# Lagrange Spaces with(γ,β)-Metric

Authors: Suresh K. Shukla; P. N. Pandey
Journal: Geometry
 (2013)

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## Abstract

We study Lagrange spaces with(γ,β)-metric, where γ is a cubic metric and β is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a (γ,β)-metric. Several other properties of such space are also discussed.


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## Body

## 1. Introduction

Finsler spaces endowed with(α,β)-metric were studied by several geometers such as Matsumoto [1, 2] and Kitayama et al. [3], and various important applications of such spaces have been observed in physics and relativity theory (cf. [4, 5]). The notion of (α,β)-metric was taken to a more general space called Lagrange space and the study was performed by the authors such as Miron [6], Nicolaescu [7, 8], and the present authors [9]. An n-dimensional Lagrange space Ln=(M,L(x,y)) is said to be endowed with (α,β)-metric if Lagrangian L(x,y) is function of α(x,y) and β(x,y) only, that is, L(x,y)=L¯(α,β),α(x,y)=aij(x)yiyj being a Riemannian metric and β(x,y)=bi(x)yi a 1-form. Recently, Pandey and Chaubey [10] discussed Lagrange spaces with (γ,β)-metric and obtained several results. They called a Lagrange space Ln=(M,L(x,y)) to be endowed with (γ,β)-metric if Lagrangian L(x,y) is a function of γ(x,y) and β(x,y) only, that is, L(x,y)=L¯(γ,β) where γ(x,y) is a cubic metric and β(x,y) is a 1-form, that is, γ3=aijk(x)yiyjyk and β(x,y)=bi(x)yi. The paper [10] by Pandey and Chaubey is full of flaws and needs to be rectified. The aim of the present paper is to develop a revised and modified theory of Lagrange spaces with (γ,β)-metric.The paper is organized as follows. In Section2, we define a Lagrange space and discuss some preliminary results required for the discussion of the following sections. It includes the notion of a Lagrange space with (γ,β)-metric. In Section 3, we discuss some properties of a Lagrange space with (γ,β)-metric and obtain the expression for the fundamental metric tensor gij and its inverse gij. In Section 4, we consider the variational problem in Lagrange spaces with (γ,β)-metric and obtain various forms of Euler-Lagrange equations. Section 5 deals with the semispray of a Lagrange space with (γ,β)-metric. In Section 6, we obtain the coefficients of nonlinear connection in a Lagrange space endowed with (γ,β)-metric. Section 7 consists of concluding remarks on the results obtained in the paper.


## 2. Preliminaries

LetM be an n-dimensional smooth manifold and let TM be its tangent bundle. Let (xi) and (xi,yi) be local coordinates on M and TM, respectively. A Lagrangian is a function L:TM→ℝ which is a smooth function on TM~=TM∖{0} and continuous on the null section. The Lagrangian L(x,y) is said to be regular if rank  (gij(x,y))=n, where
(1)gij(x,y)=12∂˙i∂˙jL,∂˙i≡∂∂yi,
is a covariant symmetric tensor called the fundamental metric tensor of the Lagrangian L(x,y). A Lagrange space is a pair Ln=(M,L(x,y)),L(x,y) being a regular Lagrangian whose metric tensor gij has constant signature on TM~.The integral of action of the LagrangianL(x,y) along a smooth curve c:[0,1]→M leads to the following Euler-Lagrange equations:
(2)EiL≡∂L∂xi-ddt(∂L∂yi)=0,yi=dxidt.The coefficients of the semisprayS of a Lagrange space Ln=(M,L(x,y)) are given by
(3)Gi(x,y)=14gih(yk∂˙h∂kL-∂hL),∂k≡∂∂xk.
The semispray S is called a canonical semispray as its coefficients depend on L(x,y) only.The coefficients of canonical nonlinear connectionN(Nji(x,y)) of a Lagrange space Ln=(M,L(x,y)) are given by
(4)Nji=∂˙jGi.In the present paper, we study a Lagrange space whose LagrangianL is a function of γ(x,y) and β(x,y) only, where
(5)γ3=aijk(x)yiyjyk,(6)β(x,y)=bi(x)yi.
Let us denote this Lagrangian by L¯. Thus
(7)L(x,y)=L¯(γ,β).The spaceLn=(M,L(x,y)) is called a Lagrange space with(γ,β)-metric (cf. [10]). Following are some examples of regular Lagrangians with (γ,β)-metric:
(8)(i)L¯(γ,β)=γ3+β,(ii)L¯(γ,β)=γ3+aγ+bβ+cβ2,a,b,c∈ℝ,(iii)L¯(γ,β)=(γ+β)2.
The Lagrange space determined by the Lagrangian in (8)(iii) is reducible to a Finsler space whereas those determined by the Lagrangians in (8)(i) and (8)(ii) are not so.For basic notations and terminology related to a Lagrange space, we refer the reader to [11, 12].


