Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), ISSN

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1 Acta Mathematca Academae Paedagogcae Nyíregyházenss ), ISSN A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY NICOLETA BRINZEI Abstract. We show that, for mechancal system wth external forces, the equatons of devatons of soluton curves of the correspondng Lagrange equatons, determne a nonlnear connecton on the second order tangent bundle. In partcular, Jacob equatons n Fnsler and Remann spaces determne such a nonlnear connecton. 1. Introducton As shown n [27], nonlnear connectons on bundles can be a powerful tool n ntegratng systems of dfferental equatons. A way of obtanng them s that of dervng them from the respectve systems of DE s, n partcular, from varatonal prncples, [2], [16], [15]. For nstance, an ODE system of order 2 on a manfold M nduces a nonlnear connecton on ts tangent bundle T M. A remarkable example s here the Cartan nonlnear connecton of a Fnsler space, whch has the property that ts autoparallel curves correspond to geodescs of the base manfold: δy dy := + N jy j = 0. Further, an ODE system of order three determnes a nonlnear connecton on the second order tangent jet) bundle T 2 M = J0 2 R, M). For nstance, Crag- Synge equatons R. Mron, [16]) lead to: d 3 x 3 + 3!G x, ẋ, ẍ) = 0, 2000 Mathematcs Subject Classfcaton. 53B40, 70H50. Key words and phrases. nonlnear connecton, 2-tangent bundle, Fnsler space, Jacob equatons. 33

2 34 NICOLETA BRINZEI a) Mron s connecton: ) 1) M j = G 1) y, M 2)j j = 1 j + M1) m M1) mj, 2) 2 SM1) where S = y x + 2y2) y 3G y s a semspray on T 2 M. 2) b) Bucătaru s connecton M 1) j = G y, M 2)j j = G 2) y j. Wth respect to the last one, f G are the coeffcents of a spray on T 2 M.e., 3-homogeneous functons), then the Crag-Synge equatons can be nterpreted as: 2) where δy2) := dy2) + M 1) j dy j δy 2) = 0, j dx j + M 2). In Mron s and Bucătaru s approaches, nonlnear connectons on T 2 M are obtaned from a Lagrangan of order 2, Lx, ẋ, ẍ), by computng the frst varaton of ts ntegral of acton. Here, we propose a dfferent approach, whch, we consder, could be at least as nterestng as the above one from the pont of vew of Mechancs - namely, we start wth a frst order Lagrangan Lx, ẋ) and compute ts second varaton. Ths way, for a mechancal system M, Lx, ẋ), F x, ẋ)) wth external force feld F, we obtan a nonlnear connecton on T 2 M, wth respect to whch the equatons of devatons of evoluton curves have a smple nvarant form. As a remark, our nonlnear connecton s also sutable for modellng the solutons of a globally defned) ODE system, not necessarly attached to a certan Lagrangan, together wth the devatons of these solutons. More precsely, n the followng our ams are: 1) to obtan the Jacob equatons for the trajectores δy = 1 2 F x, y) for extremal curves of a 2-homogeneous Lagrangan Lx, ẋ) n presence of external forces). 2) to buld a nonlnear connecton such that: w X M) Jacob feld along c δw2) = 0,

