ARCHIVUM MATHEMATICUM (BRNO) Tomus 51 (2015), 297-313 RELATIONS BETWEEN CONSTANTS OF MOTION AND CONSERVED FUNCTIONS Josef Janyska Abstract. We study relations between functions on the cotangent bundle of a spacetime which are constants of motion for geodesies and functions on the odd-dimensional phase space conserved by the Reeb vector fields of geometrical structures generated by the metric and an electromagnetic field. Introduction We assume a classical spacetime E to be an oriented and time oriented 4-dimen-sional Loretzian manifold. In literature as phase space is usually considered the cotangent bundle T*E and as infinitesimal symmetries are usually considered infinitesimal symmetries of the kinetic energy function. It is very well known, [13], that such infinitesimal symmetries are given as the Hamiltonian lifts of functions on T*E which are constants of motion for geodesies. Constants of motion which are polynomial on fibres of the cotangent bundle are given by Killing fc-vector fields, k > 1. For k = 1 the corresponding infinitesimal symmetries are the flow lifts of Killing vector fields and so 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. Moreover, if we consider coupling with an electromagnetic 2-form, constants of motion and the corresponding infinitesimal symmetries are generated by Killing-Maxwell multi-vector fields. On the other hand the phase space of general relativistic test particle can be defined either as the observer space, [3], (a part of the unit pseudosphere bundle given by time-like future oriented vectors) or as the 1-jet space SiE of motions, [11]. The metric and the electromagnetic fields then define geometrical structures given by a 1-form and a closed 2-form. As phase infinitesimal symmetries we define infinitesimal symmetries of these forms. Phase infinitesimal symmetries which are projectable on the spacetime were studied on the observer space by Iwai [3] and on 1-jet space of motions by Janyska and Vitolo [11]. In both situations projectable 2010 Mathematics Subject Classification: primary 70H40; secondary 70H45, 70H33, 70G45, 58A20. Key words and phrases: phase space, infinitesimal symmetry, hidden symmetry, gravitational contact phase structure, almost-cosymplectic-contact phase structure, Killing multi—vector field, Killing—Maxwell multi—vector field, function constant of motions, conserved function. Supported by the grant GA CR 14-02476S. doi: 10.5817/AM2015-5-297 298 J. JANYSKA symmetries are given by the flow lifts of Killing vector fields (eventually Killing vector fields which are infinitesimal symmetries of the electromagnetic field). In the paper [5] it was proved that nonprojectable (hidden) symmetries of the contact structure of the phase space generated by the metric are given by the Hamilton-Jacobi lifts of phase functions conserved by the Reeb vector field of the contact structure. Moreover, it was proved that such conserved functions are generated by Killing multi-vector fields. On the other hand if we assume the almost-cosymplectic-contact structure of the phase space given by the metric and the electromagnetic fields then in [7] it was proved that all infinitesimal symmetries are projectable and there are no hidden symmetries. In this case Killing-Maxwell multi-vector fields generate functions conserved by the Reeb vector field of the structure, but not infinitesimal symmetries. In the paper we discus relations between functions on T*E which are constants of motion and functions on SiE conserved by the the Reeb vector fields. We prove that conserved phase functions are obtained as a