On the geometry of higher order Lagrange spaces.

Transcription

1 On the geometry of hgher order Lagrange spaces. By Radu Mron, Mha Anastase and Ioan Bucataru Abstract A Lagrange space of order k 1 s the space of acceleratons of order k endowed wth a regular Lagrangan. For theses spaces we dscuss: certan natural geometrcal structures on the space of k-jets, varatonal problem assocated to a gven regular Lagrangana and the nduced semspray, nonlnear connecton, metrcal connectons, varous lfts. A specal attenton s pad to the regular Lagrangans provded by the prolongatons of the Remannan and Fnsleran structures. In the end we sketch the geometry of tme dependent Lagrangan. The geometry, whch we have developed, s drected to Mechancsts and Physcsts. The paper s a short survey of our results on the hgher order geometry. We present n a concse form the facts dscovered along years. The man ponts are hghlghted. For detals we refer to the monograph [7]. AMS Subject Classfcaton: 53C60, 53C80, 58A20, 58A30. Introducton In the last twenty years the problem of the geometrsaton of hgher order Lagrangans was ntensvely studed for ts applcatons n Mechancs, Theoretcal Physcs, Varatonal Calculus, etc. These Lagrangans are defned on k-jets spaces. The theory had as consequence the appearance of the noton of hgher order Lagrange space ntroduced by 1

2 R.Mron n [7]. The geometry of these spaces derves from the prncples of the Hgher Order Mechancs. The ntegral of acton for a Lagrangan leads to a canoncal k-semspray from whch one constructs the whole geometry of the Lagrange space L n. On the total space of k-jets bundle appears n a unque way a specal almost contact structure whch nvolves n the geometry of L n. Essental contrbuton to ths subject have brought also by M. Anastase who studed the tme dependent hgher order Lagrangans and I.Bucataru who elaborated a Ph.D. Thess n ths feld. 1 The k-osculator bundle of a manfold. In ths secton we consder the bundle of jets of order k for the mappngs from IR to a manfold M usually denoted by J0 k (M) or T k M. In order to stress that we encounter only ths jet bundle and for some hstorcal reasons we call t the k-osculator bundle and denote t by (Osc k M, π k, M) For a local chart (U, ϕ = (x )) n p M ts lfted local chart n u (π k ) 1 (p) wll be denoted by ((π k ) 1 (U), Φ = (x, y,..., y )). For each u = (x, y,..., y ) E := Osc k M, the natural bass of the tangent space T u E s { x u, y u,..., y u}. The summaton over repeated ndces wll be mpled. We have k-canoncal surjectve submerson π k : Osc k M M and πα k : Osc k M Osc α M, α {1,..., k 1} whch are locally expressed by: π k : (x, y,..., y ) (x) and πα k : (x, y,..., y ) (x, y,..., y (α) ). Each of them determne a vertcal dstrbuton V α+1 E = Ker(πα) k, where (πα) k s the tangent map assocated to πα, k α {0, 1,..., k 1}. The tensor feld: J = y dx + y (2) dy + + y dy(k 1) s called the k-almost tangent structure on E. It has the propertes: 1. J k+1 = 0, 2. ImJ α = KerJ k α+1 = V α E, 3. rank J α = (k α + 1)n, α {1, 2,..., k}. The vector felds Γ 1 = y y, Γ 2 = y y (k 1) +2y(2) y,... 2

3 k Γ = y +2y(2) + +ky are called the Louvlle y y (2) y vector felds and they are globally defned on E. A vector feld S χ(e) s called a semspray or a k-semspray on E f JS = k Γ. The local expresson of a semspray s: (1.2) S = y x +2y(2) + +ky (k+1)g y y (k 1) y, where the functons G are defned on every doman of local charts. We consder the operator: (1.3) Γ = y x + 2y(2) + + ky y y. (k 1) Γ s not a vector feld on E but f f F(E) such that Γ(f) F(E). f = 0 then y 2 Varatonal problem for the hgher order Lagrangans As for Lagrangans of order one t can be consdered the ntegral acton for a Lagrangan of order k > 1. In ths secton we show how the varatonal problem assocated to t leads to the Euler-Lagrange equatons and the Synge equatons. From the latter a semspray of order k s derved. Also we present a new verson of the Noether theorem. Several examples are ncluded. A Lagrangan of order k, (k IN ), s a mappng L : E := Osc k M IR. L s called dfferentable f t s of class C on Ẽ := E \ {0}, (0 denotes the null secton of the k-osculator bundle) and contnuous on the null secton. The Hessan matrx of a dfferentable Lagrangan L, wth respect to y, on Ẽ has the elements g j : (2.1) g j = L y y j 3