3 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 35 where d denotes drectonal dervatve wth respect to ċ and δw 2) = 1 d 2 w M dw j j + M 1) j w j. 2) For F = 0, ths nonlnear connecton has as adonal propertes: I. In Fnsler spaces M, c s a geodesc of M f and only f ts extenson T 2 M s horzontal. II. A vector feld w along a geodesc c on M s parallel along c f and only f δw = 0. Throughout the paper, by dfferentable or smooth we mean C -dfferentable. 2. Tangent bundle of frst and second order Let M be a real dfferentable manfold of dmenson n and class C ; the coordnates of a pont x M n a local chart U, φ) wll be denoted by φ x) = x ), = 1,..., n. Let T M, π, M) be ts tangent bundle and x, y ) the coordnates of a pont n a local chart. The 2-tangent bundle T 2 M, π 2, M) s the space of jets of order two at 0 of all smooth functons f : ε, ε) M, t f t)), on ε, ε), ε > 0, [19]-[24], [16], [10]). In a local chart, a pont p of T 2 M wll have the coordnates x, y, y 2) ). Ths s, x = f 0), y = f 0), y 2) = 1 f 0), = 1,..., n, 2 for some f as above. Then, T 2 M, π 2, M ) s a dfferentable manfold of class C and dmenson 3n, and T M can be dentfed wth a submanfold of T 2 M. The local coordnate changes nduced by local coordnate changes on M are, [16], [19]-[24], x = x x 1,..., x n) ) x, det x j 0 3) ỹ = x x j yj 2ỹ 2) = ỹ x j yj + 2 ỹ y j y2)j. For a curve c: [0, 1] M, t x t)) on the base manfold M, let us denote: by ĉ ts extenson to the tangent bundle T M : ĉ: [0, 1] M, t x t), ẋ t));

4 36 NICOLETA BRINZEI along ĉ, there holds: by c ts extenson to T 2 M: y = ẋ t), = 1,..., n; c: [0, 1] T 2 M, t x t), ẋ t), 1 x t)); 2 along such an extenson curve, there holds y t) = ẋ t), y 2) t) = 1 x t), = 1,..., n. 2 A tensor feld on T M or T 2 M) s called a dstngushed tensor feld, or smply, a d-tensor feld f, under a change of local coordnates nduced by a change of coordnates on the base manfold M, ts components transform by the same rule as the components of a correspondng tensor feld on M, [16]. 3. Nonlnear connectons on T M Let T M, π, M) be the tangent bundle of a dfferentable manfold M as above and x, y ) the coordnates of a pont p T M n a local chart. For smplcty, we shall also denote x, y) = x, y ) =1,n. Let dπ : T T M) T M denote the tangent lnear mappng of the projecton π : T M M and V T M) = ker dπ, the vertcal subbundle of T T M). Its fbres generate the vertcal dstrbuton V on T M of local dmenson n, V : p T M V p) T p T M), locally spanned by { y }. A nonlnear Ehresmann) connecton on T M, [16], [18], s a dstrbuton N : p T M Np) T p T M), whch s supplementary to the vertcal dstrbuton: 4) T p T M) = N p) V p), p T M. Let where: 5) δ δx = B = { } δ δx, y, x N j, = 1,..., n, yj denote a local adapted bass to the drect decomposton 4). The quanttes N j = N j x, y), [16], [18], are called the coeffcents of the nonlnear connecton N. Wth respect to local coordnate changes on T M nduced by changes of local coordnates x ) x δ ) on the base manfold M, δx transform by the rule: δ δx = xj x δ δ x j.

5 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 37 The dual bass of B s B = { dx, δy }, gven by 6) δy = dy + N jdx j. Wth respect to changes of local coordnates on T M nduced by local coordnate changes on M, there holds: δỹ = x x j δyj. Any vector feld X X T M) s represented n the local adapted bass as 7) X = X 0) δ δx + X1) y, where the components X 0) δ and X1) are d-vector felds. δx y Smlarly, a 1-form ω X T M) wll be decomposed as the sum of two d-1-forms: 8) ω = ω 0) dx + ω 1) δy. In partcular, f ĉ: t x t), y t)) s an extenson curve to T M, then ts tangent vector feld s expressed n the adapted bass as 9) ĉ = dx δ δx + δy y. In our further consderatons, an mportant role wll be played by the notons of semspray and spray, [25], [10]. A semspray S X T M) s a vector feld locally descrbed n the natural bass by S = y x 2G x, y), where the y functons G called the coeffcents of the semspray) obey, wth respect to coordnate changes nduced by a change of local coordnates x ) x ) on M, the rule: 2 G = 2 x x j Gj ỹ x j yj, = 1,..., n. If G are 2-homogeneous functons n y, then the semspray s called a spray. As shown by Grfone, [12], a semspray n partcular, a spray) on M determnes a nonlnear connecton on T M. Also, evoluton curves of mechancal systems wth external forces, can be descrbed n terms of semsprays on T M, R. Mron, [15]): Proposton 1. Let L = Lx, ẋ) be a nondegenerate Lagrangan: 2 ) L det y y j 0, and g j = 1 2 L 2 y, the nduced Lagrange) metrc tensor. Then, the equatons yj of evoluton of a mechancal system wth the Lagrangan L and the external force feld F = F x, ẋ)dx are 10) d 2 x 2 + 2G x, ẋ) = 1 2 F x, ẋ),