## 3. Fundamental Metric Tensor ofLn=(M,L(x,y))

If we differentiate (5) partially with respect to yj and use the symmetry of arst in its indices, we obtain
(9)∂˙jγ=γ-2aj(x,y),
where aj(x,y)=ajst(x)ysyt.Again differentiating (9) partially with respect to yh, using symmetry of arst(x) in its indices and simplifying, we find
(10)∂˙j∂˙hγ=2γ-2ajh(x,y)-2γ-5ajah,
where ajh=ajhsys.Differentiating (6) partially with respect to yj, we have
(11)∂˙jβ=bj(x).
Differentiating (11) partially with respect to yh, we get
(12)∂˙j∂˙hβ=0.
Thus, we have the following.Proposition 1.
In a Lagrange spaceLn with (γ,β)-metric, the following hold good:
(13)∂˙jγ=γ-2aj(x,y),∂˙j∂˙hγ=2γ-2ajh(x,y)-2γ-5ajah,∂˙jβ=bj(x),∂˙j∂˙hβ=0,
where(14)aj(x,y)=ajst(x)ysyt,ajh(x,y)=ajhsys.The moments of LagrangianL(x,y) are given by
(15)pi:=12∂˙iL.
In our case, the Lagrangian L(x,y) is a function of γ and β only (vide (7)). Therefore, we have
(16)pi=12(L¯γ∂˙iγ+L¯β∂˙iβ),
where L¯γ=∂L¯/∂γ,L¯β=∂L¯/∂β.Using (9) and (11) in (16), we obtain
(17)pi=12(γ-2L¯γai+L¯βbi).
Thus, we have the following.Theorem 2.
In a Lagrange spaceLn with (γ,β)-metric, the moments of Lagrangian L(x,y) are given by
(18)pi=ρai+ρ1bi,
where
(19)ρ=12γ-2L¯γ,(20)ρ1=12L¯β.Remarks 3.
The scalarsρ and ρ1 appearing in Theorem 2 are called the principal invariants of the space Ln.
Differentiating (19) and (20), partially with respect to yj and simplifying, we, respectively, have
(21)∂˙jρ=12γ-4(L¯γγ-2γ-1L¯γ)aj+12γ-2L¯γβbj,(22)∂˙jρ1=12γ-2L¯βγaj+12L¯ββbj,
where
(23)L¯γγ=∂2L¯∂γ2,L¯γβ=∂2L¯∂γ∂β=∂2L¯∂β∂γ=L¯βγ,L¯ββ=∂2L¯∂β2.
Thus, we have the following.Proposition 4.
The derivatives of the principal invariants of a Lagrange spaceLn with (γ,β)-metric are given by
(24)∂˙jρ=ρ-2aj+ρ-1bj,∂˙jρ1=ρ-1aj+ρ0bj,
with
(25)ρ-2=12γ-4(L¯γγ-2γ-1L¯γ),ρ-1=12γ-2L¯γβ,ρ0=12L¯ββ.The energy of LagrangianL(x,y) is defined as
(26)EL:=yi∂˙iL-L.
Using (7) in (26), we have