4 If One can see that g j (x, y,..., y ) s a symmetrc d-tensor feld. (2.2) rank g j (x, y,..., y ) = n on Ẽ we say that L(x, y,..., y ) s a regular Lagrangan, otherwse we say that L s degenerate. For the begnnng we study the hgher order dfferentable Lagrangans wthout the regularty condton (2.2). The Le dervatves of a dfferentable Lagrangan L(x, y,..., y ) wth respect to the Louvlle vector felds 1 Γ,..., k Γ determne the scalars (2.3) I 1 (L) = 1 Γ L,..., I k (L) = kγ L. These are dfferentable functons on Ẽ, called the man nvarants of the Lagrangan L, because of ther mportance n ths theory. Let us consder a smooth parametersed curve c : [0, 1] M represented n a doman of local chart by x = x (t), t [0, 1]. The ntegral of acton for L(x, y,..., y ) s (2.4) I(c) = 1 0 L(x(t), dx(t),..., 1 dt k! d k x(t) )dt. dt k Theorem 2.1 The necessary condtons for the ntegral of acton I(c) be ndependent on the parametersaton of the curve c are (2.5) I 1 (L) = = I k 1 (L) = 0, I k (L) = L. We call (2.5) the Zermello condtons. We also have: Theorem 2.2 If the dfferentable Lagrangan L of order k, k > 1, satsfes the Zermello condtons, then rank g j (x, y,..., y ) < n on Ẽ,.e. L s degenerate. Let c : [0, 1] M be a smooth curve. On the open set U we consder the curves (2.6) c ε : t [0, 1] (x (t) + εv (t)) M, 4

5 where ε s a real number, suffcently small n absolute value so that Im c ε U, V (t) = V (x(t)) beng a regular vector feld on U, restrcted to the curve c. We assume that all curves c ε have the same endponts c(0) and c wth the curve c and ther dervatves of order 1,..., k 1 concde at the ponts c(0) and c. Theorem 2.3 In order that the ntegral of acton I(c) be an extremal value for the functonal I(c ε ) t s necessary that the followng Euler Lagrange equatons hold: (2.7) E (L) := L x d dt y = dx dt,..., y = 1 d k x k! dt k L y + + ( 1)k 1 k! d k dt k L y = 0 The curves c : [0, 1] M, solutons of equatons (2.7), are called extremal curves of the ntegral acton I(c). The equalty (2.7) mples the followng result: Theorem 2.4 E (L) s a d-covector feld. Along a smooth curve c we shall assocate to the covector felds E (L) the k-covector felds 1 E (L),..., k E (L) ntroduced by Crag and Synge. Let be along a smooth curve c : [0, 1] M the operators (2.8) E= 1 E= x d dt y ( 1)k k! k ( 1) α 1 α! ( α α 1 ) dα 1 dt α 1 α=1 k d k dt k y (α), y, 2 E= ( 1) α 1 α=2 α! ( α α 2 ) dα 2 dt α 2 y, (α)... k E= ( 1) α 1 k! y, These operators act IR-lnearly on the IR-lnear space of dfferentable Lagrangans L(x, y,..., y ). 5

6 Theorem 2.5 For any dfferentable Lagrangan L(x, y,..., y ) along a smooth curve c, 1 E (L),..., k E (L) are d-covector felds. Now let us consder a nondegenerate Lagrangan L wth the fundamental tensor feld g j. The d-covector feld k 1 E (L) has the followng form (2.9) k 1 E (L) = ( 1) k 1 1 (k 1)! { L L Γ( y (k 1) where Γ s the operator (1.3). It follows y ) 2 k! g j d k+1 x }, dtk+1 Theorem 2.6 The system of dfferental equatons (Synge equatons) k 1 j (2.10) g E j (L) = 0, determnes a k-semspray S wth the coeffcents (2.11) (k + 1)G = 1 { 2 gj Γ( L y ) L }. j y (k 1)j Theorem 2.7 For any dfferentable Lagrangans L(x, y,..., y ) and F (x, y,..., y (k 1) ), along a smooth curve c, we have (2.12) E (L + df dt ) = E (L), E ( df dt ) = 0, 1 E ( df dt ) = E (F ),..., k E ( df dt ) = k 1 E (F ). Theorem 2.8 The ntegrals of acton (2.13) I(c) = I (c) = L(x, dx dt,..., 1 d k x k! dt )dt k [L(x, dx dt,..., 1 k! d k x dt k )+ + df dx (x, dt dt,..., 1 (k 1)! d k 1 x )]dt dtk 1 have the same extremal curves, for any dfferentable Lagrangan F wth the property F = 0. y 6

7 Defnton 2.1 We call energes o order k, k 1,..., 1 of the Lagrangan L(x, y,..., y ), wth respect to the curve c, the followng nvarants Ec k (L) = I k (L) 1 di k 1 (L) + + 2! dt +( 1) k 1 1 d k 1 I 1 (L) L, k! dt k 1 (2.14) E k 1 c (L) = 1 2! Ik 1 (L) + 1 3! di k 2 (L) dt + + +( 1) k 1 1 d k 2 I 1 (L), k! dt k 2 Ec k 2 (L) = 1 3! Ik 2 (L) 1 d 4! dt Ik 3 (L) + + +( 1) k 1 1 d k 3 I 1 (L), k! dt k 3... E 1 c (L) = ( 1) k 1 1 k! I1 (L). The dependence of these nvarants on the curve c s obvous. A frst result s Proposton 2.1 The followng denttes hold: (2.15) Ec k (L) d dt E c k 1 (L) = I k (L) L E k 1 c (L) d dt E c k 2 (L) = 1 2 Ik 1 (L)... E 2 c (L) d dt E 1 c (L) = ( 1) k 2 1 (k 1)! I2 (L). As we shall see, the energes E k c (L),..., E 1 c (L) are nvolved n a Noether theory of symmetres of the hgher order Lagrangans. Wth ths end n vew, we state the followng result Theorem 2.9 For any Lagrangan L(x, y,..., y ) along a smooth curve c : [0, 1] (x (t)) M we have (2.16) de k c (L) dt = E (L) dx dt 7

