Lagrange Spaces with (𝛾,𝛽)-Metric

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.

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 𝐿^𝑛=(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)=𝑎_𝑖𝑗(𝑥)𝑦^𝑖𝑦^𝑗 being a Riemannian metric and 𝛽(𝑥,𝑦)=𝑏_𝑖(𝑥)𝑦^𝑖 a 1-form. Recently, Pandey and Chaubey [10] discussed Lagrange spaces with (𝛾,𝛽)-metric and obtained several results. They called a Lagrange space 𝐿^𝑛=(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=𝑎_𝑖𝑗𝑘(𝑥)𝑦^𝑖𝑦^𝑗𝑦^𝑘 and 𝛽(𝑥,𝑦)=𝑏_𝑖(𝑥)𝑦^𝑖. 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 Section 2, 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 𝑔_𝑖𝑗 and its inverse 𝑔^𝑖𝑗. 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

Let 𝑀 be an 𝑛-dimensional smooth manifold and let 𝑇𝑀 be its tangent bundle. Let (𝑥^𝑖) and (𝑥^𝑖,𝑦^𝑖) 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  (𝑔_𝑖𝑗(𝑥,𝑦))=𝑛, where 𝑔_𝑖𝑗=1𝑥,𝑦2𝜕̇_𝑖𝜕̇_𝑗𝐿,𝜕̇_𝑖≡𝜕𝜕𝑦^𝑖, (1) is a covariant symmetric tensor called the fundamental metric tensor of the Lagrangian 𝐿(𝑥,𝑦). A Lagrange space is a pair 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦)),𝐿(𝑥,𝑦) being a regular Lagrangian whose metric tensor 𝑔_𝑖𝑗 has constant signature on 𝑇𝑀.

The integral of action of the Lagrangian 𝐿(𝑥,𝑦) along a smooth curve 𝑐∶[0,1]→𝑀 leads to the following Euler-Lagrange equations: 𝐸_𝑖𝐿≡𝜕𝐿𝜕𝑥^𝑖−𝑑𝑑𝑡𝜕𝐿𝜕𝑦^𝑖=0,𝑦^𝑖=𝑑𝑥^𝑖.𝑑𝑡 (2)

The coefficients of the semispray 𝑆 of a Lagrange space 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦)) are given by 𝐺^𝑖=1𝑥,𝑦4𝑔^𝑖ℎ𝑦^𝑘𝜕̇_ℎ𝜕_𝑘𝐿−𝜕_ℎ𝐿,𝜕_𝑘≡𝜕𝜕𝑥^𝑘. (3) The semispray 𝑆 is called a canonical semispray as its coefficients depend on 𝐿(𝑥,𝑦) only.

The coefficients of canonical nonlinear connection 𝑁(𝑁^𝑖_𝑗(𝑥,𝑦)) of a Lagrange space 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦)) are given by 𝑁^𝑖_𝑗=𝜕̇_𝑗𝐺^𝑖. (4)

In the present paper, we study a Lagrange space whose Lagrangian 𝐿 is a function of 𝛾(𝑥,𝑦) and 𝛽(𝑥,𝑦) only, where 𝛾³=𝑎_𝑖𝑗𝑘(𝑥)𝑦^𝑖𝑦^𝑗𝑦^𝑘,𝛽𝑥,𝑦=𝑏_𝑖(𝑥)𝑦^𝑖. (5)(6) Let us denote this Lagrangian by 𝐿. Thus 𝐿𝑥,𝑦=𝐿𝛾,𝛽. (7)

The space 𝐿^𝑛=(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: (i)𝐿𝛾,𝛽=𝛾³(+𝛽,ii)𝐿𝛾,𝛽=𝛾³+𝑎𝛾+𝑏𝛽+𝑐𝛽²(,𝑎,𝑏,𝑐∈ℝ,iii)𝐿𝛾,𝛽=𝛾+𝛽². (8) 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 of 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦))

If we differentiate (5) partially with respect to 𝑦^𝑗 and use the symmetry of 𝑎_𝑟𝑠𝑡 in its indices, we obtain 𝜕̇_𝑗𝛾=𝛾⁻²𝑎_𝑗𝑥,𝑦, (9) where 𝑎_𝑗(𝑥,𝑦)=𝑎_𝑗𝑠𝑡(𝑥)𝑦^𝑠𝑦^𝑡.