pull-back of constants of motion on T*E. 1. Infinitesimal symmetries of the kinetic energy function A classical spacetime is assumed to be an oriented and time oriented 4-dimensional manifold E equipped with a Lorentzian metric g of signature (1,3). We denote by (xx) local coordinates on E and by (xx, x\) the induced fibred coordinates on T*E. In what follows we shall use notation dx = dxx, d\ = dx\, d\ = and dx = S— The inverse metric will be denoted by q. OX\ J a 1.1. Canonical symplectic structure. Suppose the phase space to be the cotangent bundle T*E. Then we have the canonical symplectic 2-form to and the canonical Poisson 2-vector A given by oj = dx A dx , A = dx A dx . Let us assume the kinetic energy function H ^g ^ Xx xn . A function K on T*E is said to be a constant of motion if (1.1) 0 = {H, K} = LXhK = -LXkH = gXp xx dpK - \ dpKdpgx» xx xp . Remark 1.1. A phase function AT is a constant of motion means that its Hamil-tonian lift Xx is an infinitesimal symmetry of the kinetic energy function or that K is constant on geodesic curves since the Hamiltonian lift (1.2) XH = gxpxpdx-\ dxgpa xpxa dx . is the tangent vector field of lifts of geodesies to T*E, [13]. □ Now, let us discuss functions satisfying the equation (1.1). If K is the pull-back k of a spacetime function then K has to be a constant. Further suppose that K is CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 299 homogeneous of order k on fibres of T*E, i.e. K = KXl-Xk xXl... xXk , KXl-Xk e C°°{E). k k k Then K can be considered as a symmetric k-vector field K = KXl'"Xk d\1 ©• • - Qd\k. k Then the equation (1.1) is satisfied if and only if K satisfies the Killing equation (1.3) [$,£]=0, k where [, ] is the Schouten bracket for symmetric multi-vector fields. So AT, considered fc-vector field, is a Killing tensor field. l Remark 1.2. For k = 1 we obtain that a vector field K admits a symmetry of the kinetic energy function if and only if it is a Killing vector field. Moreover, the l Hamiltonian lift of the corresponding constant of motion is the flow lift T*K of l the vector field K to the cotangent bundle. □ So functions of the type q _ /j. 0 fc (1.4) K = K + ^KXl-Xk xXl ...xXk , K, KXl-Xk e C°°{E), k>l 0 k are constants of motion if and only if AT is a constant and K, k > 1, are Killing multi-vector fields. 1.2. (Souriau's) coupling with an electromagnetic field. Let us consider a Maxwell (electromagnetic) field F = F\^ dx A cP satisfying the Maxwell equation dF = 0. Then we consider the total (joined) 2-form, [2], J = iv + \F = dx A dx + \F\^ dx A # . We obtain the corresponding total (joined) Poisson 2-vector Aj = A + Ae = dx A dx + \FXlJL dx A . Assume a function K on T*E satisfying (1.5) 0 = {H, Ky =Lx,hK = gxp xp dxK - (± dxgpa xp xa - FpX gpa xa) dxK, where {, }i is the total (joined) Poisson bracket. According to [13] functions of the type (1.4) satisfy the equation (1.5) if and only if 1.6) 0 = ]T(±[p, K]^-°k+kFp^Kp k-l k a2---crk ^ Xai . . . Xak . k>l 300 J. JANYSKA 0 1 Corollary 1.1. For a function K = K + K x\ two identities have to be satisfied 0 = g^dpK + Fp^K?, 0=[g,K], 1 0 1 which implies that K is a Killing vector field and the identity dK + K j F = 0 is i ^ satisfied. Then K is an infinitesimal symmetry of F, i.e. L i F = 0. Moreover, K xiK = Kxdx- (d\k + dxh ±p + Fp\ h) bx = kxdx- dxh Xp dx i i which is the flow lift T*K of the vector field K. □ k k Corollary 1.2. For a function function K = KXl---Xk xXl ... x\k, k > 2, we get k k 0=[g,K]J 0 = F^U1 Ka2-a^p , k i.e. K is a Killing-Maxwell k-vector field, [2]. Moreover, the corresponding vector field X\ is not protectable on spacetime and the infinitesimal symmetry of H is K hidden. □ 2. Infinitesimal symmetries of the gravitational contact phase structure In what follows we shall consider a phase space of a general relativistic test particle considered as the 1-jet space of motions. 