Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), ISSN

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1 Acta Mathematca Academae Paedagogcae Nyíregyházenss 24 (2008), ISSN ON THE RHEONOMIC FINSLERIAN MECHANICAL SYSTEMS CAMELIA FRIGIOIU Abstract. In ths paper t wll be studed the dynamcal system of a rheonomc Fnsleran mechancal system, whose evoluton curves are gven, on the phase space T M R, by Lagrange equatons. Then one can assocate to the consdered mechancal system a vector feld S on T M R, whch s called the canoncal semspray. All geometrc objects of the rheonomc Fnsleran mechancal system one can be derved from S. So we have the fundamental noton as the nonlnear connecton N, the metrcal N-lnear connecton, etc. 1. The geometry of phases space (T M R, π, M) Let be M a smooth C manfold of fnte dmenson n, called the space of confguratons and (T M, π, M) be ts tangent bundle.the 2n-dmensonal manfold T M s called the phases space of M. We denote by (x ), = 1, 2,..., n, the local coordnates on M and by (x, y ) the canoncal local coordnates on T M. We consder the manfold T M R and we shall use the dfferentable structure on T M R as product of the manfold T M fbered over M wth R. The manfold E = T M R s a 2n + 1 dmensonal, real manfold. In a doman of a local chart U (a, b), the pont u = (x, y, t) E have the local coordnates (x, y, t). A change of local coordnates on E has the followng form: (1.1) x = x (x 1, x 2,..., x n ); ỹ = x x j yj ; t = φ(t) ( ) x wth rank x = n and φ := dφ j Mathematcs Subject Classfcaton. 53C60, 53C80. Key words and phrases. Rheonomc Fnsler space, mechancal system, semspray, nonlnear connecton. 65

2 66 CAMELIA FRIGIOIU Of course, we may take on R only one chart, that s t = t or we may consder the affne change of charts on R, that s t = at + b, a 0, a, b R. The natural bass of tangent space T u E at the pont u U (a, b) s gven by ( (1.2) x, y, ). t The transformaton of coordnates (1.1) determnes the transformatons of the natural bass as follows (1.3) x = xj x y = ỹj y ỹ j ; x j + ỹj x ỹ j t = φ t ỹ j y = xj x ; ỹ j x = 2 x j x x h yh. In [4], t s ntroduced on( the manfold E, a vertcal ) dstrbuton V, generated by n + 1 local vector felds y, 1 y,..., 2 y, n t (1.4) V : u E V u T u E. It follows: V u = V n, u V 0, n u E, the lnear space V 0, n s generated by the vector feld t u and t s an 1- dmensonal lnear subspace of the tangent ( space T u E. Also, ) the n-dmensonal lnear space V n, u generated by the felds y, 1 y,..., 2 y n u s a lnear subspace of T u E. On the manfold E there exsts a tangent structure, [1],[4],[10], gven by (1.5) J J : χ(e) χ(e), ( ) x = ( ) ( ) y ; J y = 0; J = 0; = 1, 2,..., n. t J s an ntegrable structure. [1],[4],[10]. On T M R there exsts a globally defned vector feld C = y y. It s the Louvlle vector feld. A semspray on E s a vector feld S χ(e) whch has the property (1.6) JS = C.

3 ON THE RHEONOMIC FINSLERIAN MECHANICAL SYSTEMS 67 Proposton 1 ([8]). a) Locally a semspray S has the form (1.7) S = y x 2G (x, y, t) y G0 (x, y, t) t G (x, y, t) and G 0 (x, y, t) are the coeffcents of S. b) The functons {G (x, y, t), G 0 (x, y, t)} transform under a change of coordnates (1.1), as follows: (1.8) 2 G = 2 x y j Gj ỹ x j yj ; G0 = φ G 0. The ntegrals curves of S are the solutons of the followng system of dfferental equatons dx dτ = dy y (τ); dτ + 2G (x(τ), y(τ), t(τ)) = 0 (1.9) dτ + G0 (x(τ), y(τ), t(τ)) = 0. We shall say that S s a dynamcal system on the phases manfold T M R and the equatons (1.9) are the evoluton equatons of dynamcal system S. When G 0 1, we may take t = τ and the system (1.9) reduces to the second order dfferental equaton (SODE): d 2 x 2 + 2G (x(t), y(t), t) = 0; y = dx. In the followng, we put t = y 0 and we ntroduce the Greek ndces α, β,... rangng on the set {0, 1, 2,..., n}. A non-lnear connecton n E s a smooth dstrbuton: (1.10) N : u E N u T u, E whch s supplementary to the vertcal dstrbuton V: (1.11) T u E = N u V u, u = (x, y, t) E. The local bass adapted to the descomposton (1.11), s ( ) δ (1.12) δx, y α (1.13) δ δx = x N j (x, y, t) y j N 0 (x, y, t) t. (N 0(x, y, t), N j (x, y, t)) are the local coeffcents of the non-lnear connecton N on E. The followng transformaton rule, under (1.1), hold: (1.14) Ñm j x m x = xj x m N m ỹj x ; x j x Ñ 0 j = φ N 0