6 38 NICOLETA BRINZEI where 2G = 1 2 L 2 gs y s x j yj L ) x s, yeld a semspray called the canoncal semspray of the Lagrange space M, L)) and F = g j F j, = 1,..., n. In the followng, we shall use the above results n the case when G s a spray; ths s, we shall have 2G = G y j yj. Then, [12], [2], [5], [18], the quanttes N j = G y j are the coeffcents of a nonlnear connecton on T M. Moreover, N j = N j x, y) are 1-homogeneous n y. Wth respect to the above nonlnear connecton, equatons 10) take the form: δy 11) = 1 2 F, = 1,..., n. In partcular, f there are no external forces, ths s, f F = 0, then the extremal curves t x t) of the Lagrangan L have horzontal extensons and vce-versa: horzontal extenson curves ĉ project onto soluton curves of the Euler-Lagrange equatons of L. 4. Nonlnear connectons on T 2 M Let dπ 2 : T T 2 M ) T M denote the tangent lnear mappng of the projecton π 2 : T 2 M M and V T 2 M ) = ker dπ 2, the vertcal subbundle of T T 2 M ). Its fbres generate the vertcal dstrbuton V on T 2 M of local dmen- son 2n, V : p T 2 M V p) T p T 2 M ) {, locally spanned by y, y 2) In the same way, f the projecton π1 2 : T 2 M T M s gven by x, y, y 2)) x, y ), then V 2 := ker dπ1 2 generates a dstrbuton V 2 : p T 2 M V 2 p) T p T 2 M ) { of local dmenson n, locally spanned by y 2) Then, at any p T 2 M, there exsts a chan of vector spaces }. V 2 p) V p) T p T 2 M ). Let us consder the F T 2 M ) -lnear mappng J : X T 2 M ) X T 2 M ), ) 12) J x = ) y, J y = ) y, J = 0, 2) y 2) }.

7 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 39 called the 2-tangent structure on T 2 M. J s globally defned on T 2 M and Im J = V, KerJ = V 2, J V ) = V 2. A nonlnear connecton on T 2 M, [16], s a dstrbuton on T 2 M, N : p T 2 M Np) T p T 2 M), such that 13) T p T 2 M) = N 0 p) V p), p T 2 M. By settng N 1 p) := JN 0 p)), p T 2 M, we get: the horzontal dstrbuton N 0 : p Np); the v 1 -dstrbuton N 1 : p N 1 p); the v 2 -dstrbuton V 2 : p V 2 p), and there holds T p T 2 M) = N 0 p) N 1 p) V 2 p), p T 2 M. We denote by h = v 0, v 1 and v 2 the projectors correspondng to the above dstrbutons. Let B denote a local adapted bass to the decomposton 13): B = { δ 0) := δ δx, δ 1) := δ δy, δ 2) := δ δy 2) ths s, N 0 = Spanδ 0) ), N 1 = Spanδ 1) ), V 2 = Spanδ 2) ). The elements of the adapted bass are locally expressed as 14) δ 0) = δ δx = x N j 1) y j N 2) δ 1) = δ δy = y N j 1) y 2)j δ 2) = δ δy = 2) y. 2) j y 2)j Wth respect to changes of local coordnates on T 2 M, nduced by changes x ) x ) of local coordnates on the base manfold M, for δ α), α = 0, 1, 2, there holds: δ α) = xj x δ α)j. The dual bass of B s B = { dx, δy, δy 2)}, gven by }, 15) δy 0) = dx, δy = dy + M 1) j dx j, δy 2) = dy 2) + M 1) j dy j + M 2) j dx j. The above δy α), α = 0, 1, 2, = 1,..., n, are d-1-forms on T 2 M. j The quanttes N, N j are called the coeffcents of the nonlnear connecton 1) 2) N, whle M j and M j are called ts dual coeffcents. The lnk between the two 1) 2)