8 An mmedate consequence of the last theorem s the followng law of conservaton: Theorem 2.10 For any Lagrangan L(x, y,..., y ), the energy of order k, E k c (L) s conserved along every curve whch s soluton of the Euler Lagrange equatons E (L) = 0. Defnton 2.2 A symmetry of the dfferentable Lagrangan L(x, y,..., y ) s a C dffeomorphsm ϕ : M R M R, whch preserves the varatonal prncple of the ntegral of acton I(c) from (2.4). Generally, the varatonal prncple s consdered on an open set U M. So, we can consder the noton of local symmetry of the Lagrangan L, takng ϕ as local dffeomorphsm. Therefore, n the followng consderatons we study the nfntesmal symmetres, gven on an open set U (a, b) M R n the form: (2.17) x = x + εv (x, t), ( = 1,..., n) t = t + ετ(x, t) where ε s a real number, suffcently small n absolute value so that the ponts (x, t) and (x, t ) belong to the same set U (a, b), where the curve c : t [0, 1] (x (t), t) U (a, b) s lyng. Throughout the followng calculaton terms of order ε 2, ε 3,... wll be neglected. Of course V (x, t) s a vector feld on the open set U (a, b). In the end ponts c(0) and c, we assume that V (t) := V (x(t), t) has the same values for = 1, 2,..., n. The nfntesmal transformaton (2.17) s a symmetry for the dfferentable Lagrangan L(x, y,..., y ) f and only f for any dfferentable functon F (x, y,..., y (k 1) ) the followng equaton holds: (2.18) L(x, dx dt,..., 1 d k x k! dt k )dt = = {L(x, dx dt,..., 1 k d k x dt ) + df dx (x, k dt dt,..., 1 (k 1)! d k 1 x )}dt dtk 1 8

9 Usng the operator (2.19) we can state: d V dt = V x k! d k V dt k y Theorem 2.11 A necessary and suffcent condton that an nfntesmal transformaton (2.17) be a symmetry for the Lagrangan L along the smooth curve c s that the left sde of the equalty (2.20) d V L dt + {L dτ dt [Ik (L) dτ dt + 1 2! Ik 1 (L) d2 τ dt k! I1 (L) dk τ dφ ]} = dtk dt be of the form d dt φ(x, y,..., y (k 1) ) along c. Theorem 2.12 (Noether) For any nfntesmal symmetry (2.17) (whch satsfes (2.18)) of a Lagrangan L(x, y,..., y ) and for any functon φ(x, y,..., y (k 1) ), the functon (2.21) F k (L, φ) := IV k (L) 1 d 2! dt Ik 1 V (L) + + +( 1) k 1 1 d k 1 k! dt k 1 I1 V (L)) τec k (L) + dτ dt E c k 1 (L) + +( 1) k dk 1 τ dt E 1 k 1 c (L) φ s conserved along the soluton curves of the Euler-Lagrange equaton E (L) = 0, where (2.22) IV 1 (L) = V L y, I2 V (L) = V I k V (L) = V L dv + y dt L dv + y (k 1) dt L,..., y L y d k 1 V (2) (k 1)! dt k 1 L y. 9

10 The functons F k (L, φ) n (2.21) contan the relatve nvarants I 1 V (L),..., I k V (L), the energes of order 1,..., k, E 1 c (L),..., E k c (L) and the functon φ(x, y,..., y (k 1) ). In partcular, f the Zermello condtons (2.5) are satsfed, then the energes E 1 c (L),..., E k c (L) vansh and we have a smpler form of the Noether theorem: Theorem 2.13 For any nfntesmal symmetry (2.17) of a Lagrangan L(x, y,..., y ), whch satsfes the Zermello condtons (2.5) and for any C functon φ(x, y,..., y (k 1) ), the followng functon F k (L, φ) := I k (2.21) V (L) 1 d 2 dt Ik 1 V (L) + + +( 1) k 1 1 d k 1 k! dt k 1 I1 V (L) φ, s conserved along the soluton curves of the Euler Lagrange equaton E (L) = 0. 3 Hgher order Lagrange spaces In the book ([7]), R.Mron defnes the Lagrange spaces as follows: A Lagrange space of order k s a par L n = (M, L), where M s a real n-dmensonal manfold, L : Osc k M IR s a dfferentable Lagrangan for whch the fundamental tensor gven by (2.1) satsfes (2.2) and the quadratc form ψ = g j ξ ξ j has the constant sgnature on ẼḞrstly, we prove: Theorem 3.1 ([2], [7]) If the base manfold M s paracompact then there exst the Lagrange spaces of order k, L n = (M, L), for whch the fundamental tensor g j s postvely defned. Proof. M beng a paracompact manfold, there exsts at least a Remannan structure γ j on M. Denote by D the Lev-Cvta connecton of (M, γ) and by γ jk = γ kj the local coeffcents of D. By a 10