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 cGF0L and the Planck constant ft G T* L2 T* ® (TE 0 TE), GXfl = ^gXfl. We assume time to be a one-dimensional affine space T associated with the vector space T = T 0 M. A motion is defined to be a 1-dimensional timelike submanifold s: T E. The 1st differential of the motion s is defined to be the tangent map ds: TT = T x f -> TE. We assume as phase space the open subspace SiE C Ji(E, 1) consisting of all 1-jets of motions. So elements of dixE are classes of non-parametrized curves CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 301 which have in a point x £ E the same tangent line lying inside the light cone, [8]. 7Tq : diE —> E is a fibred manifold but NOT an affine bundle! The velocity of a motion s is defined to be its 1-jet j\s: T —> di(E, 1). For each 1-dimensional submanifold s : T E and for each x G T, we have jis(x) G Hi-E- if and only if ds(x)(u) G TS(X)E is timelike, where u G T. Any spacetime chart is related to each motion s which means that s can be locally expressed by (x°, x% = sl(x0)). Then we obtain the induced fibred coordinate chart (x°, xl,xl0) on diE 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* ® TE, 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 SiE —> T* ®TE over E, such that Ro j\s = ds, for each motion s. We have g (r, r) = — c2. The coordinate expression of R is (2.1) r = ca° (d0 + x0 di), where a0 :=1/\J\g00 + 2 g0j xJ0 + g{j x^x^ . We define the time form to be the fibred morphism r = — \ g^(fl) '■ diE —> Tc?)T*£^, considered as the scaled horizontal 1-form of diE. We have the coordinate expression (2.2) T = Txdx = -^(g0X+giXxi0)dX . Note 2.1. In what follows it is very convenient to use the following notation S{ = 6{- 4 5°x and 6% = 6$ + 5» xp0. Then R = ca°6$dtM and r = p0a d\ where p0a = S^a • d Let VSiE C TdiE be the vertical tangent subbundle over E. The vertical prolongation of the contact map yields the mutually inverse linear fibred isomorphisms vT:diE^T®V?E®VdiE and v~x: diE —> V*diE (g> T* (g> VTE , where V^-i£ = kerr C TE, with coordinate expressions xq 51 dx (8) 9? , zv"1 = ca° 4 (9i - caVi TE is a generalized vector field in the sense of [12] such that 7fl • (r(X)) = 0. So, a generalized vector field X_ has to satisfy the following conditions: 1. (Projectability condition) The Hamilton-Jacobi lift (2.3) of the phase function t(X_) projects on X_. 2. (Conservation condition) The phase function t(X_) is conserved, i.e. 7fl-(r(X)) = 0. The following results were proved in [5]. Theorem 2.1. Let X = Xxdx: diE -> TE, Xx e C°°(diE), be a generalized vector field, then the following assertions are equivalent: 1. The Hamilton-Jacobi lift (2.3) projects on X_. 2. The vertical prolongation VX: VdiE -> VTE = TE © TE has values in the kernel of r. 3. In coordinates (2.4) g0p d^X? = 0 . □ Lemma 2.2. For generalized vector fields X_ and Y_ satisfying the projectability condition we have (2.5) {r(2Q, t(Z)}0 + r(X) 7* • (?(Y)) - t(Y) 7fl • (?{X)) = • D Remark 2.1. Let us remark that on the left hand side of (2.5) there is the Jacobi bracket of functions t(X_) and t(Y_). □ Theorem 2.3. Let X_ be a generalized vector field satisfying the projectability condition. Then the following assertions are equivalent: 1. The Hamilton-Jacobi lift (2.3) is an infinitesimal symmetry of the gravitational contact phase structure. 