8 40 NICOLETA BRINZEI sets of coeffcents s, [16]: 16) M 1) j = N 1) j, M 2) j = N 2) j + N 1) f N 1) fj. In the followng, the next result wll be very useful to us: Theorem 2 [16],[19]-[24]). 1. A transformaton of coordnates 3) on the dfferentable manfold T 2 M mples the followng transformaton of the dual coeffcents of a nonlnear connecton 17) x x k M k j = M x k k 1) 1) x j + ỹ x j x x k M k j = M x k k 2) 2) x j + M ỹ k k 1) x j + ỹ2) x j. 2. If on each doman of local chart on T 2 M t s gven a set of functons M1) j, M2) j ), such that, wth respect to 3), there hold the equaltes 17), then there exsts on T 2 M a unque nonlnear connecton N whch has as dual coeffcents the gven set of functons. In presence of a nonlnear connecton, a vector feld X X T 2 M ) s represented n the local adapted bass as 18) X = X 0) δ 0) + X 1) δ 1) + X 2) δ 2), wth the three rght terms whch are d-vector felds) belongng to the dstrbutons N, N 1 and V 2 respectvely. A 1-form ω X T 2 M ) wll be decomposed as 19) ω = ω 0) dx + ω 1) δy + ω 2) δy 2). Smlarly, a tensor feld T Ts r T 2 M ) can be splt wth respect to 13) nto components, whch are d-tensor felds. In partcular, f c: t x t), y t), y 2) t)) s an extenson curve, then ts tangent vector feld s expressed n the adapted bass as 20) c = dx δ 0) + δy δ 1) + δy2) δ 2). Our goal s to gve a precse meanng to the equalty v 2 c) = Berwald lnear connecton on T 2 M Let G = G x, y) be the coeffcents of a spray on T M, and N jx, y) = G y j, the coeffcents of the nduced nonlnear connecton on T M).

9 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 41 Let also L jkx, y) = N j y k = 2 G y j y k, the local coeffcents of the nduced Berwald lnear connecton on T M, [16]. Now, let on T 2 M, a lnear connecton defned by N 1) j = N j x, y1) ) as above, and arbtrary N 2) j = N 2) j x, y, y 2) ). The Berwald connecton on T 2 M, [8], s the lnear connecton defned by 21) D δ0)k δ α)j = L jkδ α), D δβ)k δ α)j = 0, β = 1, 2, α = 0, 1, 2. Ths s, wth the notatons n [16], the coeffcents of the Berwald lnear connecton are BΓN) = L jk, 0, 0). For extensons c to T 2 M of curves c: [0.1] M, we can express the v 1 component of the tangent vector feld c, gven by δy the geometrc acceleraton, [13]) by means of the Berwald covarant dervatve: 22) Dy Let T denote ts torson tensor, and: := D y ec = δy, = 1,..., n. R jk = v 1 Tδ 0)k, δ 0)j ) = δ 0)k N j δ 0)j N k, ts v 1 h, h) components. Also, let R be the curvature tensor; then Rj kl = δ 0)l L jk δ 0)k L jl + L m jkl ml L m jll mk, Pj kl = δ 1)l L 3 G jk = y j y k y l, where Rj kl δ 0) = hrδ 0)l, δ 0)k ), Pj kl δ 0) = hrδ 1)l, δ 0)k ), defne ts only nonvanshng local components, [16]. Takng nto account that L jk do not depend on y2) and that G = G x, y) are 2-homogeneous n y, t follows: 23) y j R j kl = R kl. From the 2-homogenety of G, we also have 24) Pj kly l 3 G = y j y k y l yl = 0; P j kly j = P j kly k = 0.

