We initiate a study on the geometry of an almost

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 [

Let

In this paper, we consider a more general Lagrangian as follows:

In Section

An APL-space reduces to a

If

We take

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 [

In general, the autoparallel curves of

Given a nonlinear connection

If

The

If

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

As discussed earlier, we consider the Lagrangian given by (

If the function

Differentiating (

(i) If

(ii) If

In this section, we obtain the coefficients of the canonical semispray of the APL-space

If we differentiate (

Applying (

The canonical semispray of an APL-space has the local coefficients given by

For a

The canonical semispray of a

For an AFL-space,

The canonical semispray of an AFL-space has the local coefficients given by (

In a Lagrange space, the integral curves of the Euler-Lagrange equations:

Using (

In an APL-space

For a

In a

For an AFL-space,

Thus, we have the following.

In an AFL-space, the integral curves of the Euler-Lagrange equations

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

Partial differentiation of (

The canonical nonlinear connection of an APL-space

For a

The canonical nonlinear connection of a

For an AFL-space, (

The canonical nonlinear connection of an AFL-space

Transvecting (

The autoparallel curves of the canonical nonlinear connection

The autoparallel curves of the canonical nonlinear connection

For a

The autoparallel curves of the canonical nonlinear connection of a

For an AFL-space,

The autoparallel curves of the nonlinear connection

If we compare (

Let

Using (

For any

The coefficients of the canonical metrical

For a

The coefficients of the canonical metrical

For an AFL-space,

In view of these facts, (

The coefficients of the canonical metrical

Now, we investigate some properties of the canonical metrical

The canonical metrical

(1) Using (

(2) Using (

In view of (

(3) Utilizing successively

The canonical metrical

Applying

The canonical metrical

Using

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