Again differentiating (9) partially with respect to 𝑦^ℎ, using symmetry of 𝑎_𝑟𝑠𝑡(𝑥) in its indices and simplifying, we find 𝜕̇_𝑗𝜕̇_ℎ𝛾=2𝛾⁻²𝑎_𝑗ℎ𝑥,𝑦−2𝛾⁻⁵𝑎_𝑗𝑎_ℎ, (10) where 𝑎_𝑗ℎ=𝑎_𝑗ℎ𝑠𝑦^𝑠.

Differentiating (6) partially with respect to 𝑦^𝑗, we have 𝜕̇_𝑗𝛽=𝑏_𝑗(𝑥). (11) Differentiating (11) partially with respect to 𝑦^ℎ, we get 𝜕̇_𝑗𝜕̇_ℎ𝛽=0. (12) Thus, we have the following.

Proposition 1. In a Lagrange space 𝐿^𝑛 with (𝛾,𝛽)-metric, the following hold good: 𝜕̇_𝑗𝛾=𝛾⁻²𝑎_𝑗,𝜕̇𝑥,𝑦_𝑗𝜕̇_ℎ𝛾=2𝛾⁻²𝑎_𝑗ℎ𝑥,𝑦−2𝛾⁻⁵𝑎_𝑗𝑎_ℎ,𝜕̇_𝑗𝛽=𝑏_𝑗(𝑥),𝜕̇_𝑗𝜕̇_ℎ𝛽=0, (13)
where 𝑎_𝑗𝑥,𝑦=𝑎_𝑗𝑠𝑡(𝑥)𝑦^𝑠𝑦^𝑡,𝑎_𝑗ℎ𝑥,𝑦=𝑎_𝑗ℎ𝑠𝑦^𝑠. (14)

The moments of Lagrangian 𝐿(𝑥,𝑦) are given by 𝑝_𝑖1∶=2𝜕̇_𝑖𝐿. (15) In our case, the Lagrangian 𝐿(𝑥,𝑦) is a function of 𝛾 and 𝛽 only (vide (7)). Therefore, we have 𝑝_𝑖=12𝐿_𝛾𝜕̇_𝑖𝛾+𝐿_𝛽𝜕̇_𝑖𝛽, (16) where 𝐿_𝛾=𝜕𝐿/𝜕𝛾,𝐿_𝛽=𝜕𝐿/𝜕𝛽.

Using (9) and (11) in (16), we obtain 𝑝_𝑖=12𝛾⁻²𝐿_𝛾𝑎_𝑖+𝐿_𝛽𝑏_𝑖. (17) Thus, we have the following.

Theorem 2. In a Lagrange space 𝐿^𝑛 with (𝛾,𝛽)-metric, the moments of Lagrangian 𝐿(𝑥,𝑦) are given by 𝑝_𝑖=𝜌𝑎_𝑖+𝜌₁𝑏_𝑖, (18) where 1𝜌=2𝛾⁻²𝐿_𝛾,𝜌₁=12𝐿_𝛽. (19)(20)

Remarks 3. The scalars 𝜌 and 𝜌₁ appearing in Theorem 2 are called the principal invariants of the space 𝐿^𝑛.
Differentiating (19) and (20), partially with respect to 𝑦^𝑗 and simplifying, we, respectively, have 𝜕̇_𝑗𝜌=12𝛾⁻⁴𝐿_𝛾𝛾−2𝛾⁻¹𝐿_𝛾𝑎_𝑗+12𝛾⁻²𝐿_𝛾𝛽𝑏_𝑗,𝜕̇_𝑗𝜌₁=12𝛾⁻²𝐿_𝛽𝛾𝑎_𝑗+12𝐿_𝛽𝛽𝑏_𝑗, (21)(22) where 𝐿_𝛾𝛾=𝜕²𝐿𝜕𝛾²,𝐿_𝛾𝛽=𝜕²𝐿=𝜕𝜕𝛾𝜕𝛽²𝐿=𝜕𝛽𝜕𝛾𝐿_𝛽𝛾,𝐿_𝛽𝛽=𝜕²𝐿𝜕𝛽². (23) Thus, we have the following.