10 42 NICOLETA BRINZEI 6. Jacob equatons for systems wth external forces Let us suppose that we know a pror a nonlnear connecton on the frst order tangent bundle T M, wth 1-homogeneous) coeffcents N G j x, y) = y j, comng from a spray on T M. Let c: [0, 1] M, t x t) be a curve on M, such that x are solutons for the system of ODE s 10): δẋ = 1 2 F x, ẋ), where F are the components of a d-vector feld on M. Let α: [0, 1] ε, ε) M, t, u) α t, u)) denote a varaton of c not necessarly wth fxed endponts): α t, 0) = x t), t [0, 1], y = α u=0 = dx the components of the tangent vector feld of c and w t) = α u u=0 the components of the devaton vector feld attached to the varaton α. Let α denote the followng extenson of α to the second order tangent bundle T 2 M: 25) α: [0, 1] ε, ε) T 2 M, t, u) α t, u), and αt = α, α u = α u. We have: ) ) α α h = α tδ 0), h = α u uδ 0) ; αtt, 0) = y t), αut, 0) = w, t [0, 1]. Let us denote D = D eα and D u = D eα respect to the Berwald connecton on T 2 M. Then: u α t, u), α t, u)) 2 the covarant dervatons wth Dα t = α t + N jα, α t )α j t, 26) Dα t u = α t u + N jα, α t )α j u, Dα u = α u + N jα, α t )α j u; the covarant dervatves are taken wth reference vector α, [5]).

11 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 43 By commutng partal dervatves of α, we have α t u = α u, hence that the last two covarant dervatves 26) concde: whch s, By applyng D eα 27) Dα t u = Dα u, D = D. u u agan to the above equalty, we get: D D u = D D. u In the left hand sde, we can commute covarant dervatves by means of the curvature tensor of D : D D α = R u, α ) + D D u u + D» eα, eα. u But, [ eα becomes 28), eα u ] s 0, hence the last term n the above relaton vanshes and 27) D D α = R u, α ) u + D D u. Moreover, at u = 0, we have h α u=0 = αtt, 0)δ 0) = y δ 0), and by means of 11), we get D u=0 = Dy δ 0) = 1 2 F δ 0) =: 1 2 F where F s a d-vector feld on T 2 M). Then, 28) becomes D 2 29) 2 α u u=0 = R, α ) u=0 + 1 u 2 D uf. At u = 0, we also have h α u = w δ 0). In local wrtng, by evaluatng α R, α ) u and takng nto account 24), we obtan α R, α ) u=0 = y h y k Rh u jkw j δ 0). We have thus proved

12 44 NICOLETA BRINZEI Proposton 3. The components of the devaton vector feld w = α u u=0 of the trajectores δy 30) = 1 2 F x, y), satsfy, wth respect to the Berwald lnear connecton on T 2 M, the Jacob-type equaton D 2 w 31) 2 = 1 DF 2 u u=0 + y h y k Rh jkw j. The above generalzes the usual Jacob equaton, n the case of mechancal systems wth external forces. 7. Nonlnear connecton In natural coordnates, 31) becomes: d 2 w 2 + 2N j 1 F ) dw j 2 y j 32) d + N j) + N kn k j y h y k Rh jk + L 1 kj 2 F k 1 F ) 2 x j w j = 0. Takng nto account 23), we have R hjk yh = R jk. Also, L kj = N k y j, hence the above equalty can be seen as: d 2 w 2 + 2N j 1 F ) dw j 2 y j + CN j) + N kn k j y k R jk + 1 N k 2 y j F k 1 F ) 2 x j w j = 0, where There holds: Theorem 4. 33) C = y k x k + 2y2)k y k. 1) The quanttes M 1) j x, y) = 1 2 M 2) j x, y, y 2) ) = 1 2 2N j 1 2 F ) y j, CN j) + N kn k j y k R jk + 1 N k 2 y j F k 1 F ) 2 x j are the dual coeffcents of a nonlnear connecton on T 2 M.

