ISRN GeometryVolume 2011 (2011), Article ID 505161, 16 pagesdoi:10.5402/2011/505161

Research Article

On Almost -Lagrange Spaces

P. N. Pandey and Suresh K. Shukla

Department of Mathematics, University of Allahabad, Allahabad 211 002, India

Received 12 October 2011; Accepted 13 November 2011

Academic Editors: A. Belhaj and M. Margenstern

Copyright © 2011 P. N. Pandey and Suresh K. Shukla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We initiate a study on the geometry of an almost -Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and - and -deflection tensors in an APL-space. Corresponding expressions in a -Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.

1. Introduction

In the last three decades, various meaningful generalizations of Finsler spaces have been considered. These generalizations have been found much applicable to mechanics, theoretical physics, variational calculus, optimal control, complex analysis, biology, ecology, and so forth. The geometry of Lagrange spaces is one such generalization of the geometry of Finsler spaces which was introduced and studied by Miron [1, 2]. He [1, 2] introduced the most natural generalization of Lagrange spaces named as generalized Lagrange space. Since the introduction of Lagrange spaces and generalized Lagrange spaces, many geometers and physicists have been engaged in the exploration, development, and application of these concepts [3–13]. Antonelli and Hrimiuc [14, 15] introduced a special type of regular Lagrangian called -Lagrangian. Applications of such Lagrangian have been discussed by Antonelli et al. in the monograph [16]. In the present paper, we generalize the notion of -Lagrangian and introduce the concept of almost -Lagrange spaces. We hope that the results obtained in the paper will be interesting for the researchers working on the application of Lagrange spaces in various fields of science.

Let be an -dimensional Finsler space, and let be a smooth function. The composition defines a differentiable Lagrangian. This was regarded by Antonelli and Hrimiuc [14, 15] as -Lagrangian associated to the Finsler space . They [14] proved that if the function has the following properties: then is a regular Lagrangian and thus is a Lagrange space, called a -Lagrange space.

In this paper, we consider a more general Lagrangian as follows: where is the same as discussed earlier, is a covector, and is a smooth function.

In Section 2, we show that if the function has the properties (1.1), then is a regular Lagrangian and thus the pair is a Lagrange space. We call this space as an almost -Lagrange space (shortly APL-space).

An APL-space reduces to a -Lagrange space if and only if and .

If , then the Lagrangian in (1.2) takes the form This defines a regular Lagrangian, and the pair is called an almost Finsler Lagrange space (shortly AFL-space). Such Lagrange space was introduced by Miron and Anastasiei (vide Chapter IX of [17]).

We take Henceforth, we will indicate all the geometrical objects related to by a small circle “” put over them.

In a Finsler space, the geodesics, parameterized by arc length (the extremals of the length integral), coincide with the extremals of action integral or with the autoparallel curves of the Cartan nonlinear connection [16]: where These geodesics are the integral curves of the spray [16] (i.e., (2) p-homogeneous): that is, solutions of the differential equations We have the following equalities: In a general Lagrange space , the geodesics are the extremals of the action integral and coincide with the integral curves of the semispray [17, 18] (i.e., may not be a spray): As in a Finsler space, a remarkable nonlinear connection can be considered in a Lagrange space: Such nonlinear connection is a canonical nonlinear connection [17, 18] as it depends only on the fundamental function of the Lagrange space.

In general, the autoparallel curves of are different from the geodesics of (cf. [17]).

Given a nonlinear connection on a Lagrange space , there is a unique - and -metrical -connection (cf. [17, 19]) with torsions and , called the canonical metrical -connection. This connection is linear and its coefficients are given by where is the Lagrange differentiation operator.

If is the Cartan connection of the Finsler space , then its coefficients are given by where .

The - and -deflection tensor fields and , respectively, of a Lagrange space are defined by (cf. [19]) where and , respectively, denote the - and -covariant derivatives with respect to .

If is the h-deflection tensor field and is the -deflection tensor field of the Finsler space , then where and , respectively, denote the - and -covariant derivatives with respect to .

For basic terminology and notations related to a Finsler space and a Lagrange space, we refer to the books [17, 20].