4 68 CAMELIA FRIGIOIU Conversely, a set of local functons (N 0(x, y, t), N j (x, y, t)) satsfyng (1.14) δ determnes δx, hence t unquely determnes a non-lnear connecton N. The dual bass of (1.12) s (δx, δy, δt) wth (1.15) δx = dx ; δy = dy + N jdx j ; δt = + N 0 dx. 2. Rheonomc Fnsler Spaces. Prelmnares Defnton 1. A rheonomc Fnsler space s a par RF n = (M, F (x, y, t)), for whch F : T M R R satsfy the followng axoms: (1) F s a postve scalar functon on E = T M R; (2) F s a postve 1 homogenous wth respect to the varables y ; (3) F s dfferentable on Ẽ = E \ {0} and contnuous n the ponts (x,0,t); (4) The Hessan of F, wth the entres: (2.1) g j (x, y, t) = 1 2 F 2 2 y y j s postvely defned on T M R. F s called the fundamental functon and g j (x, y, t) s the fundamental tensor of space RF n. Remark 1. (1) F s a scalar functon wth respect to (1.1). (2) g j (x, y, t) s a tensor feld wth respect to (1.1). It s covarant of order 2, symmetrc and nesngular. (3) The par (M, L = F 2 (x, y, t)) s a rheonomc Lagrange space. The geometrcal theory of rheonomc Fnsler space F n can be found n the books [8],[10]. Usng Remark 1 we can use the theory of rheonomc Lagrange spaces [1], [5], [8], for developng the geometry of rheonomc Fnsler spaces. The varatonal problem for the rheonomc Lagrangan L(x, y, t) = F 2 (x, y, t) lead us to the Euler-Lagrange equatons: d 2 x (2.2) 2 + γ jk(x, y, t) dxj dx k + g gh hj t yj = 0; y = dx γjk are the Chrstoffel symbols of the fundamental tensor g j(x, y, t). Theorem 2. The Euler-Lagrange equatons are equvalent wth the Lorentz equatons: (2.3) d 2 x 2 + γ jk(x, y, t) dxj dx k = Fj (x, dx, t)dxj F j (x, y, t) = g h g hj t s the electromagnetc tensor feld determned by the fundamental tensor feld g j.

5 ON THE RHEONOMIC FINSLERIAN MECHANICAL SYSTEMS 69 The system of equatons (2.3) locally determne a dynamcal system on the phase space T M R. We consder the followng functons on T M R 2G (x, y, t) = γ jk(x, y, t)y j y k N 0(x, y, t) = g h g hj t yj Usng the theory of the rheonomc Lagrange spaces t s obtan the canoncal spray S of RF n, as follows (2.4) S = y x (N 0(x, y, t) + N k(x, y, t)y k ) y + t wth (2.5) Nj(x, y, t) = 1 2 y j (γ rs(x, y, t)y r y s ); Nj 0 (x, y, t) = 1 2 g jk t yk. Equatons of evoluton (2.3) are the equatons of the ntegral curves of the semspray S. The semspray S determnes the Cartan non-lnear connecton N, [8], [10], wth the coeffcents (Nj (x, y, t), N j 0 (x, y, t)). Then N s a dfferentable dstrbuton on vertcal dstrbuton V,.e.: (2.6) T u T M R = N(u) V (u), u T M R. (2.7) T M R, supplementary to the Let ( δ δx, y, t ) u be the adapted bass to decomposton (2.6), wth δ δx = x N j (x, y, t) y j N 0 (x, y, t) t. The canoncal metrcal (or Cartan) N connecton CΓ(N) has the coeffcents (Fjk (x, y, t), C jα (x, y, t)) gven by the generalzed Chrstoffel symbols: Fjk = 1 ( δgsk 2 gs δx j + δg js δx k δg ) jk (2.8) δx s, Cjk = 1 ( gsk 2 gs y j + g js y k g ) jk (2.9) y s, (2.10) C j0 = 1 2 gs ( gs0 y j + g sj t g j0 y s ). 3. Rheonomc Fnsleran Mechancal Systems The dynamcal system of a nonconservatve Lagrangan mechancal system can not be correctly defned wthout geometrcal frameworks of the phases manfold T M. The Lagrangan mechancal systems, ther equatons and the assocated dynamcal systems were studed n [1],[2],[4],[5],[7],[3],[10], and the Fnsleran mechancal systems n [6],[9]. The geometrc study of the sclerhonomc