13 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 45 2) Wth respect to ths nonlnear connecton, the extensons of devaton vector felds attached to 10) have vanshng v 2 -components: 1 d 2 w M dw j j + M 1) j w j = 0. 2) Proof. 1): In the equaton 31), both the left hand sde and the rght hand sde are components of d-vector felds; by a drect computaton, t follows that, wth respect to local coordnate changes 3) on T 2 M, the quanttes M 1) j and M 2) j obey the rules of transformaton 17) of the dual coeffcents of a nonlnear connecton on T 2 M. 2): The devaton vector feld attached to the varaton α n 25) s W = α { α u u=0 u x + ) α u y α ) } 2 u 2 y 2) u=0 = w x + dw y + 1 d 2 w 2 2 y 2). In the adapted bass δ 0), δ 1), δ 2) ), ths yelds: where δw W = w δ 0) + δw δ 1) + δw2) δ 2), = dw + M j x, y)w j and 1) δw 2) = 1 d 2 w M j x, y) dwj + M 1) j x, y, y 2) )w j. 2) Takng nto account 33), the Jacob equaton 32) s re-expressed as: δw 2) = 0. In presence of the above nonlnear connecton, the extenson W to T 2 M of any Jacob feld on M, correspondng to trajectores 10) n presence of external forces, belongs to the N 0 N 1 dstrbuton. 8. Devatons of geodescs Let us examne the partcular case when F = 0. Let T M be endowed wth a spray wth coeffcents G = G x, y) and N j = G, the coeffcents of the yj assocated nonlnear connecton on T M. If F = 0, then we deal wth devatons of autoparallel curves called geodescs) δy = 0.

14 46 NICOLETA BRINZEI We get M 1) j = N j, M 2) j = 1 2 CN j) + N kn k j y j R jk); takng nto account that, n our approach, M 1) j do not depend on y 2), we notce that, n the case F = 0, our nonlnear connecton only dffers by the term y j R jk from Mron s one 1), [16]. Remark 5. Along an extenson curve c: [0, 1] T 2 M, t x t), y t) = ẋ t), y 2) t) = 1 2ẍ t)) there hold the equaltes δy = Dy, δy 2) = D2 y 2, where D denotes the covarant dervatve assocated to the Berwald connecton on T 2 M. For these curves, takng nto account the equaltes y j y k R jk = 0 whch can be obtaned by drect calculaton), t follows that, wth the assumptons made at the begnnng of ths secton, δy δy2) and have the same values as those obtaned for the connecton 1). Stll, along general curves γ on T 2 M, the value of v 2 γ) does no longer concde wth that one obtaned wth respect to 1). Remark 6. Also, for a vector feld w along the projecton c of c onto M, we have Conclusons: δw = Dw. 1) c s a geodesc f and only f ts extenson to T 2 M s horzontal. 2) For a vector feld w along a geodesc c on M, we have: a) δw b) δw2) = 0, f and only f w s parallel along ċ = y. = 0 f and only f w s a Jacob feld along c. In the case F = 0, we should menton some related results and approaches: In the geometry of T M: In the case when the base manfold M s endowed wth a lnear connecton, a lnear connecton on the tangent bundle T M, wth smlar propertes to those of 33) s gven by the complete lft C of cf. [28] and [10]). Namely, n the two cted monographs, t s shown that, f a curve σ : [0, 1] T M, t x t), w t)) s a geodesc wth respect to C,

