2-π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3: (045)=111. Victor Blãnuţã, Manuela Gîrţu

Transcription

1 FACTA UNIVERSITATIS Seres: Mechancs Automatc Control and Robotcs Vol. 6 N o pp π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3:53.511(045)=111 Vctor Blãnuţã Manuela Gîrţu Unversty of Bacău Faculty of Scences emal: vblanuta@ub.ro manuelag@ub.ro Abstract. One defnes the noton of -π structure on the phases space of a mechancal system and nvestgate ts ntegrablty. Key words: -π structures -π structures assocated to the Lagrangan mechancal systems. (005) Mathematcs Subect Classfcaton: 53C60 INTRODUCTION The theory of Fnsleran mechancal systems has been realzed by R.Mron and C. Frgou [7]. But the general theory of Lagrangan mechancal systems was realzed by R. Mron [4] and publshed also n the recent boo [6] wrtten by R. Mron and M. Bucãtaru. In the present note we study the Lagrangan case assocated to the phases space of a -π structure usng our deas whch we appled n the Fnsleran and Lagrangan cases [1] [] [9]. 1. LAGRANGIAN MECHANICAL SYSTEMS Consder a Lagrange space L n = (M L(x y)) and a Lagrangan mechancal system Σ = (M L(x y) F (x y)) F (x y) are the external forces. Followng the Mron's theory we tae the evoluton equatons of Σ d L L F y x dt = =. (1.1) x These equatons are equvalent wth the system of dfferental equatons of the second order: Receved February 6 005

2 90 V. BLÃNUŢÃ M. GÎRŢU d x dt 1 + G ( x x ) = F (1.) F g F = (1.3) and s g L L G = y s s. (1.3)' x x The system of dfferental equatons (1.) defnes a dynamcal system of the second order. R.Mron characterses ths system by means of a vector feld on the phases space TM. So he proves the followng theorem: Theorem 1.1. (Mron [4]) 1 0 The followng operator 1 S = y ( G F ) s a vector feld on the manfold TM = TM \{0} whch x 4 depend only of the Lagrangan mechancal system Σ. 0 S s a semspray on the phases space TM. 3 0 The ntegral curves of S are the soluton curves of evoluton equatons (1.). The proof of ths theorem can be found n the papers [4] [8].. CANONICAL NONLINEAR CONNECTIONS OF Σ The geometry of Lagrangan mechancal systems s determnated by the geometry of the par ( TM S ). So the nonlnear connecton N of the mechancal system Σ s gven by the coeffcents 1 1 F N = { G F } = N 4 4 (.1) N G = are the coeffcents of the canoncal nonlnear connecton N of the y assocated Lagrange space L n = (M L(x y)) of the mechancal system Σ. Now we remar that the dstrbuton N of the canoncal nonlnear connecton N gve rse to the drect decomposton: T ( u TM) = Nu Vu (.) δ Let the local adapted bass to the dstrbutons N and V:

3 -π Structures Assocated to the Lagrangan Mechancal Systems 91 δ δ 1 F = N = + x 4 (.3) δ = N x. (.3)' The dual adapted bass (dx δy ) has the 1-forms δy gven by 1 F δ y =δy dx. (.4) 4 The tensor of wea torson of the nonlnear connecton N s t We have by means at (.1) t N N = 1 F F = = 0. (.5) 4 Consequently the nonlnear connecton N of Σ s symmetrc (because t = 0 ). The curvature tensor R of system Σ s as follows δn δn R ( x y) = Thus the followng formula holds: m m 1 F F 1 δ F 1 δ F R ( x y) = R ( x y) + B m B m + (.6) N B = (.7) y. are the coeffcents of the Berwald connectons of mechancal system Σ and curvature tensor of the Cartan nonlnear connecton of Fnsler space F n. Therefore we have. R s the Proposton.1. The nonlnear connecton of the mechancal system s ntegrable f and only f the d- tensor feld R ( xy ) defned by (.6) vanshes.

4 9 V. BLÃNUŢÃ M. GÎRŢU 3. -π STRUCTURES ON THE PHASES SPACE OF Σ Followng our methods from the papers [1][] we defne the -π structure on the manfold TM for the case of the Lagrangan mechancal systems Σ. Defnton 3.1. An almost -π structure F of the mechancal system Σ s a tensor feld F of type (1.1) whch has the followng property: F ( X) = λ X X χ (TM) (3.1) λ s one of the numbers {1-1-}. But the canoncal nonlnear connecton N of the mechancal system Σ determnes by the natural way such a -π structure. δ Indeed we defne F on the adapted bass by δ δ F = λ F =λ. (3.) We remar that the defnton has the geometrcal meanng because ths respect to change of local coordnates of the manfold TM the equatons () are nvarants. F s a tensor of type (1.1) gven by δ F = λ dx +λ δy. (3.3) Usng (3.) we can prove wthout dffcultes (3.1) and (3.). Let us consder also -π structures F defned by the canoncal nonlnear connecton N of the Lagrange space L n. By means of (.) and (.3) we have F λ δ F 1 F F F = F + δy dx dx. (3.4) 4 4 Also we have δ 1 F F = λ + F 4 δ 1 F F =λ + 4 (3.5)