2. The phase function r(X) is conserved, i.e. 7fl • r(X) = 0. 3. The vector field [ja,X] is in kerr. 4. In coordinates (2.6) o = ^0ff(5pua,r + 5rw- D Corollary 2.4. For generalized vector fields X_ andY_ satisfying the projectability and the conservation conditions we have (2.7) {t(X),tQQ} = t([X,Y]). Moreover, the phase function r([X,Y]) is conserved. □ Theorem 2.5. The Lie algebra of infinitesimal symmetries of the gravitational contact phase structure is formed by the Hamilton-Jacobi lifts of phase functions t(X_), where generalized vector fields X_ satisfy the projectability and the conservation conditions (2.4) and (2.6). CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 303 Moreover, if X_ factorises through a spacetime vector field, then the corresponding infinitesimal symmetry is projectable and it is the jet flow lift di2L- If Ä is a generalized vector field which is not factorisable through a spacetime vector field, then the corresponding infinitesimal symmetry is hidden. □ 2.3. Infinitesimal symmetries of the gravitational contact phase structure generated by Killing multi-vector fields. In [5] it was proved that a k symmetric k-vector field K, k > 1, admits generalized vector field satisfying the projectability condition. Such generalized vector fields are given by (2.8) X[K] = fcf j ...j?jK -(k- 1)K(t, ...,?) A- diE -> TE, (fc — l)—times where R = —^rR. Then we obtain the induced phase function t{X[K\) = K(?) := K(t, ...,?) = KXl-Xk fAl ... ?Xk. k ^ Theorem 2.6. The phase function K(t) is conserved with respect to the gravita- ^ k ^ k tional Reeb vector field, i.e. 7fl • K(r) = 0, if and only if K is a Killing k-vector field. □ o ^o o Remark 2.2. Let AT be a spacetime function. Then ^.K = 0 if and only if K is a constant. □ Theorem 2.7. The Hamilton-Jacobi lift of a phase function 0 _ (2.9) K = K + J2k(t), k>l is an infinitesimal symmetry of the gravitational contact phase structure (—r, fifl) o k if and only if K is a constant and K, k > I, are Killing k-vector fields. □ l ^i Remark 2.3. For a Killing vector field K the Hamilton-Jacobi lift of r(K) l coincides with the jet flow lift d\K and the corresponding infinitesimal symmetry is projectable on spacetime. For k > 2 the corresponding infinitesimal symmetry is hidden. □ o k Remark 2.4. For a constant K and Killing k-vector fields K, k > 1, the conserved phase function of the type (2.9) admits the infinitesimal symmetry X[K] = K^+ ^X[K] k>l which projects on the generalized vector field X[K] = (K~y^(k- 1)K(?)) A + K + y^k? j ... jf jK fe>2 fe>2 (fc-l)-times 304 J. JANYSKA satisfying the projectability and the conservation conditions. □ 2.4. Comparison with infinitesimal symmetries of the kinetic energy function. Suppose the morphism -?: diE—>T*E. over E. In coordinates we have xx = xx , x\= t\ = gox . Remark 2.5. Let us note that the image of the mapping —r is the subset of T*E given by elements satisfying the condition G ^ x\ xn 1, where G = \ , q is the unsealed metric. □ Lemma 2.8. Let K be a constant of motion on T*E, i.e. Xh ■ K = 0, then its pull-back —t*(K) is a conserved function, i.e. 7fl • t*(K) = 0. Proof. First, it is easy to see that we have -Tef(7fl)=Xif(f(e)), where e £ SiE and XH is the vector field (1.2). Now, Y-?*(K) = i~d?