15 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 47 then ts projecton σ : t x t)) onto M s a geodesc wth respect to and Xt) = w t) s a Jacob feld along σ. x In the geometry of T 2 M: In presence of a lnear connecton on M, C. Dodson and M. Radvoovc, [11] bult a covarant dervaton law : X M) ΓT 2 M) ΓT 2 M) for sectons of the second order tangent bundle regarded as a vector bundle over M) and used t n order to defne a nonlnear connecton n the frame bundle of order 2 L 2) M. In the case when s torson-free, the covarant dervatve v X, where v = α α u u=0, and X, D ) α wth our ) δw notatons n Secton 6) would yeld our, δw2). Stll, n the cted paper, t s not establshed any lnk between the defned connecton and the Jacob equaton on M. The novelty of our approach conssts n relatng the v 2 -dstrbuton on T 2 M to devatons of geodescs of the base manfold. 9. External forces n Fnsler-locally Mnkowskan spaces Another nterestng partcular case s that of Fnsler-locally Mnkowskan spaces whose geodescs are straght lnes). Let M, Ly)) be a Fnsler-locally Mnkowskan space, [2], [5]. Then, N j = 0, L jk = 0 for the Berwald connecton), [2], [5]. In presence of an external force feld, the evoluton equatons of a mechancal system wll take the form 34) d 2 x 2 = 1 2 F x, ẋ). In ths case, wth the above notatons, our nonlnear connecton s gven by M j = 1 F 1) 4 y j, M j = 1 F 2) 4 x j. Ths s, devatons of the evoluton curves 34) can be wrtten smply: 2 δw2) d2 w 2 1 F dw j 2 y j 1 F 2 x j wj = 0. The result holds vald for any globally defned system of ordnary dfferental equatons of order 2 on M, of the form 34).

16 48 NICOLETA BRINZEI References [1] M. Anastase and I. Bucătaru. Jacob felds n generalzed Lagrange spaces. Rev. Roumane Math. Pures Appl., ): , Collecton of papers n honour of Academcan Radu Mron on hs 70th brthday. [2] P. L. Antonell, R. S. Ingarden, and M. Matsumoto. The theory of sprays and Fnsler spaces wth applcatons n physcs and bology, volume 58 of Fundamental Theores of Physcs. Kluwer Academc Publshers Group, Dordrecht, [3] V. Balan. Devatons of geodescs n fber bundles. In Proc. of the 23rd Conf. of Geom. and Topology, pages [4] V. Balan. On geodescs and devatons of geodescs n the fbered fnsleran approach. Stud. Cerc. Mat., 464): , [5] D. Bao, S.-S. Chern, and Z. Shen. An ntroducton to Remann-Fnsler geometry, volume 200 of Graduate Texts n Mathematcs. Sprnger-Verlag, New York, [6] N. Brînze Vocu). Devatons of Geodescs n the Geometry of Second Order. PhD thess, Babes-Bolya Unv., Cluj-Napoca, [7] I. Bucataru. The Jacob felds for a spray on the tangent bundle. Nov Sad J. Math., 293):69 78, XII Yugoslav Geometrc Semnar Nov Sad, 1998). [8] I. Bucataru. Lnear connectons for systems of hgher order dfferental equatons. Houston J. Math., 312): electronc), [9] C. Catz. Sur le fbré tangent d ordre 2. C.R. Acad. Sc. Pars, 278: , [10] M. de León and P. R. Rodrgues. Methods of dfferental geometry n analytcal mechancs, volume 158 of North-Holland Mathematcs Studes. North-Holland Publshng Co., Amsterdam, [11] C. T. J. Dodson and M. S. Radvoovc. Tangent and frame bundles of order two. An. Ştnţ. Unv. Al. I. Cuza Iaş Secţ. I a Mat. N.S.), 281):63 71, [12] J. Grfone. Structure presque-tangente et connexons. I. Ann. Inst. Fourer Grenoble), 221): , [13] A. D. Lews. The geometry of the Gbbs-Appell equatons and Gauss prncple of least constrant. Rep. Math. Phys., 381):11 28, [14] J. Mlnor. Morse theory. Based on lecture notes by M. Spvak and R. Wells. Annals of Mathematcs Studes, No. 51. Prnceton Unversty Press, Prnceton, N.J., [15] R. Mron. Dynamcal systems n fnsler geometry and relatvty theory. to appear. [16] R. Mron. The geometry of hgher-order Lagrange spaces, volume 82 of Fundamental Theores of Physcs. Kluwer Academc Publshers Group, Dordrecht, Applcatons to mechancs and physcs. [17] R. Mron. The geometry of hgher-order Fnsler spaces. Hadronc Press Monographs n Mathematcs. Hadronc Press Inc., Palm Harbor, FL, Wth a foreword by Ruggero Mara Santll. [18] R. Mron and M. Anastase. Vector bundles and Lagrange spaces wth applcatons to relatvty, volume 1 of Balkan Socety of Geometers Monographs and Textbooks. Geometry Balkan Press, Bucharest, Wth a chapter by Satosh Ikeda, Translated from the 1987 Romanan orgnal. [19] R. Mron and G. Atanasu. Compendum on the hgher order Lagrange spaces: the geometry of k-osculator bundles. Prolongaton of the Remannan, Fnsleran and Lagrangan structures. Lagrange spaces L kn). Tensor N.S.), 53Commemoraton Volume I):39 57, Internatonal Conference on Dfferental Geometry and ts Applcatons Bucharest, 1992).

