A Survey on Geometric Dynamics of 4-Walker Manifold

where Φ indicates the symplectic form. The triple (TQ, Φ, X) is called a Hamiltonian system on the cotangent bundle TQ. Nowadays, there are many studies about Lagrangian and Hamiltonian dynamics, mechanics, formalisms, systems and equations [1-5] and there in. There are real, complex, paracomplex and other analogues. As we know it is possible to produce different analogues in different spaces. Let 2n M be a Riemannian manifold with a neutral metric, i.e. with a semi-Riemannian metric g of signature (n, n). By a Walker n-manifold, we mean a semi-Riemannian manifold which admits a field of parallel null


Introduction
As is well known, modern differential geometry plays an important role to explain the dynamics of Lagrangians.So, if Q is an m-dimensional configuration manifold and is a regular Lagrangian function, then it is well-known that there is a unique vector field X on TQ such that dynamics equation is given by where L  indicates the symplectic form.The triple (TQ, L  , X) is called a Lagrangian system on the tangent bundle TQ.Also, modern differential geometry provides a good framework in which develop the dynamics of Hamiltonians.Therefore, if Q is an m-dimensional configuration manifold and is a regular Hamiltonian energy function, then there is a unique vector field X on T * Q such that dynamics equation is given by where Φ indicates the symplectic form.The triple (T * Q, Φ, X) is called a Hamiltonian system on the cotangent bundle T * Q. Nowadays, there are many studies about Lagrangian and Hamiltonian dynamics, mechanics, formalisms, systems and equations [1][2][3][4][5] and there in.There are real, complex, paracomplex and other analogues.As we know it is possible to produce different analogues in different spaces.Let 2n M be a Riemannian manifold with a neutral metric, i.e. with a semi-Riemannian metric g of signature (n, n).By a Walker n-manifold, we mean a semi-Riemannian manifold which admits a field of parallel null r-planes, with . 2 n r  The canonical forms of the metrics were investigated by Walker [6].Special interest manifolds are Walker manifolds of even dimensions (n = 2m) admitting a field of null planes of maximum dimensionality(r = m).An application of such a 4-dimensional Walker metric is given in [7].Since the observation of the existence of almost complex structure on Walker 4-manifolds in [8], the Walker 4-manifolds and the almost Hermitian structures on the four-manifolds have been intensively studied, e.g., [9][10][11][12][13][14], etc.In this study, we present complex (paracomplex) analogues of Lagrangian and Hamiltonian mechanical systems on 4-Walker manifold.In the end, some geometrical-physical results about the obtained complex (paracomplex) mechanical systems are also given.Throughout this paper, all mathematical objects and mappings are assumed to be smooth, i.e. infinitely differentiable and Einstein convention of summarizing is adopted.Denote by tions on M 4 , the set of vector fields on M 4 and the set of 1-forms on M 4 , respectively.

Walker Metric
A neutral metric g on a 4-manifold M 4 is said to be Walker metric if there exists a 2-dimensional null distribution D on M 4 , which is parallel with respect to g.From Walker theorem [6], there is a system of coordinates with respect to which g takes the local canonical form where a, b, c are smooth functions of the coordinates (x

Proper Almost Complex Structure φ
Let φ be a proper almost complex structure on a Walker manifold M 4 , which satisfies We easily see that the above three properties defines φ uniquely, i.e.

 
  If we write as then from (4) we can read off the nonzero components , with respect to the natural frame [14].
Similarly, we define a unique proper almost complex structure { , , , }       as follows: where d i are the abbreviated forms of dx i , i = x 1 , x 2 , x 3 , x 4 .

Lagrangian Mechanical Systems
Now, our purpose is to obtain complex Euler-Lagrange equations for relativistic, quantum and classical mechanics on 4-dimensional Walker manifold M 4 .
Let M 4 be a Walker manifold and {x 1 , x 2 , x 3 , x 4 } be its coordinate functions.Let the semispray be the vector field X determined by where X x   and the dot indicates the derivative with respect to time t.By means of the proper almost complex structure φ given by (4), the vector field is defined by which is named Liouville vector field on the Walker manifold M 4 .The maps given by such 4 , : , are said to be the kinetic energy and the potential energy of the system, respectively.Here m i , g and h stand for mass of a mechanical system having m particles, the gravity acceleration and distance to the origin of a mechanical system on Walker manifold M 4 , respectively.Then is a map that satisfies the conditions; 1) is a Lagrangian function, 2) the function determined by The function  induced by  and denoted by where d is the usual exterior derivation.For the almost complex structure  given by ( 4), the form on Walker Through a direct computation using (7), the closed 2form L   is seen to be as follows: Then the energy function .
L E  is found as follows: be an integral curve of semispray X.According to (1), using ( 7) and ( 9) then we find the following equations: Suppose that a curve By means of Theorem 2 and Theorem 3 (p.387 and p. 388) in [14], we can derive the following corollaries: such that the equations calculated in (10) then the equations in ( 10) are Now we obtain some corollaries about the equations raised in (10) thinking Remark (p.387) in [14] and Proposition 4 in [8] and Corollary 4 in [15]: Corollary 1: In (10), if c = 0 and a = b we find the equations as follows: 11) and have a solution.
, Corollary 3: If the following PDEs hold: then the equations in (10) have a solution.