*(K) = iqar*{dK) = dK(Tr(js)) = -dK(XH) = -XH ■ K. □ Now, let us assume a function (1.4) on T*E. The above function is a constant o k of motion if and only if K{x) is a constant and K{x), k > 1, are Killing fc-vector fields. The pull-back of the function (1.4) is the phase function (2.9) which is a function conserved by the gravitational Reeb vector field. Let us note that in [5] the conserved functions of the type (2.9) were obtained in a different way by using generalized vector fields (2.8). By Lemma 2.8 we get the following diagram , „ , , Hamiltonian (T*E, oj) constants of motion K, {H, K} — 0 -► ISs of H lift Killing multi-vectors „ „„, ^„ , ^, Hamilton-Jacobi , ^ „„, (01 .E, -t,09) conserved functions, 79 • (-T*(K)) = 0 ->- ISs of (-t,09) lift For Killing vector fields we obtain in both cases projectable infinitesimal symmetries which are obtained by the flow lifts. For Killing fc-vector fields, k > 2, the corresponding infinitesimal symmetries are hidden. CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 305 3. Infinitesimal symmetries of the total almost-cosymplectic-contact phase structure We assume the total (joined) almost-cosymplectic-contact structure (—r, Qi) on the phase space given naturally by the metric and an electromagnetic field. 3.1. Total almost-cosymplectic-contact phase structure. We assume an electromagnetic field to be a closed scaled 2-form on E 2 f: E^ (L1/2 (g)M1/2) ®/\t*E. Given a particle with charge q G T_1 ® L3/2 ® M1/2 the rescaled electromagnetic field f = -| f can be incorporated into the geometrical structure of the phase space, i.e. the gravitational form. Namely, we define the total (joined) phase 2-form 2 (3.1) w := n9 + nc = n9 + \f-. diE -> /\t*3iE. The pair (—r, f^) is almost-cosymplectic-contact, i.e. it is regular and and the 2-form is closed, [9]. Then the dual almost-coPoisson-Jacobi pair is (—7', A'). Here 7' = ■^^(j9 + 7e), where 7e: diE -^T* ®VdiE with the coordinate expression (3.2) 7e = -^A^FpAu°®5°, Gif = GqA. Further, Aj = A0 + Ae, where (3-3) Ae = ^y&x &0" fx, a? A 5° . 3.2. Infinitesimal symmetries of the total almost-cosymplectic-contact phase structure. We define a phase infinitesimal symmetry of the total almost-cosymplectic-contact phase structure to be a vector field X on diE such that: (1) Lx? =0; (2) LxW = 0. Remark 3.1. The conditions (1) and (2) are equivalent to [X, 7^] = 0 and [X, A'] =0. □ Lemma 3.1. A phase vector field X is an infinitesimal symmetry of Vl) if and only if it is of the form (3.4) X = df^+h^, where f is a conserved phase function, i.e. 7^ • / = 0, and h = r(X_), X_ = Tttq(X). Proof. We have the splitting TdiE = kerr © (7^), i.e. X = X + hji, where t(X) = 0 and h is a phase function. Then from t(t^) = 1 we have h = t(X). Further the phase 2-form is closed, then from = 0 we obtain 0 = Lx^V = di-Vl), which implies locally that = df for a phase function /, i.e. X = . Moreover, ^ ■ f = df = i~SV = -i W =0. □ 306 J. JANYSKA Theorem 3.2. A phase vector field (3.4), where where f is a conserved phase function, i.e. 7^ • / = 0, and h = r(X_), X_ = Tttq(X), is an infinitesimal symmetry of t if and only if f is of the form (3.5) / = t go+; for a generalized vector field X_ and a spacetime function f £ C°° (E) such that (3.6) df = XjF. Proof. By Lemma 3.1 infinitesimal symmetries of (—r, fP) are of the form (3.4), where jKf = 7fl • / + je.f = 0, and h = t(X). If X is an infinitesimal symmetry of r then by [4] (3.7) -idfijns - hi~ns+dh = 0. But and idftj W =df + ^gja d°JF^ d» = df + (g* o uT)(dvf) jf. which follows from (nb* o A«»)(#) = # - (7fl • /) f = df + (7e • /) r , o A*')(#) = -(7* • /) f + dUK & ■ Then (3.7) reads as (3.8) d{h-f) = -h%jF + (g» o vT){dvf) jF. Now, if we put t{X) = t(X_) = h, then we can rewrite (3.8) as (3.9) df = d(?(X)) + ?(X)%jF - (g» o vT){dvf) jF = d(f(2Q) - ((a°)2^op xp «S0A + ^Glx a?/) fv d" which implies that (3.10) a?