) such that dCl = 0 , luACV1 ^ 0 . The dual almost-coPoisson-Jacobi structure (pair) is given by the pair (E, A) such that [E, A] = -EM^(Lelu) , [A, A] = 2£A (A* A')(dw), where lu is the fundamental 1-form satisfying íelu — 1, i^A — 0. Here [, ] is the Schouten bracket of skew symmetric multivector fields. Remark 2.3. An almost-cosymplectic-contact pair generalizes standard cosymplectic and contact pairs. Really, ii diu = 0 we obtain a cosymplectic pair [2]. The corresponding dual pair is coPoisson pair [9] given by the pair (E, A) such that [E, A] — 0, [A, A] — 0. A contact structure (pair) is given by a pair (lu, CI) such that CI — c1lu , lu A Cln ^ 0. The dual Jacobi structure (pair) is given by the pair (E, A) such that [E, A] = 0 , [A, A] = -2E A A. n 2.4. Infinitesimal symmetries of almost-cosymplectic-contact structures. Let (lu, CI) and (E, A) be mutually dual regular structures on an odd dimensional manifold M. An infinitesimal symmetry of the structure (lu, CI) is a vector field X on M such that Lxlu — 0, LxCl — 0. Similarly, an infinitesimal symmetry of the structure (E,A) is a vector field X on AÍ such that LxE — [X,E] — 0, LXA= [X,A] = 0. hidden symmetries 301 Lemma 2.1 ([11]). Let X be a vector field on M. The following conditions are equivalent: 1. Lx^> — 0 and LxCl — 0. 2. LXE = [X, E}=0 and LXA = [X, A] = 0. □ Theorem 2.2 ([5]). A vector field X is an infinitesimal symmetry of the almost-cosy mplectic-contact structure (lo, CI) if and only if it is of local type X = df + hE, where f,he C°°(M) such that E ■ f — 0 and (2.4) idpdbj + hiE du: + dh — Q . Corollary 2.3. 1. An infinitesimal symmetry of the cosymplectic structure (lu,CL) is of local type Y — df + h E, where f G C°°{M) such that E ■ f — 0 and h is a constant. 2. Any infinitesimal symmetry of the contact structure (lu,CL) is of local type (2.5) X = df*-fE, where f e C°°(M) such that E-f = 0. Proof. 1. For a cosymplectic structure we have du> — 0 and (2.4) reduces to dh = 0. 2. For a contact structure we have du> = il and (2.4) reduces to z^jsO + dh = 0 , i.e. dh = —df. Then h = — (/ + k), where A; is a constant. The vector field (2.5) is the Hamilton-Jacobi lift of a function /. But the Hamilton-Jacobi lift of a constant k is the vector field k E which is an infinitesimal symmetry of the contact structure (lu, diu). So all infinitesimal symmetries of the contact structure form an M-algebra and they are Hamilton-Jacobi lifts of functions on M satisfying E ■ f — 0. □ Lemma 2.4. We have iEdh + lEidftdu! — E.h + A(Lelo, df) — 0 . So dh + idp dw € ker E. Proof. If we apply on the equation (2.4) we get iEdh + iEidpdu> — E ■ h — idpiEdw — E ■ h — A(df, Lelo) — 0. □ 2.5. Lie bracket of generators of infinitesimal symmetries. Any infinitesimal symmetry of the almost-cosymplectic-contact structure (lu, CI) can be identified with a pair of functions (/, h) on M such that / is conserved, i.e. E ■ f = 0, and / and h are related by the condition (2.4). The pair (/, h) is said to be a generator of the infinitesimal symmetry of the almost-cosymplectic-contact structure (lu,CI). Lemma 2.5. Suppose two infinitesimal symmetries X — dß + hE and X' — dg^ + k E of the almost-cosymplectic-contact structure (lu, CI). Then [X, X'} = d{f, ff}8 + ({/, k} - {g, h} - dLj((tft,dgt)) E . Proof. We have [X, X'} = [df ,dgi} + [df ,kE] + [hE,dgi] + [hE,kE}. 