11 straghtforward calculaton we can prove that: (3.2) z m = y m, z (2)m = 1 2 [Γzm + γ m j z z j ],, are d-vector felds. The Lagrangan z m = 1 k [Γz(k 1)m + γ m j z j z (k 1) ] L(x, y,..., y ) = γ j (x)z z j s defned on Ẽ, s a dfferentable Lagrangan, and has the fundamental tensor g j (x, y,..., y ) = γ j (x). Thus, the par (M, L) s a Lagrange space of order k. q.e.d. In a Lagrange space L n = (M, L) there exsts a k-semspray on E, wth the coeffcents gven by (2.11) whch depend only on the fundamental functon L. Ths k-spray s also called canoncal for the Lagrange space L n. A nonlnear connecton on the manfold E s a dstrbuton on E, N : u E N(u) T u E whch s supplementary to the vertcal dstrbuton V 1 : u E V 1 (u) T u E.e. we have T u E = N(u) V 1 (u), u E. For a nonlnear connecton N and for every u E the mapp (π k ),u N(u) : N(u) T π (u)m s a lnear somorphsm. Its nverse map wll be denoted by l h and s called the horzontal lft assocated N. Set = k x l h ( x ). The lnearly ndependent vector felds x,..., can be 1 x n unquely wrtten n the form: (3.3) x = The functons N j (α) x N j y j N j y j. (x, y,..., y ), (α = 1,..., k) are called the coeffcents of the nonlnear connecton N and ( ), ( = 1,..., n) s named x 11

12 the adapted bass to the horzontal dstrbuton N. Let be N α = J α (N) and = J α ( ), α = 0,..., k 1. Then the followng drect decomposton of lnear spaces y (α) x holds: (3.4) T u (E) = N 0 (u) N 1 (u) N k 1 (u) V k (u), u E. The bass adapted to ths decomposton s (3.5) { x,,..., y }. y The dual bass of the adapted bass (3.5) s gven by: (3.6) where (3.7) M j x = dx, y = dy + M j y = dy + M j =N j Mj =Nj, Mj =Nj (2) (2) + Nm Mj m (k 1) + N m M m j The set of functons (Mj (α) the nonlnear connecton N. dx j,..., dy (k 1)j + + M j (k 1),..., + + N m (2) M m j (k 2) dy j + M j dx j, + Nm Mj m. (k 1) ) α=1,k are called the dual coeffcents of Theorem 3.2 (R.Mron, [7]) In a Lagrange space L n = (M, L) there exsts a canoncal nonlnear connecton whch has the dual coeffcents (3.8) Mj = G M j (α) = 1 α y j S M j (α 1) + M m M m j, (α = 2,..., k). (α 1) Theorem 3.3 (I.Bucataru, [2]) Let S be a k-semspray wth G coeffcents. The system of functons: as (3.9) Mj = G y, M j j (2) = G y (k 1)j,..., M j 12 = G y j,

13 are the dual coeffcents of a nonlnear connecton N on the k-osculator bundle. For a nonlnear connecton N, an N-lnear connecton s a lnear connecton D on E wth the propertes D preserves by parallelsm the horzontal dstrbuton N. (2) The k tangent structure J s absolutely parallel wth respect to D. An N-lnear connecton D on E can be represented n the adapted bass (3.5), n the form (3.10) D x j D y (β)j x =Lm j x m, x = C (β) m j (α, β = 1,..., k). D x j x, D y (β)j Theorem 3.4 The followng propertes hold: y (α) =Lm j, (α=1,...,k) y (α)m y = m (α) C j (β) y, (α)m 1 There exsts a unque N-lnear connecton D on axoms: Ẽ verfyng the (3.11) g j h = 0, g j (α) h = 0, (α = 1,..., k) (3.12) T jh = L jh L hj = 0, (0) S jh =C jh C hj = 0, (α = 1,..., k). (α) (α) (α) 2 The coeffcents CΓ(N) = (L jh, C jh,..., C jh ) of ths connecton are gven by the generalsed Chrstoffel symbols: 13

14 (3.13) L m j = 1 ( gs 2 gms x + g sj j x g ) j, x s m C j = 1 ( gs (α) 2 gms y + g sj (α)j y g j (α) y (α)s ) (α = 1,..., k). 3 Ths connecton depends only on the fundamental functon L(x, y,..., y ) of the space L n. The connecton D from the prevous theorem s the canoncal metrcal N-connecton of the space L n. Its set of coeffcents was denoted by CΓ(N). Let R n = (M, γ j ) be a Remann space and z the Louvlle vector felds gven by (3.2). Thus (3.13) L(x, y,..., y ) = mcγ j (x)z z j + 2e m A (x)z + U(x) s a regular Lagrangan, called the Lagrangan of the hgher order Electro-dynamcs. Of course, m, c, e are the well known constants from Physcs, γ j (x) are the gravtatonal potentals, A (x) s the electromagnetc covector feld and U(x) a scalar. The par L n = (M, L), wth L from (3.13) s a Lagrange space of order k. 4 Horzontal lfts n the hgher order geometry I.Bucataru studed the noton of lft for L n, n hs Ph.D.Thess. Below we follow hs presentaton of ths subject, [2]. Frst, we present n a new form the vertcal and the complete lfts for a vector feld. For, X = X x χ(m), we denote by Xv k = (X π k ) ts vertcal y lft. The map l vk : χ(m) χ(e), whch s defned by l vk (X) = X v k s F(M)-lnear and s called also the vertcal lft. Ths means that for every X χ(m) and f F(M) we have l vk (fx) = (f π k )l vk (X). For 14