2. Almost -Lagrange Spaces

As discussed earlier, we consider the Lagrangian given by (1.2) in which the function satisfies (1.1). We prove that it is a regular Lagrangian and the pair is a Lagrange space which we term as an almost -Lagrange space (APL-space in short).

Theorem 2.1. If the function satisfies the conditions (1.1), then , given by (1.2), is a regular Lagrangian and is a Lagrange space.

Proof. Differentiating (1.2) partially with respect to , we get Again differentiating (2.1) partially with respect to , we obtain which, in view of (1.4), provides Now In view of (2.4), (2.3) takes the form Under the hypothesis, the matrix is invertible and its inverse is (see Lemma 6.2.2.1, page 891 in [20]) This proves the theorem.

Remarks 1. (i) If and in (1.2), then expression (2.5) remains unchanged. Hence, the symmetric metric tensor of a -Lagrange space is the same as that of an APL-space.(ii) If , then and . Hence, the symmetric metric tensor of an AFL-space coincides with that of the associated Finsler space.

3. Semispray, Integral Curves of Euler-Lagrange Equations

In this section, we obtain the coefficients of the canonical semispray of the APL-space and deduce corresponding expressions for a -Lagrange space and an AFL-space. Next, we obtain the differential equations whose solution curves are the integral curves of Euler-Lagrange equations in an APL-space. We deduce corresponding differential equations for a -Lagrange space and an AFL-space.

If we differentiate (1.2) partially with respect to , we have Differentiating (3.1) partially with respect to , we obtain which, in view of (2.4), takes the form Using (3.1) and (3.3) in (1.10), we have where is electromagnetic tensor field of the potentials .

Applying (2.6) in (3.4) and using , and (by Euler’s theorem on homogeneous functions), we obtain Using (1.7) in (3.6) and simplifying, we get Thus, we have the following.

Theorem 3.1. The canonical semispray of an APL-space has the local coefficients given by where are the local coefficients of the spray of .

For a -Lagrange space, and . Hence, from (3.5), we have . Therefore, (3.7) reduces to Thus, we may state the following.

Corollary 3.2 (see [14]). The canonical semispray of a -Lagrange space becomes a spray and coincides with that of the associated Finsler space.

For an AFL-space, (see Remark (ii)). Hence, (3.7) takes the form Thus, we have the following.

Corollary 3.3 (see [17, 20]). The canonical semispray of an AFL-space has the local coefficients given by (3.10).

In a Lagrange space, the integral curves of the Euler-Lagrange equations: are the solution curves of the equations [20] Using (3.7) in (3.12), we obtain where .

Using (1.9) (a) in (3.13), we have Thus, we have the following.

Theorem 3.4. In an APL-space , the integral curves of the Euler-Lagrange equations are the solution curves of (3.14).

For a -Lagrange space, equations (3.14) take the following simple form: This enables us to state the following.

Corollary 3.5 (see [14]). In a -Lagrange space, the integral curves of the Euler-Lagrange equations are the solution curves of (3.15).

For an AFL-space, . Therefore, equations (3.14) become where .

Thus, we have the following.

Corollary 3.6 (see [17, 20]). In an AFL-space, the integral curves of the Euler-Lagrange equations are the solution curves of (3.16).

4. Nonlinear Connection, Autoparallel Curves

In this section, we find the coefficients of the nonlinear connection of an APL-space and obtain the differential equations of the autoparallel curves of the nonlinear connection. Corresponding results have been deduced for a -Lagrange space and an AFL-space.

Partial differentiation of (2.5) with respect to yields Using (3.7) in (1.11) and taking (1.9) (b), (2.6), (4.1), , , and into account, we obtain If we take the last expression becomes that is, where Thus, we have the following.

Theorem 4.1. The canonical nonlinear connection of an APL-space has the local coefficients given by (4.5).

For a -Lagrange space, we have and and hence . Therefore, (4.5) reduces to Thus, we have the following.

Corollary 4.2 (see [14]). The canonical nonlinear connection of a -Lagrange space coincides with the nonlinear connection of the associated Finsler space.

For an AFL-space, (4.3) reduces to and hence (4.6) gives Therefore, (4.5) takes the form Thus, we have the following.

Corollary 4.3 (see [17, 20]). The canonical nonlinear connection of an AFL-space has the local coefficients given by (4.10).

Transvecting (4.5) by and using , we obtain where .