Hamiltonian Mechanical Systems
This section is devoted to present complex Hamiltonian equations and Hamiltonian mechanical systems for relativistic, quantum and classical mechanics constructed on Walker manifold M 4 .
Assume that a Liouville form and a 1-form on Walker manifold M 4 are shown  and  , respectively .
Let us consider Using ( 5), we have Assume that the vector field X determined by ( 6 and Furthermore, the differential of Hamiltonian energy function is obtained by With respect to (2), if ( 16) and ( 17) are equaled, the Hamiltonian vector field is found as follows:

X c H a H b H H c H a H b H H c H a H b H H c H a H b H H
In the local coordinates, taking , , , Using ( 19), ( 18), (20), it holds As is known that if Φ is a closed form on Walker manifold M 4 , then Φ is also a symplectic structure on Walker manifold M 4 [14].Hence, the equations introduced in (21) are named complex Hamiltonian equations on Walker manifold M 4 if Φ is a symplectic structure.
 is named a complex Hamiltonian mechanical system on Walker manifold M 4 .We obtain the following corollary considering the equations found in (21) using Remark (p.387) in [14] and Proposition 4 in [8] and Corollary 4 in [15].
Corollary 4: In (21), if c = 0 and a = b we find the equations as follows: Taking Theorem 2 and Theorem 3 (p.387 and p. 388) in [14], we can derive some corollaries as follows: Corollary 5: If a and b satisfy the following PDEs: , 16 and have a solution.
Corollary 6: If the following PDEs hold: then the equations in (21) have a solution.

Paracomplex Lagrangian and Hamiltonian Mechanical Systems
In this section, we produce a paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on a Walker manifold M 4 .
If we write as j j then from (24) we can read off the nonzero components 4 1 ,  as follows: with respect to the natural frame .
  , , ,     According to the proper almost paracomplex structure  , the 2-form L   is given by as follows:

L d d
By means of L   , using similar method in Section 3, we get .
For the proper almost paracomplex structure *  , the 2-form    is given by By using similar method in section 4, by means of

Discussions
From above, complex and paracomplex Lagrangian mechanical systems have intrinsically been described on a Walker manifold M 4 : The paths of semispray X on Walker manifold M 4 are the solutions of complex and paracomplex Euler-Lagrange equations raised in (10) and (25).Also, complex and paracomplex Hamiltonian mechanical systems have intrinsically been described on a Walker manifold M 4 .The paths of Hamilton vector field X on the Walker manifold M 4 are the solutions of complex and paracomplex Hamiltonian equations raised in (21) and ( 27).
One can be proved that the here obtained equations are very important to explain the space-time mechanicalphysical problems.Therefore, the found equations are only considered to be a first step to realize how complex (paracomplex) structures on a Walker manifold has been used in solving problems in different physical areas.
For further research, the complex (paracomplex) Lagrangian and Hamiltonian mechanical equations derived here are suggested to deal with problems in electrical, magnetical and gravitational fields of relativistic, quantum and classical mechanics of physics.

4 M
) is Hamiltonian vector field associated with Hamiltonian energy function H. Then we find

R
be an integral curve of the Hamiltonian vector field X, i.e.,


Associated to any Walker metric (3) we consider aproper almost para-Hermitian structure  defined by 25)such that the equations calculated in (25) are named a paracomplex Euler-Lagrange equations constructed on the Walker manifold M 4 if the 2-form L paracomplex mechanical system on the Walker manifold M 4 .Now, let us consider the proper almost paracomplex structure   : 27) Hence, the equations introduced in (27) are named a paracomplex Hamiltonian equations on the Walker manifold M 4 if the 2-form L   is symplectic structure.Then the triple  4 , ,  are named complex Euler-Lagrange equations constructed on Walker manifold M 4 if 2-form L 