/ = a?f(2Q = -co «0(g°p x" + g°p , where GQip = GQip + (a0)2 g$i g[]p, and we can rewrite (3.9), by using the identity GloXG°ip = 5xp+(a0)2g0p5S, as (3.11) df = d(?(X)) + {XX + Gl0x G°0p dPX") FXp d» . If we consider the condition that the vector field (3.4), where df is given by (3.11), projects on X_ which is equivalent with (2.4) we get (3.12) df = d(?(X)) + XX d^ = d(?(X)) +XjF. So we get (3.5), where / £ Cco{E) such that (3.6) is satisfied. □ CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 307 Theorem 3.3. All phase infinitesimal symmetries of the total phase structure are vector fields of the type (3.13) X = d(?(X) + /)"' + ?(X) f where X_ is a generalized vector field and f £ C°°(E) satisfying the following conditions: 1) df = XjF. 2) (Projectability condition) The vector field (3.13) projects on X_. 3) (Conservation condition) The phase function r(X_) + f is conserved, i.e. 7j • (r(2Q + /) = 0. Proof. It follows from Lemma 3.1 and Theorem 3.2. □ Remark 3.2. Let us note that to find a pair (X, /) satisfying the projectability condition 2) of the above Theorem 3.3, it is sufficient to find a generalized vector field satisfying the projectability condition (2.4) of Theorem 2.1. It follows from the fact that df^ is a vertical vector field. Further, if the condition 1) and 2) are satisfied, then the conservation condition 7^ ' (t(2Q + f) = 0 is equivalent with the conservation condition given by (2.6) in Theorem 2.3 which follows from 7fl • / = —7e • t(X). □ Lemma 3.4. Let (X_,f) and (Y_,h) be pairs of generalized vector fields and spa-cetime functions such that the projectability condition of Theorem 3.3 is satisfied. Let X and Y are phase vector fields given by (3.13). Then (3.14) ?([X, Y]) = ?([X,Y}) = c0 a0 G°ox {Xf> dpYx - Yf dpXx). Proof. For a pair (X, /) satisfying the projectability condition (2.4) we obtain (3.15) X = Xxdx-Gl0p[--\dpf L Co au + X.a daGQp + G ~ d°Pf dPh) + Gl0p GJ0a (daG°0p — dpG°0o. + -^öFpa) d^hdjf] , and, for generalized vector fields X, Y_ and spacetime functions / and h satisfying the projectability condition (2.4), we obtain {9(20,9{Y)}] = co a0 [GqX (Y_p dpXx - Xp dpYx) + («0)2 G°0p Yf (gox Sp dpXX + \ Xx dxg00) - (a0)2 G°ox Xx (g0p Sp dpY» + \ Yf dpgm) + t£p K XxYf-^ Sp (gox Fpp - g0p FpX) Xx Yf] = ?([X, Y]) - 9(20 7fl • 9(Y) + ?(Y) 7fl • 9(20 \ F(X, Y) - \ f (20 F(% Y) + \ ?(Y) F(% X), {?(X), hy = Xp dph + (a0)2 g0a X° Sp dph = X_.h — t(2L)R- h, {/, hy = o, where goo = ^o ^o 0V' which follows from Lemma 3.4 and t(Y) (7fl • f (20) = co (a0)3 G°0p Yf (gox Sp0 dpXx + \ Xf dpg00) . □ Theorem 3.6. All infinitesimal symmetries of the total almost-cosymplectic-contact phase structure (—r, fP) are protectable. Proof. Let us consider two infinitesimal symmetries of the almost-cosymplectic-con-tact phase structure (—r, £V) X = d(?(X) + /)"' + ?(X) 7j, Y = d(?(Y) + hfi+ ?(Y) 7j , where the pairs (X, /) and (Y_, h) satisfy the conditions 1), 2) and 3) of Theorem 3.3. Then the Lie bracket [X, Y] is also an infinitesimal symmetry of (—r, fP). By [5] (Lemma 2.5) we have [X, Y] = d{f(20 + /, 9(Y) + h}ij + {{9(20 + f, 9(Y)Y - {tQO + h,9(X)y + ns(d(?(X) + ff\d(?(Y) + h)V)) 7j • But [X, Y] is an infinitesimal symmetry of (—r, fP) if and only if the difference {f(20 + /, 9(Y) + h}j - {{9(20 + f, 9(Y)y - {9(Y) + k ?(20V + n*(d(?(20 + ff\d(?(Y) + h)V)) is a spacetime function. The above difference is (3.20) -{f (20, 9(Y)y - fi«(d(f (20 + ff\d(?(Y) + h)M). CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 309 But Qs = Q} — Qe. Then from the duality and T4{d{?{x) + ffi)=x-?{x)% we get u){d{?