15 X = X x χ(m) and S a k-semspray, the vector feld Xc χ(e) defned by: (4.1) X c = (X π k ) x + 1 1! S(X π k ) y k! Sk (X π k ) y s called the complete lft of the vector feld X. A drect consequence of X = X (x) s S α (X ) = S α (X ) for every two semsprays S and S and so the complete lft of a vector feld X s ndependent on the choce of the semspray S. For the vertcal and complete lfts the followng formulae hold: (4.2) J k (X c ) = X v k, (fx) c = k α=0 1 α! Sα (f)j α (X c ), f F(M), X χ(m). It s seen from the second formula (4.2) that the map X χ(m) X c χ(e) s not a F(M)-lnear map. Next, we modfy ths map such that the new map wll be F(M)-lnear. Defnton 4.2 It s called horzontal lft a F(M)-lnear map l h : χ(m) χ(e), for whch we have: (4.3) J k l h = l vk. Every horzontal lft l h determnes a nonlnear connecton N = Iml h on the k-osculator bundle and conversely, every nonlnear connecton N determnes a horzontal lft. Let D be a lnear connecton on M. We denote by D c ts complete lft. Ths s unquely determned by: (4.4) D c X cy c = (D X Y ) c. For ths lnear connecton we have also (4.5) D c J α (X c )Y c = D c X cj α (Y c ) = J α ((D X Y ) c ), α = 0, k. Theorem 4.1 Let S be a semspray on E. For X χ(m) we de- 15

16 fne X v k 1,..., X v 1, X h χ(e) by: X v k 1 = J k 1 (X c ) 1 1! Dc S(X v k ), (4.6) X v k 2 = J k 2 (X c ) 1 1! Dc SX v k 1 1 2! Dc SX v k,..., X h = X c 1 1! Dc SX v 1 1 2! (Dc ) 2 SX v 2 1 k! (Dc ) k SX v k. The maps l vα, l h : χ(m) χ(e), α = 1, 2,..., k defned by l vα (X) = X v α, l h (X) = X h are F(M)-lnear and verfy J k l h = l vk, J α l h = l vα. These maps are ndependent on the choce of the semspray S. For N = Iml h let be N 1 = J(N), N 2 = J 2 (N),..., N k 1 = J k 1 (N). x = l h( x ) and y = l (α) v α ( x We set ), α {1,..., k 1}. On ths way we get for every u E a bass { x u, y u,..., y u, (k 1) for T u E whch s adapted to the drect decomposton (3.4) y u} 5 The prolongaton of the Remannan and Fnsleran structures to the hgher order jets bundle In ths secton we shall gve a soluton for the dffcult problem of the prolongaton to the manfold Osc k M of the Remannan and Fnsleran structures, defned on the base manfold M, [7]. Let R n = (M, g) be a Remannan space, g beng a Remannan metrc defned on M, havng the local coordnates g j (x), x U M. We show n the proof of the Theorem 3.1 that the Remann structure g determnes a regular Lagrangan L. Let S be the canoncal semspray wth the local coeffcents G gven by (2.11). From Theorem 3.2 or 3.3 we obtan the dual coeffcents of a nonlnear connecton N assocated to the Lagrange space L n = (M, L). Now, we can use the canoncal nonlnear connecton N wth the dual coeffcents (M j,..., M j ) and adapted cobass (dx, y,..., y ) gven by (3.6). 16

17 Theorem 5.1 The par Prol k R n = ( Osc k M, G ), where (5.1) G = g j (x)dx dx j + g j (x)y y j + + g j (x)y y j, s a Remannan space of dmenson (k + 1)n, whose metrc structure G depends only on the structure g j (x) of the apror gven Remann space R n = (M, g). The exstence of the Remannan space Prol k R n = ( Osc k M, G) solves the enuncated problem. Ths space s called the prolongaton of order k of the space R n = (M, g). Also, we say that G s Sasak N lft of the Remannan structure g. The prolongaton of a Fnsler structure can be ntroduced n a smlar way. Let F (x, y ) be a fundamental functon of a Fnsler space F n = (M, F ) and (5.2) g j (x, y ) = F 2 y y j, ts fundamental tensor feld. The problem s to determne a Remannan structure G on Osc k M whch depends only on the fundamental tensor g j (x, y ) of the Fnsler space F n. Denote by γ jm(x, y ) the Chrstoffel symbols of the d-tensor feld g j (x, y ). Then (5.3) G = 1 2 γ jmy j y m are the coeffcents of the canoncal 2-spray of the Fnsler space F n. The Cartan nonlnear connecton on T M has the followng coeffcents: (5.4) G j = G y j Now we can prove Theorem 5.2 There exst nonlnear connectons on Ẽ determned only by fundamental tensor g j (x, y ) of the Fnsler space F n =(M, F ). 17

18 One of them has the followng dual coeffcents: M j = G j ) (ΓG j + G m mj M (5.5) M j = 1 (2) 2... ( ), M j = 1 k Γ M (k 1) j + G m M (k 1) m j where Γ s the operator (1.3). Theorem 5.3 The par Prol k F n = ( Osc k M, G), k 2, where (5.6) G = g j (x, y )dx dx j + g j (x, y )y y j + + +g j (x, y )y y j s a Remannan space of dmenson (k+1)n, whose metrc structure G depends only on the fundamental tensor feld g j (x, y ) of the apror gven Fnsler space F n = (M, F ). Ths Remannan space Prol k F n = ( Osc k M, G) s called the prolongaton of order k of the Fnsler space F n. We say that G s Sasak N lft of the metrc tensor g j. The dfferental geometry of Prol k F n can be studed usng the general theory of the k-osculator bundle (Osc k M, π, M) endowed wth the nonlnear connecton N, from (2.5), and wth the Remannan structure G. For a Remannan space R n = (M, γ) let us consder the d- Louvlle vector felds z gven by (3.2), z = γ j z j and n(x, y,..., y ) 1 a smooth functon on Osc k M. Then (5.7) g j (x, y,..., y 1 ) = γ j (x) + (1 n ( x, y,..., y ) )z z j s a symmetrc nondegenerate d-tensor feld. Ths tensor s called the metrc tensor of the hgher order relatvstc optcs. The geometry of the generalsed hgher order metrc space Gl n = (M, g) can be studed usng the technques from the hgher order Lagrange geometry. 18