6 70 CAMELIA FRIGIOIU Fnsleran mechancal systems gven by equatons wth the external forces a pror gven was studed n [4], [9]. Defnton 2. A rheonomc Fnsleran mechancal system s a trple Σ = (M, F 2 (x, y, t), σ(x, y, t)) F (x, y, t) s the fundamental functon of a rheonomc Fnsler space RF n = (M, F (x, y, t)) and σ(x, y, t) = σ (x, y, t) y s a vertcal vector feld called the external force of Σ. A rheonomc Lagrange space RL n = (M, L(x, y, t)) reduces to a Fnsler space RF n = (M, F (x, y, t)) f the Lagrangan functon s second order homogeneous wth respect to the velocty coordnates. A frst consequence of the homogenety conon s the energy of a Fnsler space concdes wth the square of the fundamental functon of the space: (3.1) E F 2(x, y, t) = y F 2 and t s verfed the next equalty (3.2) y F 2 = 2F 2 F 2 = F 2 = g j (x, y, t)y y j, df 2 = dx E (F 2 ) F 2 t, E (F 2 ) = F 2 x d ( ) F 2 y. Takng nto account the varatonal problem of the ntegral acton of L(x, y, t) = F 2 (x, y, t) we ntroduce the evoluton equatons of Σ by: The evolutons equatons of the rheonomc Fnsleran mechancal system Σ are the followng Lagrange equatons: ( d L (3.3) y ) L x = σ (x, y, t); y = dx σ (x, y, t) = g j (x, y, t)σ j (x, y, t). One can wrte an equvalent form of Lagrange equatons (3.3) as a system of second order dfferental equatons, gven by (3.4) d 2 x 2 + 2Γ (x, y, t) = 1 2 σ (x, y, t), 2Γ = 2G (x, y, t) + N 0(x, y, t), 2G (x, y, t) = γjk (x, y, t)yj y k and N0(x, y, t) = 1 2 gh. ty h The equatons (3.4) are called equatons of evoluton of the mechancal system Σ. The solutons of these equatons are called evoluton curves of the mechancal system Σ. 2 L

7 ON THE RHEONOMIC FINSLERIAN MECHANICAL SYSTEMS 71 Wth respect to (1.1), the functons Γ : (3.5) 2 Γ (x, y, t) = (2G (x, y, t) 1 2 σ (x, y, t)) + N 0(x, y, t) transform as We can prove: 2 Γ(x, y, t) = 2 Γ j (x, y, t) x x j ỹ x j yj. Theorem 3. a) S gven by: (3.6) S = y x 2 Γ (x, y, t) y + t s a semspray on T M R. b) S s a dynamcal system on T M R dependng only on the rheonomc Fnsleran mechancal system Σ. We call ths semspray the evoluton semspray of the mechancal system Σ. c) The ntegral curves of S are the evoluton curves of Σ gven by (3.3). We can say: The geometry of the rheonomc Fnsleran mechancal system Σ s the geometry of the par (RF n, S), RF n s a rheonomc Fnsler space and S s the evoluton semspray. The varaton of the knetc energy E F 2 along the evoluton curves of the rheonomc mechancal system Σ, s gven by: de F 2 = y σ (x, y, t) F 2 t. The knetc energy of the Fnsler space RF n s not conserved along the evoluton curves of the mechancal system. Now we can consder some geometrc objects determned by the evoluton semspray S and we wll refer to these as the geometrc objects of the mechancal system Σ. a) The non-lnear connecton N of mechancal system Σ has the coeffcents ( N j, N j 0): (3.7) N j = Nj 1 σ 4 y j = Ğ y j ; 0 N j = L ty j, wth Ğ = G (x, y, t) 1 2 σ (x, y, t). N s the canoncal non-lnear connecton of mechancal system Σ. The adapted bass to the dstrbutons N and V = V n V 0 s gven by (3.8) { δ x, y, t }