Proposition 4. The derivatives of the principal invariants of a Lagrange space 𝐿^𝑛 with (𝛾,𝛽)-metric are given by 𝜕̇_𝑗𝜌=𝜌₋₂𝑎_𝑗+𝜌₋₁𝑏_𝑗,𝜕̇_𝑗𝜌₁=𝜌₋₁𝑎_𝑗+𝜌₀𝑏_𝑗, (24) with 𝜌₋₂=12𝛾⁻⁴𝐿_𝛾𝛾−2𝛾⁻¹𝐿_𝛾,𝜌₋₁=12𝛾⁻²𝐿_𝛾𝛽,𝜌₀=12𝐿_𝛽𝛽. (25)

The energy of Lagrangian 𝐿(𝑥,𝑦) is defined as 𝐸_𝐿∶=𝑦^𝑖𝜕̇_𝑖𝐿−𝐿. (26) Using (7) in (26), we have 𝐸_𝐿=𝑦^𝑖𝐿_𝛾𝜕̇_𝑖𝛾+𝐿_𝛽𝜕̇_𝑖𝛽−𝐿. (27) Since 𝛾 and 𝛽 are positively homogeneous of degree one in 𝑦^𝑖, by virtue of Euler’s theorem on homogeneous functions, we have 𝑦^𝑖𝜕̇_𝑖𝛾=𝛾,𝑦^𝑖𝜕̇_𝑖𝛽=𝛽. (28) In view of (28), (27) takes the form 𝐸_𝐿=𝛾𝐿_𝛾+𝛽𝐿_𝛽−𝐿. (29) 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 tensor 𝑔_𝑖𝑗(𝑥,𝑦) of a Lagrange space with (𝛾,𝛽)-metric. Using (7) in (1), we have 𝑔_𝑖𝑗=12𝐿_𝛾𝛾𝜕̇_𝑖𝛾+𝐿_𝛾𝛽𝜕̇_𝑖𝛽𝜕̇_𝑗𝛾+𝐿_𝛾𝜕̇_𝑖𝜕̇_𝑗𝛾+𝐿_𝛽𝛾𝜕̇_𝑖𝛾+𝐿_𝛽𝛽𝜕̇_𝑖𝛽𝜕̇_𝑗𝛽+𝐿_𝛽𝜕̇_𝑖𝜕̇_𝑗𝛽. (30) In view of Proposition 1, (30) takes the form 𝑔_𝑖𝑗𝑥,𝑦=2𝜌𝑎_𝑖𝑗+𝜌₋₂𝑎_𝑖𝑎_𝑗+𝜌₋₁𝑎_𝑖𝑏_𝑗+𝑎_𝑗𝑏_𝑖+𝜌₀𝑏_𝑖𝑏_𝑗. (31) Equation (31) can be written as 𝑔_𝑖𝑗𝑥,𝑦=2𝜌𝑎_𝑖𝑗+𝑐_𝑖𝑐_𝑗, (32) where 𝑐_𝑖=𝑞₋₁𝑎_𝑖+𝑞₀𝑏_𝑖 (33) and 𝑞₋₁,𝑞₀ satisfy (a)𝑞₀𝑞₋₁=𝜌₋₁,(b)𝑞₋₁²=𝜌₋₂,(c)𝑞²₀=𝜌₀. (34) 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 of 𝑔_𝑖𝑗.

Theorem 7. The inverse 𝑔^𝑖𝑗 of the fundamental metric tensor 𝑔_𝑖𝑗 of a Lagrange space with (𝛾,𝛽)-metric is given by 𝑔^𝑖𝑗=1𝑎2𝜌^𝑖𝑗−12𝜌+𝑐²𝑐^𝑖𝑐^𝑗, (35) where (a)𝑐^𝑖=𝑎^𝑖𝑟𝑐_𝑟,(b)𝑐²=𝑎^𝑖𝑗𝑐_𝑖𝑐_𝑗. (36)

Proof. Utilizing Lemma 6.1.2.1 of [11] for the nonsingular matrix 𝑔_𝑖𝑗 given by (32) we have the result.