5 -π Structures Assocated to the Lagrangan Mechancal Systems 93 The condton of ntegrablty for the -π structure F s gven by N X Y X Y X Y X Y X Y F ( ) = F [ ] + [ F F ] F[ F ] F[ F ] = 0 (3.6) N F s the Nenhus tensor. Let us calculate the ntegrablty condtons N F (XY) = 0 by consderng the followng values for the par (XY): δ δ δ. h h h At frst tme we have δ δ F = λ F = λ (3.7) δ δ δ and for h h x x δ δ and h one obtans δ δ R h h x x = δ δ δ N = = B h = 0 h h h (3.8) N R h and B h have the expresson (.1)(.6) and (.7). Consequently the followng partal results are vald: F δ δ R δ h h = λ δ F F 0 h = δ δ N δ F F =λ h h δ δ N δ F x x δ δ h F h = λ δ N F h = λ h δ N F F h h =λ δ F F = 0 h F 0 h = δ δ F F = λ R h h F F =λ R h h

6 94 V. BLÃNUŢÃ M. GÎRŢU N δ F F h = λ h Applyng them we conclude that N δ F F h h =λ δ δ N N h δ NF h = λ Rh + h y y δ N δ N h NF R h =λ h h x y x y y δ δ N N h δ NF R h =λ h + h (3.9) N Nh Tang nto account the expresson of the torson tensor t h = the h followng result follows: Theorem 3.1. An almost -π structure F of the mechancal system Σ s ntegrable f and only f the curvature tensor R s gven by: m m 1 F F 1 δ F 1 δ F R ( x y) = R ( x y) + B m B m + (3.10) s equal to zero. 4. THE METRICAL -π STRUCTURE ASSOCIATED TO MECHANICAL SYSTEM Σ Let us consder the N-lft G of the fundamental tensor g of L m G = g dx dx + g δy δ y (4.1) We remar that G s a Remannan structure on TM whch depend only of the mechancal system Σ. By means of (.4) G can be wrtten n the form: 1 F 1 F h G = gdx dx + g δy dx y dx. δ h 4 4 We have (4.) Theorem 4.. The par (F G) s an almost metrc - π structure of the mechancal system Σ.

7 -π Structures Assocated to the Lagrangan Mechancal Systems 95 Proof. It s suffcent to prove the formula G(FX FY) = λ G(X Y) usng the adapted δ bass. We get δ G g = 0 x G δ = G g δ x y = δ δ δ δ G F G g G F =λ =λ =λ δ G F = 0 F δ δ G F G g G F =λ =λ =λ q.e.d y y REFERENCES 1. Blănuţă V. Hassan B.T. Metrcal homogeneous -π structure determned by a Fnsler metrc n tangent bundle. Kluwer Academc Publshers FTPH (003) Blănuţă V. Gîrţu M. Yawata M. Natural n-almost -π structures on Cartan spaces of order (to appear). 3. Mron R. The Geometry of Hgher Order Lagrange Spaces. Applcatons to Mechancs and Physcs. Kluwer Academc Publshers FTPH no. 8 (1997). 4. Mron R. Anastase M. The Geometry of Lagrange Spaces. Theory and Applcatons. Kluwer Academc Publshers FTPH no 59(1994). 5. Mron R. A Lagrangan theory of relatvty (III). Anal.St.Unv "Al.I.Cuza" Iaş XXXII S.1 Math. ff3 (1986) Mron R. Bucătaru M. The Fnsler-Lagrange Geometry. Applcatons to Dynamcal Systems (to appear) 7. Mron R. Frgou C. Fnsleran Mechancal Systems. Algebras Groups and Geometres Nr Mron R HrmucD. Shmada H. Sabău V. The Geometry of Hamlton and Lagrange Spaces Kluwer Academc Publshers FTPH nr. 118 (000). 9. Sandovc A. Blănuţă V. A class of metrcal almost -π structures on tangent bundle. Hadronc Press.Ims.Palm.Harbor F.C (U.S.A.) (00). -π STRUKTURA PRIDRUŽENA LAGRANE-OVOM MEHANIČKOM SISTEMU Vctor Blãnuţã Manuela Gîrţu Defnsano e retane dnamčog sstema -π structure u faznom prostoru mehančog sstema zučavana e negova ntegrablnost. Klučne reč: -π structure -π structure prdružene Lagrange- ovom mehančom sstemu.