19 6 The hgher order Fnsler spaces Sxty-fve years ago, A. Kawaguch gave a frst defnton of Fnsler spaces of order k. But hs axom of homogenety leads to the so-called Zermello condtons. These have as consequence the fact that the fundamental tensor of the space s sngular, for k > 1. A new defnton of hgher order Fnsler space was formulated by the frst author wth Sorn Sabău, [11], takng nto account a new knd of homogenety for the fundamental functon of the consdered space. At the begnnng we study the noton of homogenety on the fbres of E = Osc k M. The group of homothetes of the real feld, H = {h t t IR + }, h t (a) = ta, a IR, acts as a Le group of transformatons on the manfold Ẽ = Osck M \ {0} as follows: (6.1) h t (x, y,..., y ) = (x, ty,..., t k y ), t IR +. Ths acton preserves the fbres of E. Now we can ntroduce the followng defnton: A functon f : E IR s called r homogeneous, (r Z), on the fbres of E f: (6.2) f h t = t r f, t IR + Theorem 6.1 A functon f : Osc k M IR s r-homogeneous on the fbres of E f and only f we have: kγ f = rf where s the Le operator of dervaton. Ths noton can be extend to the vector felds X χ(e) and q- forms on E. It follows that a vector feld X χ(e) s r-homogeneous on the fbres of E f, and only f kγ X = (r 1 )X. If X s gven n the form X = X (0) x + + X y then t s r-homogeneous on the fbres of E f, and only f the functons X (0),.., X are r 1,.., r 1 + k homogeneous, respectvely. 19

20 For nstance k Γ s 1-homogeneous on the fbres of E. A k-spray S s 2-homogeneous f, and only f ts coeffcents G are (k+1)-homogeneous on the fbres of E. Now the noton of hgher-order Fnsler space can be ntroduced [11]. Defnton 6.1 A Fnsler space of order k 1 s a par F n = (M, F ) determned by a real C -dfferentable manfold M of dmenson n and a functon F : Osc k M R havng the propertes: 1. F s of C -class on Ẽ and contnuous on the null secton; 2. F s postve; 3. F s k-homogeneous on the fbres of E; 4. The Hessan wth the elements: (6.3) g j = 1 2 F 2 2 y y j s postvely defned on Ẽ. Of course, g j from (6.3) s a d-tensor feld, called fundamental for the space F n, and F s called the fundamental functon of space F n. Evdently, f k = 1, F n = (M, F ) s a classcal Fnsler space. Some mmedate propertes 1. F 2 s 2k-homogeneous; F 2 2. s a k-homogeneous covector feld; y 3. g j s 0-homogeneous. Consequently, we have: (6.4) kγ F 2 = 2kF 2, kγ F 2 y = k F 2 y, k Γ g j = 0. 7 Tme dependent Lagrangans The case when a Lagrangan of order k > 1 explctly depends on tme was consdered by M. Anastase n [1]. In ths secton the noton of tme-dependent k-spray s ntroduced and charactersed. Then t 20

21 s shown that any tme-dependent k-spray nduces a nonlnear connecton and any tme dependent Lagrangan of order k determnes a k-spray va the Euler-Lagrange equatons. The explct appearance of tme s modelled by consderng the manfold E = IR Osc k M projected over IR M by π(t, u) = (t, x), x = π k (u), u Osc k M. The local coordnates on E are those on Osc k M together wth a new one t IR wth the meanng of absolute tme. Let πh k : IR Osc k M R Osc h M, h < k, h, k IN, be gven by (t, x, y,..., y ) (t, x, y,..., y (h) ), π0 k := π k and V 1 = ker(π k ), V 2 = ker(π1) k,...,v k = ker(πk 1) k, where (πh) k means the dfferental (tangent map) of the mappng πh. k Now, consder the lnear operators J, J : T u E T u E defned wth respect to the natural bass as follows: (7.1) J( ) = 0, J( t x ) =,..., J( y y ) =, J( (k 1) y y ) = 0 J( t ) = Γ, k J( x ) =,..., J( y y ) =, J( ) = 0. (k 1) y y A drect calculaton gves Proposton 7.1 a) J( k Γ) = k 1 Γ,..., J( 2 Γ) = 1 Γ, J( 1 Γ) = 0. b) J} J {{ J} = 0. The same holds for J. k+1 tmes c) J s an ntegrable k tangent structure. We notce that J s not ntegrable as k tangent structure. A tme-dependent vector feld on Osc k M s a smooth mappng X : IR Osc k M T (Osc k M), (t, u) X (t, u) T u (Osc k M), u Osc k M. It nduces a vector feld on IR Osc k M by settng X(t, u) = t + X (t, u). Defnton 7.1 A tme dependent k semspray s a vector feld S = t + S, where S s a tme dependent vector feld on Osc k M verfyng J S= k Γ. 21