302 j. janyska By [9] we have [df\ dg*} = dA(df, dg)* - Mdf*, dg*) E = d{f, g}* - dw(df\ dg*) E, [E,df] = K{LELo,df) E = iEidfidLo E which implies [X, X'} = d{f, g}8 + ({/, k} — {g, h} — dw(df*, dg*) + h(E-k + K{Lelo, dg)) -k{E-h + K{Lelo, df))) E and from Lemma 2.4 we obtain Lemma 2.5. □ Infinitesimal symmetries form a Lie algebra with respect to the Lie bracket. This defines the Lie bracket on pairs of functions (2.6) [(/, h);(g,k)] = ({/, g}, {/, k} - {g, h} - dio(df*, dg*)) . Really, this bracket is antisymmetric and satisfies the conditions for generators of infinitesimal symmetries. Namely, from [9] and properties of the almost-coPoisson--Jacobi structure we have (2.7) E ■ {/, g} = {E ■ /, g} + {f,E-g} + i[E,A]df A dg = 0 , i.e. the sheaf of functions satisfying E.f — 0 is closed with respect to the Poisson bracket. The condition (2.4) corresponds to id{f,g}tduj + ({/, k} - {g, h} - du)(dj\dg^)) iEdu +d{f, k} - d{g, h} - d{du(df*,dg*)) = 0 which follows from L^x^x']1-0 — {LxLx' — Lx'Lx) L2 ® (T*E ® T*E), with signature (—h ++); we suppose spacetime to be time oriented. The contravariant metric is denoted by g: E —> L-2 ® (TE ® TE). A spacetime chart is defined to be a chart (xA) = (x°,xl ) e C°°(U, M x M3), U C E is open, of E, which fits the orientation of spacetime and such that the vector field do is timelike and time oriented and the vector fields 81,82, 83 are spacelike. Greek indices A, /i,... will span spacetime coordinates, while Latin indices i,j,... will span spacelike coordinates. In the following, we shall always hidden symmetries 303 refer to spacetime charts. The induced local bases of TE and T*E are denoted, respectively, by (d\) and (dx). We have the coordinate expressions ff = 9\»dx ®(F , with — G C°°(E, L2 ® 1R), cy = aA • T ® (T*E ® T*.E), 6?^ = ^ gx^, and the associated contravariant re-scaled metric G^^gi-E^T*® (T.E T.E), GgM = ^ gx^, where = Hqu0, Hq G L2®M. Eventually, we consider the unsealed metric G-— (^)2g: £7 —> T*E E . The isi differential of the motion s is defined to be the tangent map ds: tt = Txf ^ TE. We assume as phase space the open subspace 3iE C Ji(E, 1) consisting of all 1-jets of motions. The velocity of a motion s is defined to be its 1-jet j±s: T —> 3i(E, 1). For each 1-dimensional submanifold s: T <^-> E and for each x G T, we have j'is(x) G if and only if ds(x)(u) G TS^E is timelike, where w G T. Any spacetime chart (x°,a;J) is related to each motion s which means that s can be locally expressed by (x°,xl — sl(x0)). Then we obtain the induced fibred coordinate chart (x°, xl, x0) on 3iE such that x0 o s = dos1. Moreover, there exists a time unit function T —> T such that the 1st differential of s, considered as the map ds: T —> T* ® T.E, is normalized by g(ds, ds) — — c2 , for details see [8]. We shall always refer to the above fibred charts. We define the contact map to be the unique fibred morphism a: 3iE —> T* ®TE over E, such that a° jis — ds, for each motion s. We have g (a, a) — — c2. The coordinate expression of a is (3.1) a—ca° (do + Xq df), where a0 := 1/-^/|gr0o + 2g0j x30 + gtj xl0 x30\ . The map a: 3\E ->• T* ® TE is injective. Indeed, it makes 3\E C f * ® TE the fibred submanifold over E characterised by the constraint gx^Xg x^ — — (co)2 . We define the time form to be the fibred morphism r — —\ 9^{p) '■ 3\E —> T®T*.E, considered as the scaled horizontal 1-form of 3\E. We have the coordinate expression (3.2) T = Txdx = -^(gox+gixx0)dx. The complementary contact map 8: 3\E Xe TE —> TE is given by 8 = id — a ® t. 304 j. janyska Note 3.1. In what follows it is very convenient to use the following notation 5lx — 5\ — Xq 5® and 5$ — 5$ + 6£ x1^. Then a — ca° 5$ <9M and t — —^-gox dx, where g0\ = g^xS^. n Let V3iE C T3\E be the vertical tangent subbundle over E. The vertical prolongation of the contact map yields the mutually inverse linear fibred isomorphisms vT : t* ® VTE —> VSiE and v'1: V3iE t* VTE , where VTE — kerr C TE, with the coordinate expressions (3.3) vT = -L- 5\ dx ® 0? , v~x = ca0 T*E TTE of the bundle TE —> E. Its coordinate expression is of the type K — dx ® (d\ + K\ M iM <9„), with K^x = Kx\eC™(E). We denote by K[g] the Levi Civita connection, i.e. the torsion free linear space-time connection such that \7g — 0. We define a phase connection to be a connection of the bundle 3iE —> _E. A phase connection can be represented, equivalently, by a tangent valued form T: 3iE ->• T*.E ® T3i£?, which is projectable over 1: E ->• T*.E ® T.E, or by the complementary vertical valued form i/[T]: 3\E —> T*3\E ® V3iE, or by the vector valued form uT\T] :^v~x o u[T]: 3\E ->• T*3\E (T* VTE). Their coordinate expressions are r = rfA ® (9A + iY0 0°), !/[r] = (4 - ry0 dx) ® a?, ^T[r] = ca°K-rAJ0rfA)® (^-ccA-^o + xgdp)), with rA'0er(p). We can prove [8] that there is a natural map x '■ K l—> T between linear spacetime connections -KT and phase connections T, with the coordinate expression T\l0 — & K^p ft" Op J^x a «o ■ 3.4. Dynamical phase connection. The space of 2-jets of motions 3?E can be naturally regarded as the affine subbundle 3?E cT*® T3iE, which projects on a: 3iE T* ® TE. A dynamical phase connection is defined to be a 2nd-order connection, i.e. a section 7: 3\E —> 32E, or, equivalently, a section 7: 3\E —> T* ® T2\E, which projects on /i- The coordinate expression of a dynamical phase connection is of the type (3.4) 7 = ca°(^ + 4ai + 7ol0^), with ToJerftE). If 7 is a dynamical phase connection, then we have 7 jt = 1. The contact map a and a phase connection T yield the section 7 = 7 [a, T] := fljT: —> T* ® T3iE, which turns out to be a dynamical phase connection, with the coordinate expression 70Q — Tp105q. In particular, a linear spacetime connection K yields the dynamical phase connection j :— j[a, K] '-—a with hidden symmetries 305 the coordinate expression 7o0 — 5p KapT 5g 5q. For the Levi Civita connection we get the gravitational dynamical phase connection j[g] — Ajx(if[g]). 3.5. Phase 2-form and 2-vector. The rescaled metric G and a phase connection r yield the 2-form 51 , called phase 2-form, and the vertical 2-vector A , called phase 2-vector, 2 (3.5) 0:=0[G,r]:=Gj(^[r] A 6) : 3iE /\ t*2iE , 2 (3.6) A:=A[G,r] :=Gj(rA^T) : 3iE f\T3iE , with the coordinate expressions (3.7) 51 = Co a0 G^M ( (l1/2 ®m1/2)® f\t*E. Given a charge q, the rescaled electromagnetic field f — (q/2h) f can be incorporated into the geometrical structure of the phase space, i.e. the gravitational 2-form. Namely, we define the joined (total) phase 2-form 2 51 := n\g] + ^f = n\g] + 5ic ■. 3iE f\t*3iE . Of course dft — 0 but 51 is exact if and only if f is exact. We recall [8] that a unique connection T on 3\E —> E can be characterized through the total 2-form 51 by the formula (3.5). Namely the joined (total) phase connection T — T[g] + Te , where Te := - l-{uToGi2){f + 2TA{ji^f)) with the coordinate expression Te = -(l/(2c0a°))G0M (FA/i - (a°)2g0\ FpfI 5g) dx ® df, here G'2 means that we apply G" on the second index. The total phase connection then admits the joined (total) phase 2-vector A — A[g] + Ac and the joined (total) dynamical connection 7 — j[g] + 7C given by Ac = \ Alt ((i/Tofff)(g)(i/Tofff))(F) and 7' =: j f : 3\E —> t* ®V3iE . 306 j. janyska Here 7C is the Lorentz force. 3.7. Dynamical structures of the phase space. In what follows we shall use the unsealed time form r — r. First, let us consider the gravitational objects 7[s] = ^7[ff],n[ff] andA[ff]. Theorem 3.1. ([8]) We have: (1) fl[g] — —dr, i.e. (—t, fl[g]) is a contact pair. (2) fi[g],A[g}] = 0 and [A[g},A[g]] = 2^[g}AA[g], i.e. (-%], A[g}) is a (regular) Jacobi pair. Moreover, the contact pair (—t, £l[g\) and the (regular) Jacobi pair (—^[g], A[g}) are mutually dual. n According to Theorem 3.1 the metric g defines on the phase space the natural contact structure which will be called the gravitational contact phase structure. Dually the metric defines the gravitational Jacobi phase structure. Remark 3.1. Let us remark that in the standard literature the phase space (the observer space) is defined as a part of the unit pseudosphere bundle formed by timelike and future oriented vectors. The contact 1-form is then obtained either as the restriction of the 1-form a = g\^ xx cP to the phase space [4] or by a direct construction described in [3]. n Further let us consider the joint objects 7 — ^[g] + 7% O — il[g] + Oc and A = A[ff]+A*. Theorem 3.2. ([8]) We have: (1) t A 513 ^0, dfl — 0, i.e. (—t, 51) is an almost-cosymplectic-contact pair. (2) [7, A] = 7 A A'(Z^f) and [A, A] = -27 A (A* A*)(rff), i.e. (-7, A) is a (regular) almost-coPoisson-Jacobi pair. Moreover, the pairs {—t, 51) and (—7, A) are mutually dual. U According to Theorem 3.2 the metric g and the electromagnetic 2-form F define on the phase space the almost-cosymplectic-contact structure which will be called the joined (total) almost-cosymplectic-contact phase structure. Dually the metric and the electromagnetic 2-form define the joined (total) almost-coPoisson-Jacobi phase structure. 4. Hidden symmetries of the gravitational contact phase structure In [11] projectable infinitesimal symmetries of the gravitational contact structure (eventually of the total almost-cosymplectic-contact structure) on the phase space were characterised. It was proved that all such symmetries are obtained as the flow 1-jet lifts of Killing vector fields (eventually the flow 1-jet lifts of Killing vector fields which are infinitesimal symmetries of the electromagnetic 2-form). In what follows we shall study infinitesimal symmetries of the gravitational contact structure of the phase space which are not projectable. Such symmetries are not generated by infinitesimal symmetries of the spacetime and they are usually called hidden symmetries. hidden symmetries 307 4.1. Infinitesimal symmetries of the gravitational contact phase structure. In what follows we assume the gravitational contact and the gravitational Jacobi structures (—t, $7) :— (—T,il[g\) and (—7, A) :— (—7[g],A[g]) given by the metric. Let us recall that in this situation O — —dr. Then by Corollary 2.3 infinitesimal symmetries of the gravitational contact phase structure are the Hamilton-Jacobi lifts X — df$ + / 7, where / is a conserved phase function, i.e. 7 ■ / = 0. Moreover, / = f (X) = ?