{x) + /)«, d(f 00 + hf]) = -{r{X) + f, ?0O + hY , ne(d(f (20 + /)"', d(f(y) + hfi) = \f{x - ?{x) %y- ?(y) a) which implies n*(d(?(x) + /)"', d(f(y) + hf) = - {r(20 + /, r(Y_) + hy - ±F(x, y) + If(x) F(%, y) + ±?(y)F(x,a). Then (3.20) is {fgo, hy + {/, ?(y)y + \f{x, y) - if(20 20 - \?QQ f(x, a) and, from (3.18), it can be rewritten as x-h-?(x)a-h-y-f + ?(y)a-f + lF(xjy) - §f (20 f{% y) - If (10 F(X, A) • Finally, from (3.6) and FjIjF = |-F(X,H), we get that the difference is equal to —hF(x, y) which is a spacetime function if and only if x_ and y_ are spacetime vector fields. So all infinitesimal symmetries of the almost-cosymplectic-contact structure (—f, sv) are projectable and there are no nonprojectable (hidden) symmetries. □ Remark 3.4. Let us note that projectable infinitesimal symmetries of of the almost-cosymplectic-contact phase structures were classified in [11]. It was proved that all projectable infinitesimal symmetries are vector fields of the type (3.13) where X is a spacetime Killing vector field and / is a spacetime function such that the condition (3.6) is satisfied. In this case the vector field (3.13) reduces to (3.16). But if X is a Killing vector field, then IjG = 0 which in coordinates reads as xa dagx/1 + gxa dfj,2La + G>CT d\2C = 0, i.e. x_a da&lp + gqa dpx° = — gqpa Sq d^JC7 which implies that the vector field (3.16) can be rewritten as (3.21) x = xx dx + gl0p g°pa 5% cL2T 3° = xxdx + 6% 5% d^xa 0° = xxdx+ (doJC + xp0 dpx{ - 4 d0x° - xi xp0 dp x°) $ which is the 1-jet flow lift of x_ to diE. □ 310 J. JANYSKA 3.3. Conserved functions and Killing-Maxwell multi-vector fields. Now, o let us consider a phase function (2.9) given by a spacetime function f = K and k symmetric multi-vector fields K, k > 1. If we consider the phase vector field k k>l then this vector field coincides with the vector field (3.13) for the generalized vector field X[K] = Tttq1 (dK^ + K(?) 7j) k i k>l \ _ k _ k (3.22) = K -y^(k-l)K(?)A+y^kfj . jtjK. k>2 k>2 (fc-l)-times ^ fc ^ Really, we have r(X_[K]) = ^2k>1 K(t). Such generalized vector field satisfies the projectability condition and we have to find conditions for the function (2.9) to be conserved by the joined Reeb vector field, i.e. jKK = 0. Lemma 3.7. We have o 7e • K = 0, 7e.K(f) = -k ir^i-^-i FXkp?Xl ...?Xk , k>l, where FXkp = gaXk Fap. Proof. We have r-rP = —S°0Fap mc which implies, with Sq = - -^t^ gauJ , 7e • K(?) = k ir^i-^-i (7e.fp) fAl ... ?xk_, h2 k ~ — —b _ TS-p^l—^k-l fi r7°"Afe T\ T\ — ft. I\ Fapg l\1---l\k- |—| Lemma 3.8. Let us suppose a phase function B = BXl---Xk t\x ...r\k, k > 1, BXl-Xk e C°°{E). Then B = 0 if and only if BXl--Xk = 0 for all X1,...,Xk. Proof. B = 0 if and only if Bx^-Xk gXlPl ... gXkPk 8?1 ... 8p0k = 0 which is a polynomial function on fibres of diE. Then BXl'"Xk g\lPl ...g\kPk = 0 for all indices p\,..., pk and from regularity of the metric we get Lemma 3.8. □ CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 311 Theorem 3.9. A phase function (2.9) is conserved by the joined Reeb vector field, i.e. 7^ • K = 07 if and only if (3.23) gpx dpK + Kp Fxp = 0, k fc+1 (3.24) y(AiKA2...Afe + 1) + (fc + !) # p(Ai...Afe ^Afe + 1)^ = q ^ /or = 1, 2,... . Proof. From Lemma 3.7 we have 7' • K = 7fl • K + 7e • K =--^ [(l From k k y(A1KA2...Afe+1) = l^;j^]Ai...Afe+1 and Lemma (3.8) we obtain Theorem 3.9. □ Corollary 3.10. The vector field X[K] = dK^ + h^, where the phase func- o tion K is given by (2.9) and h = K — K, is an infinitesimal symmetry of the almost-cosymplectic-contact pair (—r, f^) if and only if the conditions (3.23) and o ^ (3.24) are satisfied and dK = X_[K] jF. □ o 1 ^ Remark 3.5. Let