22 It s not dffcult to see that J S= k Γ mples S= y x +2y(2) y + +ky y (k 1) (k+1)g (t, x, y,..., y ), where the form y of the last term was chosen for the sake of convenence. Thus we get Proposton 7.2 A tme-dependent k-semspray s of the form (7.2) S = t + y x + 2y(2) + + ky y y (k 1) (k + 1)G (t, x, y,..., y ) y. Proposton 7.3 A vector feld S on E = IR Osc k M s a tmedependent k-semspray f and only f JS = k Γ, J(S) = 0. Let Ψ = dx y dt, Ψ (2) = dy 2y (2) dt,..., Ψ = dy (k 1) ky dt be 1 forms on E. The Propostons 7.2 and 7.3 yeld Proposton 7.4 A vector feld S on IR Osc k M s a tme-dependent k-semspray f and only f (7.3) dt(s) = 1, Ψ (S) = 0,..., Ψ (S) = 0. Let c:t x (t) be a curve on M and c(t)=(t, x (t), dx dt, d k x k! dt ) ts prolongaton to IR k Osck M. We have d 2 x dt 2,..., Proposton 7.5 The curve c s an ntegral curve of a tme dependent d c k semspray S.e. dt = S( c) f and only f the functons t x (t) are solutons of the system of dfferental equatons 1 d k+1 x ( (k + 1)! dt + k+1 G t, dx dt, 1 d 2 x 2! dt,..., 1 2 k! d k x ) = 0. dt k It s known that a tme ndependent k-semspray nduces a nonlnear connecton, [7]. It s our am to show that ths also happens for tme dependent k-semsprays. 22

23 In ths framework, by a nonlnear connecton s meant a dstrbuton u H u E on E = IR Osc k M, supplementary to the vertcal dstrbuton u V u E = ker(π k ),u, that s T u E = H u E V u E, u E. Let v be the vertcal projector on T u E and l h : IR T π(u) M T u E the horzontal lft (v l h = 0, dπ l h = dentty). Settng x = l h ( x ), t = l h( ) t follows that t x and x t are vertcal vector felds. Thus the local bass ( t, t ) of the horzontal x dstrbuton can be put n the form (7.4) t = t N j 0 x = x N j y N j j 0 (2) y j N j (2) y N j (2)j 0 y (2)j N j y j, y j, where the sgns were chosen for convenence. Next, we shall use a nonlnear connecton as gven by ts dual coeffcents (M) to be ntroduced below. Let be N 0 = span( x ), N 1 = J(N 0 ),..., N k 1 = J k 1 (N 0 ) and let us put = J( y x ), y = J 2 ( ),..., (2) x y = J k 1 ( ). We (k 1) x have the decomposton (7.5) T u E = span( t ) N 0 N 1 N k 1 V k, and the local vector felds ( t, x,,..., y y, ) provde (k 1) y a bass of T u E, adapted to ths decomposton. The dual of ths bass s (dt, dx, y,..., y ) wth (7.6) y = dy + Mj y (2) = dy (2) + Mj dx j + M 0 dt, dy j + Mj (2)... y = dy + M j dy (k 1)j + M j (2) 23 dx j + M 0 (2) dt, dy (k 2)j M j dx j + M 0 dt,

24 The functons (M (α) ), α = 1,..., k, n (7.6) are called the dual coeffcents of the gven nonlnear connecton. The dualty of the above bass yelds (7.7) M j M j (2). M j M 0 M 0 =Nj, =N j (2) =Nj =N0 =N0 + N s + N s Mj s, Nj s (k 1) =N0 (2) (2) Ms, M 0 + N s 0 (k 1) + + N s + N s 0 M s Mj s (2) (k 2), N s 0 (2) + N s Mj s (k 1) M (k 2)s + N s 0, Ms (k 1). These formulae can be reversed n order to express the functons (N (α) ) wth the help of the functons (M (α) ). In many cases a nonlnear connecton appears as gven by the dual coeffcents (M (α) ). It follows that for any fxed t the coeffcents (Mj ), α = 1,..., k, (α) defne a nonlnear connecton on Osc k M, whle (M0 ), appear as components of a vector feld on Osc k M. Thus for any fxed t the (α) vector feld S, whch s nothng but a k semspray on Osc k M, determnes the coeffcents (Mj ), α = 1,..., k, accordng to the followng formulae, (α) establshed by R.Mron, [7]: (7.8) Mj = y j G, Mj = 1 (2) 2 ( SM j. Mj = 1 k (S M j (k 1) + M s + M s Nj s ), M s j (k 1) ). 24