Remarks 8. In [10], Pandey and Chaubey obtained the following: 𝜕̇_𝑖𝛾=𝛾⁻¹𝑦_𝑖,𝜕̇_𝑖𝜕̇_𝑗𝛾=2𝛾⁻¹𝑎_𝑖𝑗𝑥,𝑦−𝛾⁻³𝑦_𝑖𝑦_𝑗,𝑦_𝑖=𝑎_𝑖𝑗𝑥,𝑦𝑦^𝑗,𝑝_𝑖=𝜌𝑦_𝑖+𝜌₁𝑏_𝑖, (37)(38) where 𝜌 and 𝜌₁ are, respectively, given by (19) and (20), 𝜕̇_𝑖𝜌=𝜌₋₂𝑦_𝑖+𝜌₋₁𝑏_𝑖,𝜕̇_𝑖𝜌₁=𝜌₋₁𝑦_𝑖+𝜌₀𝑏_𝑖, (39) 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), 𝐸_𝐿=𝛾⁻¹𝐿_𝛾+𝛽𝐿_𝛽−𝐿,𝑔_𝑖𝑗𝑥,𝑦=2𝜌𝑎_𝑖𝑗+𝑐_𝑖𝑐_𝑗, (40)(41) where 𝑐_𝑖=𝑞₋₁𝑦_𝑖+𝑞₀𝑏_𝑖 and 𝑞₋₁,𝑞₀ satisfy 𝑞₀𝑞₋₁=𝜌₋₁,(𝑞₋₁)²=𝜌₋₂,𝑞²₀=𝜌₀, and 𝑔^𝑖𝑗=1𝑎2𝜌^𝑖𝑗−11+𝑐²𝑐^𝑖𝑐^𝑗, (42) where 𝑐^𝑖=(1/2)𝜌⁻¹𝑎^𝑖𝑟𝑐_𝑟 and 𝑐²=𝑐^𝑖𝑐_𝑖.
In view of corresponding results obtained by us, these results are erroneous.

4. Euler-Lagrange Equations

Using (7) in (2), we obtain 𝐸_𝑖𝐿≡𝜕𝐿𝜕𝑥^𝑖−𝑑𝜕𝑑𝑡𝐿𝜕𝑦^𝑖=0,𝑦^𝑖=𝑑𝑥^𝑖.𝑑𝑡 (43) For the Lagrangian 𝐿 given by (7), we have 𝑑𝜕𝑑𝑡𝐿𝜕𝑦^𝑖=𝐿_𝛾𝛾𝑑𝛾+𝑑𝑡𝐿_𝛾𝛽𝑑𝛽𝑑𝑡𝜕𝛾𝜕𝑦^𝑖+𝐿_𝛽𝛾𝑑𝛾+𝑑𝑡𝐿_𝛽𝛽𝑑𝛽𝑑𝑡𝜕𝛽𝜕𝑦^𝑖+𝐿_𝛾𝑑𝑑𝑡𝜕𝛾𝜕𝑦^𝑖+𝐿_𝛽𝑑𝑑𝑡𝜕𝛽𝜕𝑦^𝑖. (44) In view of 𝜕_𝑖𝐿=𝐿_𝛾𝜕_𝑖𝛾+𝐿_𝛽𝜕_𝑖𝛽 and (44), (43) takes the form 𝐸_𝑖𝐿≡𝐿_𝛾𝐸_𝑖𝛾+𝐿_𝛽𝐸_𝑖𝛽−𝐿_𝛾𝛾𝑑𝛾+𝑑𝑡𝐿_𝛾𝛽𝑑𝛽𝑑𝑡𝜕𝛾𝜕𝑦^𝑖−𝐿_𝛽𝛾𝑑𝛾+𝑑𝑡𝐿_𝛽𝛽𝑑𝛽𝑑𝑡𝜕𝛽𝜕𝑦^𝑖. (45) Since 𝐸_𝑖𝛾³=3𝛾²𝐸_𝑖𝛾−3𝜕𝛾𝜕𝑦^𝑖𝑑𝛾²,𝑑𝑡 (46) we get 𝐸_𝑖𝛾=13𝛾⁻²𝐸_𝑖𝛾³+𝛾⁻²𝜕𝛾𝜕𝑦^𝑖𝑑𝛾².𝑑𝑡 (47) From 𝐸_𝑖(𝛽)=(𝜕𝛽/𝜕𝑥^𝑖)−(𝑑/𝑑𝑡)(𝜕𝛽/𝜕𝑦^𝑖), we have 𝐸_𝑖𝛽=2𝐹_𝑖𝑟𝑦^𝑟,𝑦^𝑟=𝑑𝑥^𝑟,𝑑𝑡 (48) where 𝐹_𝑖𝑟=12𝜕𝑏_𝑟𝜕𝑥^𝑖−𝜕𝑏_𝑖𝜕𝑥^𝑟 (49) is the electromagnetic tensor field of the potentials 𝑏_𝑖.