(X). Here X = Ttv^(X) : 3\E —> TE is a fibred morphism over E. So any infinitesimal symmetry is the Hamilton-Jacobi lift (4.1) ^ = d(?(A))8+?(A)7 of the phase function t(2l) where X_: 3\E —> TE such that 7 • (t(A)) — 0. Remark 4.1. Let us remark that X_ = Tir^X): 3iE —> TE is a generalized (1st order) vector field in the sense of Olver [16]. □ In what follows we shall characterize generalized vector fields X_ admitting infinitesimal symmetries of the gravitational contact phase structure. So, such generalized vector fields have to satisfy the following two conditions: 1. (Projectability condition) The Hamilton-Jacobi lift (4.1) of the phase function t(X) projects on X_. 2. (Conservation condition) The phase function t(X_) is conserved, i.e. 7 • (jUO) = 0 . We shall start with the projectability condition. Theorem 4.1. Let X_: 3iE —> TE be a generalized vector field, then the following assertions are equivalent: 1. The Hamilton-Jacobi lift X — d(r(X))^ + t(X) 7 projects on X_. 2. The vertical prolongation VX: V3iE VTE — TE © TE has values in the kernel of r. 3. In coordinates (4.2) G°0pd?Xp = 0. Proof. 1. ^ 3. Let X_ — X_x d\ , XxeC°°(3iE), be a generalized vector field. Then we have the coordinate expression (4.3) f = nX) = -c0a°G°0pXO. For the function (4.3) we have d(r(20)8 = (2CA + (a0)2 g0pXf öx + G°0p djX?) 3X — Gqp [X_a daGop + Goa dpX_a + (a0)2 goa Xf (<5q 9wGqp — \ dpGQ0) + Gq" Gqo- djXf (ö^Gqp — 9pGqw)] 308 j. janyska and t{X) 7 = -{a°)2g0pXp 6X dx + (a°)2g0pXp &0° (5o dA - \dA) $ , where Gq0 — G^v5x5q (the same notation we use for g00 — g\u5$ 5q). So we obtain (4.4) X = d(?(20)* + f(2Q 7 = (*A + G[SP GJ0A a?2Cp) dx — G0P ^o-Gqp + Ggg. 9pXCT + Ggg. Gq" X? {duG^p — <9pGqw)] 9? and this vector field projects on X. if and only if the condition (4.5) <5°p#oA0°£p = O is satisfied. Further let us consider the generalized vector field X : 3\E —> TE with the coordinate expression x = G°0pGlxd°3xpdx. This generalized vector field is r-vertical, i.e. X : 3iE —> VTE, and if we use the isomorphism Gb : TE t ® T* E we obtain the T-valued 1-form x}> = &%p5l d°Xpu0®d», u0 is a base of T, such that Xb(a) = 0, i.e. Xb : 3iE ->• T VT*.E. Further, if we use the isomorphism v~x : T ® V^-E —> V*3iE, we obtain from (3.3) ^(Xb) = c0 a0 £gp 9?2CP 4 . Since v~x o Gb : V^-E —> V*3\E is an isomorphism we obtain that ^1Tl(Xb) = 0 if and only if X — 0, i.e. (4.2) is satisfied if and only if (4.5) is satisfied. 2. & 3. It follows from i/"1^) = -f(VX). □ If the vector field (4.1) projects on X_ then we get the coordinate expression (4.6) X = d(r(X))8 + rQQ 7 = *A ^ - <%p [Xf 8aG°0p + G^ 9pXff] 0° . Next, let us characterize conditions for the phase function t(X_) to be conserved. Theorem 4.2. For a generalized vector field X_ the phase function t(X_) is conserved if and only if the condition (4.7) 0 = -6% 5P (Gl dpXf+±Xf daG%)+G0^ Gtf d%X» (5% daG°0p-± dpG°m) is satisfied. Proof. Let us consider the condition 7 • / — 0 which is equivalent with ^(3P/ + r[jUa?/)-o. Then for the function (4.3) we obtain 7-(t(20) = -(a°)2^o $o (