25 Now t follows that (7.9) M0 = t G, M0 = (2) t t G,..., M0 = t G }{{ t} k tmes transform as the components of a covector so these functons can be taken as coeffcents for defnng a nonlnear connecton. Thus we get Theorem 7.1 A tme dependent k-semspray S = t + S nduces a nonlnear connecton on E whose dual coeffcents (Mj ) and (M0 ), (α) (α) α = 1,..., k, are gven by (7.8) and (7.9). Fnally we notce the possblty to consder a connecton smlar to the Berwald connecton from Fnsler geometry. Ths s a lnear connecton D n the vertcal bundle over E defned wth respect to the adapted bass as follows: (7.10) D t D x j D y = (α) Lh 0 y, (α)h Lh 0 = y = (α) Lh j y, (α)h Lh j = = 0. y (α) y (β)j y N h 0, y N h j, Ths connecton can be vewed as defned by a k semspray va the equatons (7.11) N h 0 =M0 h = t Gh, N h =M h = y Gh. It can be seen that ths connecton s flat when there exst coordnates on M n whch G (t, x, y,..., y ) = A j(t)y j + G (t, x, y,..., y (k 1) ). A regular tme-dependent Lagrangan s a smooth functon L:IR Osc k M IR, (t, x, y,..., y ) L(t, x, y,..., y ) wth the property that the 25

26 matrx (g j (t, x, y,..., y )) := ( 1 2 of the varatonal problem y t1 t 0 L) has rank n. The solutons y j L dt = 0 are gven by the well-known Euler-Lagrange equatons E (L):= L (7.12) x d dt ( L d2 )+ y dt ( L dk ) 2 ( 1)k y (2) dt ( L k y y = dx dt,..., y = dk x dt. k Besdes E, vewed as an operator on L, one consders the followng operators )=0, (7.13) 1 k E = α=1( 1) α 1 ( ) α d α α! α 1 dt α y, (α) 2 k E = α=2( 1) α 1 ( ) α d α 2 α! α 2 dt α 2 y, (α)... ( k 1 E = ( 1) k 1 1 (k 1)! y d ), (k 1) dt y k E = ( 1) k 1 k! y. These operators were ntroduced( n [7]. Each one appled to L and computed along of the curve σ = t, x (t), 1 dx 1! dt, 1 d 2 x 2! dt,..., 1 d k x ) 2 k! dt, k whch extends to E a soluton t σ(t) = (t, x (t)) of (7.12), provde a d-covector feld. All these are called the Crag Synge covectors. Proposton 7.6 Along of the curve σ, the d-covector k 1 E (L) takes the form (7.14) ( k 1 E (L) = ( 1) k 1 1 (k 1)! L Γ( y (k 1) y L) 2 k! g j d k+1 x ), dt k+1 where Γ = + Γ, Γ = y t x + 2y(2) + + ky y y. 26

27 Theorem 7.2 A regular tme dependent Lagrangan determnes a k- semspray. The Theorem 7.1 and Proposton 7.6 yeld Corolarry 7.1 Every regular tme dependent Lagrangan determnes a nonlnear connecton. In the presence of a nonlnear connecton the space E = IR Osc k M carres besdes the k-tangent structures J and J a new structure whch plays an mportant role n the geometry of E. We defne t as an F(M) lnear map IF χ(e) χ(e) gven on the adapted bass by (7.15) IF ( ) = 0, IF ( t x ) =, IF ( y y ) = IF ( ) = 0, α = 1, 2,..., k 1. y (α) x, By a drect calculaton one fnds (7.15) IF 3 + IF = 0 Let ξ α belongs to N α, α = 1,..., k 1 orthogonal and of lengths 1 wth respect to the metrc G on E gven by: (7.16) G = e 2σ dt dt + g j dx dx + g j y y j + + g j y y j. It follows IF (ξ α ) = 0. Let η α defned by Usng (7.15) one gets η α (X) = G(X, ξ α ), X χ(e). (7.17) IF 2 = Id + η α ξ α + dt t. Ths s a reason to call IF a (k 1) almost contact structure on Osc k M. The ntegrablty of t can be also studed. 27

28 References [1] Anastase, M., Geometry of hgher order sprays. New Fronters n Algebras, Groups and Geometres.(Gr.T. Tsagas, Ed. ), Hadronc Press, Palm Harbor, FL , U.S.A., 1996, p [2] Bucataru, I., Prolongaton of the Remannan, Fnsleran and Lagrangan structures to the hgher order structures. Ph.D. Thess, Al.I.Cuza Unversty, Ias, [3] Crampn, M., Jet Bundle Technques n Analytcal Mechancs, Quadern del Cons. Naz. delle Rcerche, Gruppo Naz. d Fsca Matematca no. 47, [4] Krupka, D. and Janyška, J., Lectures on Dfferental Invarants, Unv. Brno, [5] Krupkova, O., The Geometry of Ordnary Varatonal Equatons. Sprnger, Lecture Notes n Math, no.1678, [6] Léon, M. de and Rodrgues, P., Generalsed Classcal Mechancs and Felds Theory, North Holland, [7] Mron, R., The Geometry of Hgher Order Lagrange Spaces. Applcatons to Mechancs and Physcs. Kluwer Academc Publsher, FTPH no 82, [8] Mron, R., The geometry of Hgher-Order Fnsler spaces. Hadronc Press, [9] Mron, R., Anastase, M. The Geometry of Lagrange Spaces: Theory and Applcatons. FTPH 59, Kluwer Academc Publshers, 1994, 285p. [10] Mron, R., Anastase, M. Vector Bundles and Lagrange Spaces wth Applcatons to relatvty. Geometry Balkan Press, Monographs and Textbooks no.1, 1997, 217p. [11] Mron, R., Sabău, S.V., The Fnsler spaces of order k 1. Algebras, Groups and Geometres - Hadronc Press (to appear). 28

29 Authors address: Al.I.Cuza Unversty, Ias Department of Mathematcs, 6600 Iaş, Romana 29