Using (47) and (48) in (45), we obtain 𝐸_𝑖𝐿=2312𝛾⁻²𝐿_𝛾𝐸_𝑖𝛾³1+22𝛾⁻²𝐿_𝛾𝜕𝛾𝜕𝑦^𝑖𝑑𝛾²1𝑑𝑡+42𝐿_𝛽𝐹_𝑖𝑟𝑦^𝑟−𝜕𝛾𝜕𝑦^𝑖𝐿_𝛾𝛾𝑑𝛾+𝑑𝑡𝐿_𝛾𝛽𝑑𝛽−𝑑𝑡𝜕𝛽𝜕𝑦^𝑖𝐿_𝛽𝛾𝑑𝛾+𝑑𝑡𝐿_𝛽𝛽𝑑𝛽.𝑑𝑡 (50) Thus, we have the following.

Theorem 9. The Euler-Lagrange equations of a Lagrange space with (𝛾,𝛽)-metric are of the following form: 𝐸_𝑖𝐿≡23𝜌𝐸_𝑖𝛾³+2𝜌𝜕𝛾𝜕𝑦^𝑖𝑑𝛾²𝑑𝑡+4𝜌₁𝐹_𝑖𝑟𝑦^𝑟−𝜕𝛾𝜕𝑦^𝑖𝐿_𝛾𝛾𝑑𝛾+𝑑𝑡𝐿_𝛾𝛽𝑑𝛽−𝑑𝑡𝜕𝛽𝜕𝑦^𝑖𝐿_𝛽𝛾𝑑𝛾+𝑑𝑡𝐿_𝛽𝛽𝑑𝛽𝑑𝑡=0,𝑦^𝑖=𝑑𝑥^𝑖.𝑑𝑡 (51)

For the natural parametrization of the curve 𝑐∶𝑡∈[0,1]↦𝑥^𝑖(𝑡)∈𝑀 with respect to the cubic metric 𝑎_𝑖𝑗𝑘(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 𝐸_𝑖𝐿≡23𝜌𝐸_𝑖𝛾³+4𝜌₁𝐹_𝑖𝑟𝑦^𝑟−𝜕𝛾𝜕𝑦^𝑖𝐿_𝛾𝛽𝑑𝛽−𝑑𝑠𝜕𝛽𝜕𝑦^𝑖𝐿_𝛽𝛽𝑑𝛽𝑑𝑠=0. (52)

If 𝛽 is constant on the integral curve 𝑐 of the Euler-Lagrange equations with natural parametrization, then (52) takes the form 𝐸_𝑖𝐿≡23𝜌𝐸_𝑖𝛾³+4𝜌₁𝐹_𝑖𝑟𝑦^𝑟=0. (53) Thus, we have the following.

Theorem 11. If 𝛽 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: 𝐸_𝑖𝐿≡23𝛾𝜌𝐸_𝑖𝛾³+2𝜌3𝛾𝜕𝛾𝜕𝑦^𝑖𝑑𝛾²𝑑𝑡+2𝜌₁𝐹_𝑖𝑟𝑦^𝑟−𝜕𝛾𝜕𝑦^𝑖𝐿_𝛾𝛾𝑑𝛾+𝑑𝑡𝐿_𝛾𝛽𝑑𝛽−𝑑𝑡𝜕𝛽𝜕𝑦^𝑖𝐿_𝛽𝛾𝑑𝛾+𝑑𝑡𝐿_𝛽𝛽𝑑𝛽𝑑𝑡=0. (54)
In case of natural parametrization, their result is 23𝛾𝜌𝐸_𝑖𝛾³+2𝜌3𝛾𝜕𝛾𝜕𝑦^𝑖𝑑𝛾²𝑑𝑡+2𝜌₁𝐹_𝑖𝑟𝑦^𝑟−𝜕𝛽𝜕𝑦^𝑖𝐿_𝛽𝛾𝑑𝛾+𝑑𝑡𝐿_𝛽𝛽𝑑𝛽𝑑𝑡=0. (55)
If 𝛽 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 23𝛾𝜌𝐸_𝑖𝛾³+2𝜌3𝛾𝜕𝛾𝜕𝑦^𝑖𝑑𝛾²𝑑𝑡+2𝜌₁𝐹_𝑖𝑟𝑦^𝑟=0. (56)
In view of Theorem 9, 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 𝐺^𝑖=1𝑥,𝑦4𝑔^𝑖ℎ𝑦^𝑘𝜕̇_ℎ𝜕_𝑘𝐿−𝜕_ℎ𝐿. (57) Since 𝛾³=𝑎_𝑖𝑗𝑘(𝑥)𝑦^𝑖𝑦^𝑗𝑦^𝑘 and 𝛽=𝑏_𝑖(𝑥)𝑦^𝑖, we have 𝜕_ℎ𝛾=𝐴_ℎ𝛾⁻²,𝜕_ℎ𝛽=𝐵_ℎ, (58) where 𝐴_ℎ=13𝜕_ℎ𝑎_𝑖𝑗𝑘𝑦^𝑖𝑦^𝑗𝑦^𝑘,𝐵_ℎ=𝜕_ℎ𝑏_𝑖𝑦^𝑖. (59) Using (58), (19), and (20) in 𝜕_𝑘𝐿=𝐿_𝛾𝜕_𝑘𝛾+𝐿_𝛽𝜕_𝑘𝛽, we get 𝜕_𝑘𝐿=2𝜌𝐴_𝑘+2𝜌₁𝐵_𝑘. (60) Differentiating (60) partially with respect to 𝑦^ℎ and simplifying, we have 𝜕̇_ℎ𝜕_𝑘𝐿=2𝜌₋₂𝑎_ℎ+𝜌₋₁𝑏_ℎ𝐴_𝑘+2𝜌𝐴_𝑘ℎ+2𝜌₋₁𝑎_ℎ+𝜌₀𝑏_ℎ𝐵_𝑘+2𝜌₁𝑏_𝑘ℎ, (61) where (a)𝐴_𝑘ℎ=𝜕̇_ℎ𝐴_𝑘,(b)𝑏_𝑘ℎ=𝜕̇_ℎ𝐵_𝑘. (62) Using (60) and (61) in (57), we obtain 𝐺^𝑖=12𝑔^𝑖ℎ𝜌₋₂𝐴₀+𝜌₋₁𝐵₀𝑎_ℎ+𝜌₋₁𝐴₀+𝜌₀𝐵₀𝑏_ℎ+𝜌𝐴_0ℎ+𝜌₁𝑏_0ℎ−𝜌𝐴_ℎ+𝜌₁𝐵_ℎ, (63) where (i)𝐴₀=𝐴_𝑘𝑥,𝑦𝑦^𝑘,(ii)𝐵₀=𝐵_𝑘𝑥,𝑦𝑦^𝑘,(iii)𝐴_0ℎ=𝐴_𝑘ℎ𝑥,𝑦𝑦^𝑘,(iv)𝑏_0ℎ=𝑏_𝑘ℎ𝑥,𝑦𝑦^𝑘. (64) 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 of 𝑔^𝑖ℎ𝑔_𝑖𝑠=𝛿^ℎ_𝑠, with respect to 𝑦^𝑗, yields 𝜕̇_𝑗𝑔^𝑖ℎ=−2𝑔^𝑟ℎ𝐶^𝑖_𝑟𝑗. (65) If we partially differentiate the quantities appearing in (25) and (64) with respect to 𝑦^𝑗, we find the following quantities: 𝜕̇_𝑗𝜌₋₂=𝜇₋₃𝑎_𝑗+𝜇₋₂𝑏_𝑗,𝜕̇_𝑗𝜌₋₁=𝜇₋₂𝑎_𝑗+𝜇₋₁𝑏_𝑗,𝜕̇_𝑗𝜌₀=𝜇₋₁𝑎_𝑗+𝜇₀𝑏_𝑗,𝜕̇_𝑗𝐴₀=𝐴_𝑗+𝐴_0𝑗,𝜕̇_𝑗𝐵₀=𝔖_(𝑠𝑗)𝜕_𝑠𝑏_𝑗𝑦^𝑠,𝜕̇_𝑗𝐴_0ℎ=2𝐴_0ℎ𝑗+𝐴_𝑗ℎ,𝜕̇_𝑗𝑏_0ℎ=𝑏_𝑗ℎ, (66) where 𝜇₋₃=12𝛾⁻⁶𝐿_𝛾𝛾𝛾−6𝛾⁻¹𝐿_𝛾𝛾+10𝛾⁻²𝐿_𝛾,𝜇₋₂=12𝛾⁻⁴𝐿_𝛾𝛾𝛽−2𝛾⁻¹𝐿_𝛾𝛽,𝜇₋₁=12𝛾⁻²𝐿_𝛾𝛾𝛽,𝜇₀=12𝐿_𝛽𝛽𝛽,𝐴_0ℎ𝑗=𝐴_𝑟ℎ𝑗𝑦^𝑟,𝐴_𝑟ℎ𝑗=𝜕_𝑟𝑎_ℎ𝑗 (67) and 𝔖_(𝑠𝑗) denotes interchanging indices 𝑠 and𝑗 and taking sum. Also, we have 𝜕̇_𝑗𝑎_ℎ=2𝑎_𝑗ℎ. (68) Now, applying (63) in (4), we get 𝑁^𝑖_𝑗=12𝜕̇_𝑗𝑔^𝑖ℎ𝜌₋₂𝐴₀+𝜌₋₁𝐵₀𝑎_ℎ+𝜌₋₁𝐴₀+𝜌₀𝐵₀𝑏_ℎ+𝜌𝐴_0ℎ+𝜌₁𝑏_0ℎ−𝜌𝐴_ℎ+𝜌₁𝐵_ℎ+12𝑔^𝑖ℎ𝜕̇_𝑗𝜌₋₂𝐴₀+𝜌₋₂𝜕̇_𝑗𝐴₀+𝜕̇_𝑗𝜌₋₁𝐵₀+𝜌₋₁𝜕̇_𝑗𝐵₀𝑎_ℎ+𝜌₋₂𝐴₀+𝜌₋₁𝐵₀𝜕̇_𝑗𝑎_ℎ+𝜕̇_𝑗𝜌₋₁𝐴₀+𝜌₋₁𝜕̇_𝑗𝐴₀+𝜕̇_𝑗𝜌₀𝐵₀+𝜌₀𝜕̇_𝑗𝐵₀𝑏_ℎ+𝜕̇_𝑗𝜌𝐴_0ℎ+𝜕̇_𝑗𝜌₁𝑏_0ℎ+𝜌𝜕̇_𝑗𝐴_0ℎ+𝜌₁𝜕̇_𝑗𝑏_0ℎ−𝜕̇_𝑗𝜌𝐴_ℎ+𝜌𝜕̇_𝑗𝐴_ℎ+𝜕̇_𝑗𝜌₁𝐵_ℎ+𝜌₁𝜕̇_𝑗𝐵_ℎ. (69) Using (24), (62)–(64), (65), (66) and (68) in (69) and simplifying, we obtain 𝑁^𝑖_𝑗=−2𝐶^𝑖_𝑟𝑗𝐺^𝑟+12𝑔^𝑖ℎ×𝜌₋₂𝐴_𝑗+𝐴_0𝑗𝑎_ℎ+𝐴_0ℎ−𝐴_ℎ𝑎_𝑗+2𝐴₀𝑎_𝑗ℎ+𝜌₋₁𝐴_𝑗+𝐴_0𝑗𝑏_ℎ+𝐴_0ℎ−𝐴_ℎ𝑏_𝑗+𝑎_𝑗𝑏_0ℎ+𝔖_(𝑠𝑗)𝜕_𝑠𝑏_𝑗𝑦^𝑠𝑎_ℎ+2𝐵₀𝑎_𝑗ℎ−𝑎_𝑗𝐵_ℎ+𝜌₀𝔖_(𝑠𝑗)𝜕_𝑠𝑏_𝑗𝑦^𝑠𝑏_ℎ−𝑏_𝑗𝐵_ℎ+𝑏_𝑗𝑏_0ℎ+𝜌₁𝔚_(𝑗ℎ)𝑏_𝑗ℎ+𝜌2𝐴_0ℎ𝑗+𝐴_𝑗ℎ−𝐴_ℎ𝑗+𝜇₋₃𝐴₀𝑎_𝑗𝑎_ℎ+𝜇₋₂𝐴₀𝑏_𝑗𝑎_ℎ+𝑎_𝑗𝐵₀𝑎_ℎ+𝐴₀𝑏_ℎ+𝜇₋₁𝑏_𝑗𝐵₀𝑎_ℎ+𝐴₀𝑏_ℎ+𝑎_𝑗𝐵₀𝑏_ℎ+𝜇₀𝐵₀𝑏_𝑗𝑏_ℎ, (70) where 𝔐_(𝑗ℎ){𝑏_𝑗ℎ}=𝑏_𝑗ℎ−𝑏_ℎ𝑗.

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.

Acknowledgments

The authors are thankful to the reviewer for his/her valuable comments and suggestions. S. K. Shukla gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), India.