(27)EL¯=yi(L¯γ∂˙iγ+L¯β∂˙iβ)-L¯.
Since γ and β are positively homogeneous of degree one in yi, by virtue of Euler’s theorem on homogeneous functions, we have
(28)yi∂˙iγ=γ,yi∂˙iβ=β.
In view of (28), (27) takes the form
(29)EL¯=γL¯γ+βL¯β-L¯.
Thus, we have the following.Theorem 5.
In a Lagrange space with(γ,β)-metric, the energy of the Lagrangian L(x,y) is given by (29).Now, we find out expression for the fundamental metric tensorgij(x,y) of a Lagrange space with (γ,β)-metric. Using (7) in (1), we have
(30)gij=12[(L¯γγ∂˙iγ+L¯γβ∂˙iβ)∂˙jγ+L¯γ∂˙i∂˙jγ211+(L¯βγ∂˙iγ+L¯ββ∂˙iβ)∂˙jβ+L¯β∂˙i∂˙jβ].
In view of Proposition 1, (30) takes the form
(31)gij(x,y)=2ρaij+ρ-2aiaj+ρ-1(aibj+ajbi)+ρ0bibj.
Equation (31) can be written as
(32)gij(x,y)=2ρaij+cicj,
where
(33)ci=q-1ai+q0bi
and q-1,q0 satisfy
(34)(a)q0q-1=ρ-1,(b)(q-1)2=ρ-2,(c)q02=ρ0.
Thus, we have the following.Theorem 6.
The fundamental metric tensor of a Lagrange space with(γ,β)-metric is given by (32).The following result gives the expression for the inverse ofgij.Theorem 7.
The inversegij of the fundamental metric tensor gij of a Lagrange space with (γ,β)-metric is given by
(35)gij=12ρ(aij-12ρ+c2cicj),
where
(36)(a)ci=aircr,(b)c2=aijcicj.Proof.
Utilizing Lemma6.1.2.1 of [11] for the nonsingular matrix gij given by (32) we have the result.Remarks 8.
In [10], Pandey and Chaubey obtained the following:
(37)∂˙iγ=γ-1yi,∂˙i∂˙jγ=2γ-1aij(x,y)-γ-3yiyj,yi=aij(x,y)yj,(38)pi=ρyi+ρ1bi,
where ρ and ρ1 are, respectively, given by (19) and (20),
(39)∂˙iρ=ρ-2yi+ρ-1bi,∂˙iρ1=ρ-1yi+ρ0bi,
where ρ-2=(1/2)γ-2(L¯γγ-γ-1L¯γ),ρ-1=(1/2)γ-1L¯γβ and ρ0 is the same as given in (25),
(40)EL¯=γ-1L¯γ+βL¯β-L¯,(41)gij(x,y)=2ρaij+cicj,
where ci=q-1yi+q0bi and q-1,q0 satisfy q0q-1=ρ-1,(q-1)2=ρ-2,q02=ρ0, and
(42)gij=12ρaij-11+c2cicj,
where ci=(1/2)ρ-1aircr and c2=cici.
In view of corresponding results obtained by us, these results are erroneous.