8 72 CAMELIA FRIGIOIU (3.9) δ δx = x N j(x, y, t) y j N j 0 (x, y, t) t + 1 σ j 4 y y j. The Le brackets of the local vector felds from ths bass are as follows: [ ] δ δx j, δ δx h = R jh y + R jh 0 t ; (3.10) [ ] δ δx j, = N j t t [ δ δy j, (3.11) R jh = y + N j 0 t ] y h = [ δ t ; δx j, ] y h = N j y h [ y j, ] [ = t t, ] = 0, t δ N j δx h δ N h δx j ; R δ N jh 0 j 0 = δx h The dual bass {dx, δy, δt} s gven by δ N 0 h δx j. (3.12) δy = dy + N jdx j 1 σ 4 y j dxj ; δt = + N 0 dx and we have d(dx ) = 0; y + N 0 j y h t ; (3.13) d( δy ) = 1 2 R jhdx h dx j + N j y h δy h dx j + N j t δt dx j ; d( δt) = 1 2 R 0 jhdx h dx j + N 0 j y h δy h dx j + N 0 j t δt dx j. We can prove the followng theorem Theorem 4. a)the canoncal non-lnear connecton N s ntegrable f and only f R jh = 0 and R 0 jh = 0. b)the canoncal metrcal N connecton of the rheonomc mechancal system Σ, CΓ( N), has the coeffcents gven by the generalzed Chrstoffel symbols: ( L jk = 1 δghk 2 gh δx j + δg jh δx k δg ) jk δx h (3.14) C jk = 1 2 gh ( ghk y j + g jh y k C j0 = 1 2 gh g jh t. g ) jk y h

9 ON THE RHEONOMIC FINSLERIAN MECHANICAL SYSTEMS 73 c)the h and v covarant dervaton wth respect CΓ( N) of Louvlle vector feld C = y lead us to ntroduce followngs h and v deflecton tensors of CΓ( N): y (3.15) D j = y j ; d α = y α. We may also ntroduce the h and v electromagnetc tensors (3.16) Fj = 1 ( D j D 2 ) j ; fj = 1 ( dj 2 d ) j D j = g r Dr j, d α = g r dr α. Let us consder the helcodal tensor of Σ: (3.17) σ j = 1 2 ( ) σ y j σj y. We obtan f j = 0 and the followng theorem Theorem 5. Between the h electromagnetc tensor of the rheonomc Fnsleran mechancal system F j, the h electromagnetc tensor of the rheonomc Fnsler space F j and the helcodal tensor σ j of Σ the followng relaton holds: (3.18) Fj = F j σ j. The proof s not dffcult. References [1] M. Anastase. The geometry of tme-dependent Lagrangans. Math. Comput. Modellng, 20(4-5):67 81, [2] M. Crampn and F. A. E. Pran. Applcable dfferental geometry, volume 59 of London Mathematcal Socety Lecture Note Seres. Cambrdge Unversty Press, Cambrdge, [3] 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, [4] C. Frgou. On the geometrzaton of the rheonomc Fnsleran mechancal systems. Analele Ştnţfce ale Unverstăt Al. I. Cuza dn Iaş, LIII: , [5] J. Kern. Lagrange geometry. Arch. Math. (Basel), 25: , [6] J. Klen. Espaces varatonnels et mécanque. Ann. Inst. Fourer (Grenoble), 12:1 124, [7] O. Krupková. The geometry of ordnary varatonal equatons, volume 1678 of Lecture Notes n Mathematcs. Sprnger-Verlag, Berln, [8] R. Mron, M. Anastase, and I. Bucataru. The geometry of Lagrange spaces. In Handbook of Fnsler geometry. Vol. 1, 2, pages Kluwer Acad. Publ., Dordrecht, [9] R. Mron and C. Frgou. Fnsleran mechancal systems. Algebras Groups Geom., 22(2): , [10] 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, 2001.

10 74 CAMELIA FRIGIOIU Camela Frgou, Unversty Dunarea de Jos, Faculty of Scences, Department of Mathematcs, Domneasca 47, Galat, Romana E-mal address: