Finslerian Nonholonomic Frame For Matsumoto (α,β)-metric

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1 Internatonal Journal of Mathematcs and Statstcs Inventon (IJMSI) E-ISSN: P-ISSN: ǁ Volume 2 ǁ Issue 3 ǁ March 2014 ǁ PP Fnsleran Nonholonomc Frame For Matsumoto (α,)-metrc Mallkarjuna Y. Kumbar, Narasmhamurthy S.K., Kavyashree A.R.. Department of P.G. Studes and Research n Mathematcs,Kuvempu Unversty, Shankaraghatta ,Shmoga, Karnataka, INDIA. ABSTRACT: The whole pont of classcal dynamcs s to show how a system changes n tme; n other words, how does a pont on the confguraton space change once we gve ntal condtons? For a system wth nonholonomc constrants, the state after some tme evoluton depends on the partcular path taken to reach t. In other words, one can return to the orgnal pont n confguraton space but not return to the orgnal state. The man am of ths paper s, frst we determne the Fnsler deformatons to the Matsumoto (α, ) metrc and we construct the Fnsleran nonholonomc frame. Further we obtan the Fnsleran nonholonomc frame for specal (α, )- metrc.e., F = c 1 + α 2, (c 1 0). KEY WORDS: Fnsler space, (α, ) metrcs, GL metrc, Fnsleran nonolonomc frame. I. INTRODUCTION In 1982, P.R. Holland ([1][2]), studes a unfed formalsm that uses a nonholonomc frame on space-tme arsng from consderaton of a charged partcle movng n an external electromagnetc feld. In fact, R.S. Ingarden [3] was frst to pont out that the Lorentz force law can be wrtten n ths case as geodesc equaton on a Fnsler space called Randers space. The author Bel R.G. ([5][6]), have studed a gauge transformaton vewed as a nonholonomc frame on the tangent bundle of a four dmensonal base manfold. The geometry that follows from these consderatons gves a unfed approach to gravtaton and gauge symmetres. The above authors used the common Fnsler dea to study the exstence of a nonholonomc frame on the vertcal subbundle V TM of the tangent bundle of a base manfold M. Consder a j (x), the components of a Remannan metrc on the base manfold M,a(x, y) > 0 two functons on TM and B(x, y) = B (x, y)dx a vertcal 1-form on TM. Then g j (x, y) = a(x, y)a j (x) + b(x, y)b (x)b j (x) (1.1) s a generalzed Lagrange metrc, called the Bel metrc. We say also that the metrc tensor g j s a Bel deformaton of the Remannan metrc a j. It has been studed and appled by R. Mron and R.K. Tavakol n General Relatvty for a(x, y) = exp 2σ(x, y). The case a(x, y) = 1 wth varous choces of b and B was ntroduced and studed by R.G. Bel for constructng a new unfed feld theory [6]. In ths paper, the fundamental tensor feld mght be taught as the result of two Fnsler deformatons. Then we can determne a correspondng frame for each of these two Fnsler deformatons. Consequently, a Fnsleran nonholonomc frame for a Matsumoto (α, ) metrc and specal (α, ) metrc.e., F = c 1 + α 2 (c 1 0) wll appear as a product of two Fnsler frames formerly determned. As f c 1 = 0 then t takes form of Kropna metrc case. 73 P a g e

2 II. PRELIMINARIES An mportant class of Fnsler spaces s the class of Fnsler spaces wth (α, ) metrcs [11]. The frst Fnsler spaces wth (α, ) metrc were ntroduced by the physcst G. Randers n 1940, are called Randers spaces [4]. Recently, R.G. Bel suggested to consder a more general case, the class of Lagrange spaces wth (α, ) metrc, whch was dscussed n [12]. A unfed formalsm whch uses a nonholonomc frame on space tme, a sort of plastc deformaton, arsng from consderaton of a charged partcle movng n an external electromagnetc feld n the background space tme vewed as a straned mechansm studed by P.R. Holland [1][2].If we do not ask for the functon L to be homogeneous of order two wth respect to the (α, ) varables, then we have a Lagrange space wth (α, ) metrc. Next we look for some dfferent Fnsler space wth (α, ) metrcs. Defnton 2.1. : A Fnsler space F n = (M, F(x, y)) s called (α, ) metrc f there exsts a 2-homogeneous functon L of two varables such that the Fnsler metrc F: TM R s gven by, F 2 (x, y) = L(α(x, y), (x, y)), α 2 (x, y) = a j (x)y y j, α s a Ramannan metrc on M, and (x, y) = b (x)y s a 1 form on M. 2 F 2 Consder g j = 1 the fundamental tensor of the Randers space (M, F), takng nto 2 y y j account the homogenety of α and F we have the followng formulae: p = 1 α y α = a j ; p y j = a j p j = α ; y l = 1 L y L = g j ; l y j = g j L = p y j + b ; l = 1 L p ; l l j = p p = 1; l p = α L ; p l = L ; b α p = ; b α l =. L wth respect to these notatons, the metrc tensors a j and g j are related by [13], g j = L α a j + b p j + p b j + b b j α p p j = L α a j p p j + l l j. Theorem 2.1: For a Fnsler space (M, F) consder the matrx wth the entres: Y j = α L (δ j l l j + α L p p j ) (2.4) defned on TM. Then Y j = Y j, j 1,2,., n s an nonholonomc frame. y Theore m 2.2: Wth respect to frame the holonomc components of the Fnsler metrc tensor (a α ) s the Randers metrc (g j ).. e g j = Y α Y j a α. (2.5) Throughout ths secton we shall rse and lower ndces only wth the Remannan metrc a j (x).e., y = a j y j, b = a j b j, and so on. For a Fnsler space wth (α, ) metrc F 2 (x, y) = L α(x, y), (x, y) we have the Fnsler nvarenrs 13. ρ 1 = 1 L ; ρ 2a α 0 = 1 2 L ; ρ = 1 2 L ; ρ 2α α = 1 2 L 1 L : (2.6) 2a 2 α 2 α α, subscrpts, 1,0, 1 gves us the degree of homogenety of these nvarants. For a Fnsler space wth (α, ) metrc 74 P a g e

3 we have: ρ 1 + ρ α 2 = 0. (2.7) wth respect to these notatons, we have that the metrc tensor g j of a Fnsler space wth (α, ) metrc s gven by 13 : g j (x, y) = ρa j (x) + ρ 0 b (x) + ρ 1 b (x)y j + b j (x)y + ρ y y j (2.8) From (2.8) we can see that g j s the result of two Fnsler deformatons: ) a j j = ρa j + 1 (ρ ρ 1 b + ρ y )(ρ 1 b j + ρ y j ) ) j g j = j + 1 (ρ ρ 0 ρ ρ 2 1 )b b j. (2.9) The Fnsleran nonholonomc frame that corresponds to the frst deformaton (2.9) s, accordng to the Theorem n 10, gven by: x j = ρδ j 1 B2 B2 ( ρ ± ρ + ) ρ ρ 1 b + ρ y (ρ 1 b j + y j ) (2.10) B 2 = a j ρ 1 b + ρ y ρ 1 b j + ρ y j = ρ 1 b 2 + ρ 1 ρ. The metrc tensors a j and j are related by: j = X k X l j a kl (2.11) Agan the frame that corresponds to the second deformaton (2.9) s gven by: Y j = δ j 1 1 ± 1 + ρ C 2 C 2 ρ 0 ρ ρ 2 b b j, (2.12) 1 C 2 = j b b j = ρb (ρ ρ 1 b 2 + ρ ) 2. The metrc tensors j and g j are related by the formula: g mn = Y m Y n j j. (2.13) Theorem 2. 3: Let F 2 (x, y) = L α(x, y), (x, y) be the metrc functon of a Fnsler space wth (α, ) metrc for whch the condton (2.7) s true. Then v j = X k k Y j s a Fnsleran nonholonomc frame wth X k and Y k j are gven by (2.10) and (2.12) respectvely. III. FINSLERIAN NONHOLONOMIC FRAME FOR (α, ) metrc In ths secton, we consder two Fnsler spaces wth (α, ) metrcs, such as Matsumoto metrc and specal (α, ) metrc. e., F = c 1 + α 2 then we construct Fnsleran nonholonomc frame for these. 3.1 FINSLERIAN NONHOLONOMIC FRAME FOR MATSUMOTO (α, ) METRIC: In the frst case, for a Fnsler space wth the fundamental functon L = F 2 = (α ) 2, the Fnsler nvarants (2.6 ) are gven by: ρ 1 = α2 (α), ρ (α ) 3 0 = 3α3 ρ (α ) 4 1 = α2 (α 4), ρ (α ) 4 = (4α ). (3.1) (α ) 4 B 2 = α2 (α 4) 2 (b 2 α 2 ) (α ) 8 α4 75 P a g e