## 4. Euler-Lagrange Equations

Using (7) in (2), we obtain
(43)Ei(L¯)≡∂L¯∂xi-ddt(∂L¯∂yi)=0,yi=dxidt.
For the Lagrangian L¯ given by (7), we have
(44)ddt(∂L¯∂yi)=(L¯γγdγdt+L¯γβdβdt)∂γ∂yi+(L¯βγdγdt+L¯ββdβdt)∂β∂yi+L¯γddt(∂γ∂yi)+L¯βddt(∂β∂yi).
In view of ∂iL¯=L¯γ∂iγ+L¯β∂iβ and (44), (43) takes the form
(45)Ei(L¯)≡L¯γEi(γ)+L¯βEi(β)-(L¯γγdγdt+L¯γβdβdt)∂γ∂yi-(L¯βγdγdt+L¯ββdβdt)∂β∂yi.
Since
(46)Ei(γ3)=3γ2Ei(γ)-3∂γ∂yidγ2dt,
we get
(47)Ei(γ)=13γ-2Ei(γ3)+γ-2∂γ∂yidγ2dt.
From Ei(β)=(∂β/∂xi)-(d/dt)(∂β/∂yi), we have
(48)Ei(β)=2Firyr,yr=dxrdt,
where
(49)Fir=12(∂br∂xi-∂bi∂xr)
is the electromagnetic tensor field of the potentials bi.Using (47) and (48) in (45), we obtain
(50)Ei(L¯)=23(12γ-2L¯γ)Ei(γ3)+2(12γ-2L¯γ)∂γ∂yidγ2dt+4(12L¯β)Firyr-∂γ∂yi(L¯γγdγdt+L¯γβdβdt)-∂β∂yi(L¯βγdγdt+L¯ββdβdt).
Thus, we have the following.Theorem 9.
The Euler-Lagrange equations of a Lagrange space with(γ,β)-metric are of the following form:
(51)Ei(L¯)≡23ρEi(γ3)+2ρ∂γ∂yidγ2dt+4ρ1Firyr-∂γ∂yi(L¯γγdγdt+L¯γβdβdt)-∂β∂yi(L¯βγdγdt+L¯ββdβdt)=0,yi=dxidt.For the natural parametrization of the curvec:t∈[0,1]↦xi(t)∈M with respect to the cubic metric aijk(x),γ(x,dx/dt)=1. Thus, we have the following.Theorem 10.
In the natural parametrization, the Euler-Lagrange equations of a Lagrange space with(γ,β)-metric are
(52)Ei(L¯)≡23ρEi(γ3)+4ρ1Firyr-∂γ∂yiL¯γβdβds-∂β∂yiL¯ββdβds=0.Ifβ is constant on the integral curve c of the Euler-Lagrange equations with natural parametrization, then (52) takes the form
(53)Ei(L¯)≡23ρEi(γ3)+4ρ1Firyr=0.
Thus, we have the following.Theorem 11.
Ifβ is constant along the integral curve of the Euler-Lagrange equations with natural parametrization, then the Euler-Lagrange equations of the Lagrange space with (γ,β)-metric are given by (53).Remarks 12.
Pandey and Chaubey [10] obtained the following form of Euler-Lagrange equations:
(54)Ei(L¯)≡23γρEi(γ3)+23γρ∂γ∂yidγ2dt+2ρ1Firyr-∂γ∂yi(L¯γγdγdt+L¯γβdβdt)-∂β∂yi(L¯βγdγdt+L¯ββdβdt)=0.
In case of natural parametrization, their result is(55)23γρEi(γ3)+23γρ∂γ∂yidγ2dt+2ρ1Firyr-∂β∂yi(L¯βγdγdt+L¯ββdβdt)=0.
Ifβ is constant along the integral curve c of the Euler-Lagrange equations (with natural parametrization), the Euler-Lagrange equations obtained by Pandey and Chaubey [10] are given by
(56)23γρEi(γ3)+23γρ∂γ∂yidγ2dt+2ρ1Firyr=0.
In view of Theorem9, Theorem 10, and Theorem 11, all the above discussed results of Pandey and Chaubey [10] are erroneous.


## 5. Canonical Semispray

In this section, we obtain the coefficients of the canonical semispray of a Lagrange space with(γ,β)-metric.Using (7) in (3), we obtain
(57)Gi(x,y)=14gih(yk∂˙h∂kL¯-∂hL¯).
Since γ3=aijk(x)yiyjyk and β=bi(x)yi, we have
(58)∂hγ=Ahγ-2,∂hβ=Bh,
where
(59)Ah=13(∂haijk)yiyjyk,Bh=(∂hbi)yi.
Using (58), (19), and (20) in ∂kL¯=L¯γ∂kγ+L¯β∂kβ, we get
(60)∂kL¯=2ρAk+2ρ1Bk.
Differentiating (60) partially with respect to yh and simplifying, we have
(61)∂˙h∂kL¯=2(ρ-2ah+ρ-1bh)Ak+2ρAkh+2(ρ-1ah+ρ0bh)Bk+2ρ1bkh,
where
(62)(a)Akh=∂˙hAk,(b)bkh=∂˙hBk.
Using (60) and (61) in (57), we obtain
(63)Gi=12gih[(ρ-2A0+ρ-1B0)ah+(ρ-1A0+ρ0B0)bh12gih2+ρA0h+ρ1b0h-(ρAh+ρ1Bh)],
where
(64)(i)A0=Ak(x,y)yk,(ii)B0=Bk(x,y)yk,(iii)A0h=Akh(x,y)yk,(iv)b0h=bkh(x,y)yk.
Thus, we have the following.Theorem 13.
The local coefficients of canonical semispray of a Lagrange space with(γ,β)-metric are given by (63).