The autoparallel curves of the canonical nonlinear connection of a Lagrange space are given by the following system of differential equations (vide [20]): Equations (4.12), in view of (4.11), take the form Thus, we have the following.

Theorem 4.4. The autoparallel curves of the canonical nonlinear connection of an APL-space are given by the system of differential equations (4.13).

For a -Lagrange space, and hence . Therefore, (4.13) reduces to Thus, we have the following.

Corollary 4.5 (see [14]). The autoparallel curves of the canonical nonlinear connection of a -Lagrange space are given by the system of differential equations (4.14).

For an AFL-space, and hence, by virtue of , we have . Therefore, equations (4.12) take the form Thus, we deduce the following.

Corollary 4.6 (see [17, 20]). The autoparallel curves of the nonlinear connection of an AFL-space are given by the system of differential equations (4.16).

If we compare (3.14), (3.15), and (3.16), respectively, with (4.13), (4.14), and (4.16), we observe that, in an APL-space as well as in an AFL-space, solution curves of Euler-Lagrange equations do not coincide with the autoparallel curves of the canonical nonlinear connection whereas in a -Lagrange space they do. Therefore, in a -Lagrange space, geodesics are autoparallel curves whereas in an APL-space and in an AFL-space they are not so.

5. Canonical Metrical -Connection

Let be the canonical metrical -connection of the APL-space , and let be the Cartan connection of the associated Finsler space . In this section, we obtain the expressions for the coefficients of and we investigate some properties of . We deduce corresponding results for a -Lagrange space and an AFL-space.

Using (4.1) in (1.13) and taking (1.15) into account, we find

For any -class function , taking , we have which, in view of (see proposition 9.4, page 1037 of [20]), gives Since (see proposition 9.4, page 1037 of [20]), we have If we operate on (2.5) and utilize (5.3) and (5.4), it follows that In view of , (4.5), and , we get which, on account of (4.1) and (5.5), becomes Using (5.7) in (1.12) and taking (1.14) and into account, we obtain Equations (5.1) and (5.8) enable us to state the following.

Theorem 5.1. The coefficients of the canonical metrical -connection of an APL-space are given by (5.1) and (5.8).

For a -Lagrange space, . Hence, (5.1) remains unchanged whereas (5.8) reduces to Thus, we have the following.

Corollary 5.2 (see [14]). The coefficients of the canonical metrical -connection of a -Lagrange space are given by (5.1) and (5.9).

For an AFL-space, , and . Therefore, we have and .

In view of these facts, (5.1) reduces to whereas (5.8) gives the following: where is given by (4.9). Thus, we have the following.

Corollary 5.3 (see [17, 20]). The coefficients of the canonical metrical -connection of an AFL-space are given by (5.10) and (5.11).

Now, we investigate some properties of the canonical metrical -connection of an APL-space and deduce the corresponding properties for a -Lagrange space and an AFL-space.

Theorem 5.4. The canonical metrical -connection of an APL-space has the following properties: where , where , where .

Proof. (1) Using (5.8) and (4.5) in (1.16), we have which, in view of (1.18), reduces to Next, if we use (2.5) in , then it follows that Now, applying successively , (4.5), and in and keeping (5.8) and (5.18) in view, we have Differentiating partially with respect to , we have Also, In view of (5.3), we have Using (5.20), (5.21), and (5.22) in (5.19), we obtain which, in view of (5.4), gives the desired result.(2) Using (5.1) in (1.17), we get where .In view of (5.20) and (5.21), it follows, from , that that is, as is totally symmetric.(3) Utilizing successively , (4.5), and in , we get Using (1.2) and (2.1) in (5.26), we have which, in view of (5.3), gives Using and (5.18) in (5.28) and keeping (4.5) in view, we find If we take , then the last expression takes the form Next, using (2.1) in , we get which, in view of (5.18), gives the required result.

Corollary 5.5 (see [14]). The canonical metrical -connection of a -Lagrange space has the following properties: where ,

Proof. Applying , , and in Theorem 5.4, we have the corollary.

Corollary 5.6. The canonical metrical -connection of an AFL-space has the following properties: where ,

Proof. Using , and in Theorem 5.4, we have the corollary.

Acknowledgment

S. K. Shukla gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), India.

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