Equations of geodesics in two dimensional Finsler space with special (α,β )- metric

The equation of geodesic in a two-dimensional Finsler space is given by Matsumoto and Park for Finsler space with a (α,β )metric in the year 1997 and 1998. Further Park and Lee studie d the above case for generalized Kropina metric in the year 20 00. Recently Chaubey and his co-authors studied the same for som e pecial(α,β )metric in 2013 and 2014. In continuation of this the purpose of present paper is to express the differential equa tions of geodesics in a two-dimensional Finsler space with s ome special Finsler(α,β )metric.


Introduction
In 1994,M. Matsumoto [6] studied the equation of geodesic in two dimensional Finsler spacesin detail. After that 1997, Matsumoto and Park [1] obtained the equation of geodesics in two dimensional Finsler spaces with the Randers metric (L = α + β ) and the Kropina metric L = ( α 2 β ), and in 1998, they have [2] obtained the equation of geodesic in twodimensional Finsler space with the slope metrics, i.e. Matsumoto metric given by L = α 2 (α−β ) , by considering β as an infinitesimal of degree one and neglecting infinitesimal of degree more than two they obtained the equations of geodesic of two-dimensional Finsler space in the form y ′′ = f (x, y, y ′ ), where (x, y) are the co-ordinate of two-dimensional Finsler space. Further Park and Lee [3] studied the above case for generalized Kropina metric in the year 2000. In continuation of this Chaubey and his co-authors [7,8] are studied the same case for the different special (α, β ) -metric and illustrated their main results in the different figures. In the present paper we have shown that under the same conditions, the geodesic of the two-dimensional space with following metrics:
Since L(x 1 , x 2 ; y 1 , y 2 ) is (1) p-homogeneous in (y 1 , y 2 ) we have L ( j)(i) y i = 0 which imply the existence of a function, so called the Weierstrass invariant W (x 1 , x 2 ; y 1 , y 2 ) [1,2,8] given by In a two-dimensional associated Riemannian space R 2 = (M 2 , α) with respect to L = α and ?α 2 = a i j (x 1 , x 2 )y i y j , the Weierstrass invariant W r of R 2 is written as Further L j are still (l) p-homogeneous in (y 1 , y 2 ), so that we get The geodesic equations in F 2 along curve C : x i = x i (t) are given by [1].

Equation of Geodesics in a two dimensional Finsler with (α, β ) -metric space
In [2,4,5] a two dimensional Finsler space where α (i) = a ir y r α and the subscriptions α, β of L are the partial derivatives of L with respect to α, β respectively. Then we have in Γ * .
From (1) and (7) we have On other hand, from (6) we have Similarly to the case of L(x 1 x 2 ; y 1 y 2 ) and α(x 1 , x 2 ) , we get the Weierstrass invariant w(α, β ) as follows: Substituting (10) in (9), we have From (8) and (11) we have If we put y i Substituting (12) and (13) in (4), we have where β ;i = b r;i y r . The relation of W,W r and w is written as follows: (14) is expressed as follows: Thus we have the following.

Theorem 1.
In a two-dimensional Finsler space F 2 with an (α, β )− metric, the differential equation of a geodesic is given by (16).
Suppose that α be positive−definite. Then we may refer to an isothermal coordinate system ( that is a 11 = a 22 = a 2 , a 12 = 0 and (y 1 , y 2 ) = (ẋ,ẏ). From α 2 = a i j (x)y i y j we get αα (i)( j) = a i j − a ir a js y r y s α 2 . Therefore we have αα (1)(1) = ( aẏ E ) 2 and W r = a E 3 . Furthermore the Christoffel symbols are given by Next Substituting (17) where b 0;0 = b r sy r y s = Thus we have the following.

Theorem 2.
In a two dimensional Finsler spaceF 2 with an (α, β )metric, if we refer to an isothermal coordinate system (x, y) such that α = aE , then the differential equation of a geodesic is given by (21) and (22). (21), we obtain the differential equation of a geodesic in an isothermal coordinate system (x,y) with respect to α as follows:
It seems quite complicated from, but y ′′ is given as a fractional expression in y ′ . Thus we have the following Theorem 3. Let F 2 be two-dimensional space with special Finsler metric. If we refer to a local coordinate system (x, y) with respect to α, then the differential equation of a geodesic y = y(x) of F 2 is of the form where f (x, y, y ′ ) is a quadratic polynomial in y ′ and g(x, y, y ′ ) is a polynomial in y ′ of degree at most five.
In order to find the concrete form, we treat the case of which the associated Riemannian space is Euclidean with orthonormal coordinate system. Then a = 1 and a x = a y = 0. If we take scalar function b such that Thus we have the following Corollary 1. Let F 2 be a two -dimensional Finsler space with a special metric. If we refer to an orthonormal coordinate system (x, y) with respect to α and b 1 ∂ y for a scalar b, then the differential space of geodesic y = y(x) of F 2 is given by (27).

Equation of Geodesics in a two dimensional Finsler with special
Substituting (28) in (21), we obtain the differential equation of a geodesic in an isothermal coordinate system (x,y) with respect to α as follows: In the particular case for the t of curve C is chosen x of (x, y) , thenẋ = 1,ẏ = y ′ ,ẍ = 0,ÿ = y ′′ , E = 1 + (y ′ ) 2 .
It seems quite complicated from, buty ′′ is given as a fractional expression in y ′ . Thus we have the following.
Theorem 4. Let F 2 be two-dimensional space with special Finsler metric. If we refer to a local coordinate system (x, y) with respect to α, then the differential equation of a geodesicy = y(x) of F 2 is of the form where f (x, y, y ′ ) is a quadratic polynomial in y ′ and g(x, y, y ′ ) is a polynomial in y ′ of degree at most five.
In order to find the concrete form, we treat the case of which the associated Riemannian space is Euclidean with orthonormal coordinate system. Then a = 1 and a x = a y = 0. If we take scalar function b such that b 1 = b x , b 2 = b y then b 1y − b 2x = 0.Therefore (30) is reduces to Thus we have the following.

Corollary 2.
Let F 2 be a two -dimensional Finsler space with a special metric. If we refer to an orthonormal coordinate system (x, y) with respect to α and b 1 ∂ y for a scalar b, then the differential space of geodesic y = y(x) of F 2 is given by (32).

Equation of Geodesics in a two dimensional Finsler with special
Substituting (33) in (21), we obtain the differential equation of a geodesic in an isothermal coordinate system (x,y) with respect to α as follows: In the particular case for the t of curve C is chosen x of (x, y) , thenẋ = 1,ẏ = y ′ ,ẍ = 0,ÿ = y ′′ , E = 1 + (y ′ ) 2 .
It seems quite complicated from, buty ′′ is given as a fractional expression in y ′ . Thus we have the following, Theorem 5. Let F 2 be two-dimensional space with special Finsler metric. If we refer to a local coordinate system (x, y) with respect to α, then the differential equation of a geodesicy = y(x) of F 2 is of the form y ′′ = g(x, y, y ′ ) f (x, y, y ′ ) , where f (x, y, y ′ ) is a quadratic polynomial in y ′ and g(x, y, y ′ ) is a polynomial in y ′ of degree at most five.
Corollary 3. Let F 2 be a two -dimensional Finsler space with a special metric. If we refer to an orthonormal coordinate system (x, y) with respect to α and b 1 = ∂ b ∂ x , b 2 = ∂ b ∂ y for a scalar b, then the differential space of geodesic y = y(x) of F 2 is given by (37).