## 6. Canonical Nonlinear Connection

In this section, we obtain the local coefficients of the canonical nonlinear connection of a Lagrange space with(γ,β)-metric.Partial differentiation ofgihgis=δsh, with respect to yj, yields
(65)∂˙jgih=-2grhCrji.
If we partially differentiate the quantities appearing in (25) and (64) with respect to yj, we find the following quantities:
(66)∂˙jρ-2=μ-3aj+μ-2bj,∂˙jρ-1=μ-2aj+μ-1bj,∂˙jρ0=μ-1aj+μ0bj,∂˙jA0=Aj+A0j,∂˙jB0=𝔖(sj){∂sbj}ys,∂˙jA0h=2A0hj+Ajh,∂˙jb0h=bjh,
where
(67)μ-3=12γ-6[L¯γγγ-6γ-1L¯γγ+10γ-2L¯γ],μ-2=12γ-4[L¯γγβ-2γ-1L¯γβ],μ-1=12γ-2L¯γγβ,μ0=12L¯βββ,A0hj=Arhjyr,Arhj=∂rahj
and 𝔖(sj) denotes interchanging indices s andj and taking sum. Also, we have
(68)∂˙jah=2ajh.
Now, applying (63) in (4), we get
(69)Nji=12(∂˙jgih)((ρ-2A0+ρ-1B0)ah+(ρ-1A0+ρ0B0)bh(∂˙jgih)21+ρA0h+ρ1b0h-(ρAh+ρ1Bh))+12gih[((∂˙jρ-2)A0+ρ-2(∂˙jA0)+(∂˙jρ-1)B0(∂˙jgih)22+ρ-1(∂˙jB0))ah(∂˙jgih)2+(ρ-2A0+ρ-1B0)∂˙jah(∂˙jgih)2+((∂˙jρ-1)A0+ρ-1(∂˙jA0)(∂˙jgih)2221+(∂˙jρ0)B0+ρ0(∂˙jB0))bh(∂˙jgih)2+(∂˙jρ)A0h+(∂˙jρ1)b0h+ρ∂˙jA0h+ρ1∂˙jb0h(∂˙jgih)2-((∂˙jρ)Ah+ρ(∂˙jAh)(∂˙jgih)222+(∂˙jρ1)Bh+ρ1(∂˙jBh))].
Using (24), (62)–(64), (65), (66) and (68) in (69) and simplifying, we obtain
(70)Nji=-2CrjiGr+12gih×[ρ-2((Aj+A0j)ah+(A0h-Ah)aj+2A0ajh)11+ρ-1((Aj+A0j)bh+(A0h-Ah)bj+ajb0h11+ρ-12+𝔖(sj){∂sbj}ysah+2B0ajh-ajBh)11+ρ0(𝔖(sj){∂sbj}ysbh-bjBh+bjb0h)11+ρ1𝔚(jh){bjh}+ρ(2A0hj+Ajh-Ahj)11+μ-3A0ajah+μ-2(A0bjah+aj(B0ah+A0bh))11+μ-1(bj(B0ah+A0bh)+ajB0bh)11+μ0B0bjbh],
where 𝔐(jh){bjh}=bjh-bhj.Thus, we have the following.Theorem 14.
The local coefficients of the canonical nonlinear connection of a Lagrange space with(γ,β)-metric are given by (70).


## 7. Conclusions

We have developed the theory of Lagrange spaces with(γ,β)-metric. The paper presents a significant generalization of the theory of Lagrange spaces with (α,β)-metrics (cf. [6–8]). The expressions for the geometric objects obtained in the paper may be useful in further work on the spaces under consideration. The importance of the results lies in the study of canonical metrical d-connection, curvatures, and torsions in such spaces. The expressions for canonical semispray and nonlinear connection, obtained, respectively, in Section 5 and Section 6, may be applicable in geodesic correspondences between two Lagrange spaces with different (γ,β)-metrics on the same underlying manifold. It is a matter of later investigations to look into the aforesaid applications of the results obtained in the paper.



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