us assume a (special) phase function K = K + K(r). Then the conditions (3.23) and (3.24) are reduced to dpK - Ka Fap = 0, V(Ali:A2)=0 1 0 1 and we obtain the result of [11], i.e. K is a Killing vector field and K and K 0 1^ 1 are related by the formula dK = K jF which implies that K is an infinitesimal symmetry of F. Moreover, the corresponding infinitesimal symmetry is the flow l l lift diK which projects on K, see Remark 3.4. Let us note that in this case the condition (3.6) coincides with the condition (3.23). □ k ^ Remark 3.6. Let us assume a phase function K = K(t) , k > 2. Then the conditions (3.23) and (3.24) are reduced to k k (3.25) y(AiKA2...Afe+1) = 0; Kp{\i-\k-i p\k)p = o k and we obtain that AT is a Killing-Maxwell k-vector field, [2]. The condition (3.6) has the form jF = 0 which from (3.22) is equivalent with (3.26) 0 = J2((k- 1)K(?) AjF-k (f j ... jf jK) j F) . k>2 (fc — l)—times 312 J. JANYŠKA Now, from r = G^t and Lemma 3.8 we obtain the coordinate expression of (3.26) in the form k (3.27) ^(Ai...AfcFAfc+1^ = Qj ±y ±^ For a nonvanishing A;-multi vector field the equations (3.27) are satisfied only for F = 0 but in this case the geometrical phase structure is contact. For a nonvanishing electromagnetic field the equations (3.27) are satisfied only for vanishing A;-multi vector field and we have no induced infinitesimal hidden symmetry. □ Remark 3.7. As an example let us assume the canonical Killing-Maxwell 2-vector 2 ^ field KXp = Gxp for the unsealed metric, [13]. Then the above conditions (3.27) are in the form KpXl Fn 0. GpxFPß = 0, G{XlX2FXa\,, = 0. which implies F = 0 . □ 3.4. Comparison with infinitesimal symmetries of the kinetic energy function. Lemma 3.11. Let K be a function on T*E constant of motion, i.e. X\j ■ K = 0, then its pull-back —t*(K) is a conserved function, i.e. 7' • t*(K) = 0. Proof. The proof is the same as the proof of Lemma 2.8 by observing that -Te?(^) = Fpx?p(e)dxJ eediE, and by using equation (1.5). □ If we consider the electromagnetic field, then in both approaches we get the same results for projectable infinitesimal symmetries, see Corollary 1.1 and Remark 3.5. On the other hand, Killing-Maxwell multi-vector fields of rank > 2 admits hidden infinitesimal symmetries of H on T*E. On SiE Killing-Maxwell multi-vector fields admit functions conserved by the Reeb vector field of the joined structure but to obtain infinitesimal symmetries of the joined almost-cosymplectic-contact structure we need a further strong condition (3.26) which implies either F = 0 and the structure is reduced to the gravitational one or there are no hidden infinitesimal symmetries. We can summarize the results in the following diagram (T*E,J) constants of motion K, {H, K}J — 0 Killing—Maxwell multi-v.f.s k = 1 (3iE , —t , O') conserved functions, 7^.(—t*K) — 0 ISs of H projectable ISs of (—t,0') k > 2 no hidden ISs of (—r, 0') CONSTANTS OF MOTION AND CONSERVED FUNCTIONS 313 References [1] Crampin, M., Hidden symmetries and Killing tensors, Reports Math. Phys. 20 (1984), 31-40. DOI: 10.1016/0034-4877(84)90069-7 [2] Duval, C, Valent, G., Quantum integrability of quadratic Killing tensors, J. Math. Phys. 46 (5) (2005), 053516. DOI: 10.1063/1.1899986 [3] Iwai, T., Symmetries in relativistic dynamics of a charged particle, Ann. Inst. H. Poincaré Sect. A (n.S.) 25 (1976), 335-343. [4] Janyška, J., Special phase functions and phase infinitesimal symmetries in classical general relativity, AIP Conf. Proc. 1460, XX Internát. Fall Workshop on Geometry and Physics, 2011, pp. 135-140. 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