17 A SPECIAL NONLINEAR CONNECTION IN SECOND ORDER GEOMETRY 49 [20] R. Mron and G. Atanasu. Compendum sur les espaces Lagrange d ordre supereur: La geometre du fbre k-osculateur. Le prolongement des structures Remannennes, Fnslerennes et Lagrangennes. Les espaces L k)n. Unv. Tmşoara, Semnarul de Mecancă, 40:1 27, [21] R. Mron and G. Atanasu. Lagrange geometry of second order. Math. Comput. Modellng, 204-5):41 56, Lagrange geometry, Fnsler spaces and nose appled n bology and physcs. [22] R. Mron and G. Atanasu. Dfferental geometry of the k-osculator bundle. Rev. Roumane Math. Pures Appl., 413-4): , [23] R. Mron and G. Atanasu. Hgher order Lagrange spaces. Rev. Roumane Math. Pures Appl., 413-4): , [24] R. Mron and G. Atanasu. Prolongaton of Remannan, Fnsleran and Lagrangan structures. Rev. Roumane Math. Pures Appl., 413-4): , [25] R. Mron, D. Hrmuc, H. Shmada, and S. V. Sabau. The geometry of Hamlton and Lagrange spaces, volume 118 of Fundamental Theores of Physcs. Kluwer Academc Publshers Group, Dordrecht, [26] M. Rahula. New problems n dfferental geometry, volume 8 of Seres on Sovet and East European Mathematcs. World Scentfc Publshng Co. Inc., Rver Edge, NJ, [27] M. Rahula. Vektornye polya smmetr. Tartu Unversty Press, Tartu, Chapter 2 by the author, D. Boularas and H. Lepp; Chapter 3 by the author and D. Tseluko; Chapter 4 by the author and V. Retšno; Chapter 5 by the author and Z. Navckas; Appendx II by the author and T. Mullar. [28] K. Yano and S. Ishhara. Tangent and cotangent bundles: dfferental geometry. Marcel Dekker Inc., New York, Pure and Appled Mathematcs, No. 16. Translvana Unversty, Brasov, Romana E-mal address: nco.brnze@rdslnk.ro