4 Usng (3.1) n (2.10) we have, X j = α 2 (α) (α ) 3 δ j α2 b 2 α 2 2 Agan usng (3.1) n (2.12) we have, α 2 (α) (α ) 3 ± α2 ( α 2 +2α b 2 α 3 4b 2 α 2 ) (α ) 4 b y α 2 b j y j α 2. (3.2) Y j = δ j 1 1 ± (α )3 C 2 b b C 2 α 2 j ; (3.3) C 2 = α2 (α 2)b 2 (α ) 3 (α 4) b2 α (α ) 4. α4 Theorem 3.4: Consder a Fnsler space L = (α ) 2, for whch the condton (2.7) s true. Then V j = X k k Y j s a Fnsleran nonholonomc frame wth X k and Y j k are gven by (3.2) and (3.3) respectvely. 3.2 FINSLERIAN NONHOLONOMIC FRAME FOR SPECIAL (α, ) METRIC. e., F = c 1 + α2 : In the second case, for a Fnsler space wth the fundamental functon L = F 2 = c 1 + α22, the Fnsler nvarants (2.6) are gven by: Usng (3.4) n (2.10) we have, ρ 1 = 2(c α 2 ) 2, ρ 0 = c α 4 4, ρ 1 = 4α 2, ρ 3 = 4 B 2 = 16α2 (α 2 b 2 2 ) 6 X j = 2, (3.4) 2(c α 2 ) 2 δ j 6 2 c1 2 +α 2 2 ± 2 4 c1 α α 4 b Agan usng (3.4) n (2.12) we have, 4y 2 α 2 (α 2 b 2 2 ) 4α 2 b 4y j 4a 2 b j (3.5) were Y j = δ j 1 C 2 1 ± 1 C2 4 α 4 c b b j (3.6) C 2 = 2(c α 2 ) 2 + 4(α2 b 2 2 ) P a g e

5 Theorem 3.5: Consder a Fnsler space L= c 1 + α 2 2, for whch the condton (2.7) s true.then V j = X k Y j k s a Fnsleran nonholonomc frame wth X k and Y j k are gven by (3.5)and (3.6) respectvely. IV. CONCLUSION Nonholonomc frame relates a sem-remannan metrc (the Mnkowsk or the Lorentz metrc) wth an nduced Fnsler metrc. Antonell P.L., Bucataru I. ([7][8]), has been determned such a nonholonomc frame for two mportant classes of Fnsler spaces that are dual n the sense of Randers and Kropna spaces [9]. As Randers and Kropna spaces are members of a bgger class of Fnsler spaces, namely the Fnsler spaces wth (α, ) metrc, t appears a natural queston: Does how many Fnsler space wth (α, ) metrcs have such a nonholonomc frame? The answer s yes, there are many Fnsler space wth (α, ) metrcs. In ths work, we consder the two specal Fnsler metrcs and we determne the Fnslerannonholonomc frames. Each of the frames we found here nduces a Fnsler connecton on TM wth torson and no curvature. But, n Fnsler geometry, there are many (α, ) metrcs, n future work we can determne the frames for them also. REFERENCES [1] Holland. P.R., Electromagnetsm, Partcles and Anholonomy. Physcs Letters, 91 (6), (1982). [2] Holland. P.R., Anholonomc deformatons n the ether: a sgnfcance for the electrodynamc potentals. In: Hley, B.J. Peat, F.D. (eds.), Quantum Implcatons. Routledge and Kegan Paul, London and New York, (1987). [3] Ingarden, R.S., On asymmetrc metrc n the four space of general rela-tvty. Tensor N.S., 46, (1987). [4] Randers, G., On asymmetrc metrc n the four space of general rela-tvty. Phys. Rev., 59, (1941). [5] Bel, R.G., Comparson of unfed feld theores. Tensor N.S., 56, (1995). [6] Bel, R.G., Equatons of Moton from Fnsler Geometrc Methods. In: Antonell, P.L. (ed), Fnsleran Geometres. A meetng of mnds. Kluwer Academc Publsher, FTPH, no. 109, (2000). [7] Antonell, P.L. and Bucataru, I., On Hollands frame for Randers space and ts applcatons n Physcs,. preprnt. In: Kozma, L. (ed), Steps n Dfferental Geometry. Proceedngs of the Colloquum on Dfferental Geometry, Debrecen, Hungary, July 25-30, Debrecen: Unv. Debrecen, Insttute of Mathematcs and Informatcs, (2001). [8] Antonell, P.L. and Bucataru, I., Fnsler connectons n an holonomc geometry of a Kropna space,.to appear n Nonlnear Studes. [9] Hrmuc, D. and Shmada, H., On the L-dualty between Lagrange and Hamlton manfolds, Nonlnear World, 3(1996), Mallkarjuna Y. Kumbar, Narasmhamurthy S.K. and Kavyashree A.R. [10] Ioan Bucataru, Radu Mron, Fnsler-Lagrange Geometry. Applcatons to dynamcal systems, CEEX ET 3174/ , CEEX M III 12595/2007 (2007). [11] Matsumoto, M., Theory of Fnsler spaces wth (α, ) metrcs. Rep. Math. Phys. 31(1991), [12] Bucataru I., Nonholonomc frames on Fnsler geometry. Balkan Journal of Geometry and ts Applcatons, 7 (1), (2002). [13] Matsumoto, M., Foundatons of Fnsler Geometry and Specal Fnsler Spaces, Kashesha Press, Otsu,Japan, P a g e