^{*}

A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ 2/n . It is well-known that semi-Riemannian geometry has an important tool to describe spacetime events. Therefore, solutions of some structures about 4-Walker manifold can be used to explain spacetime singularities. Then, here we present complex and paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on 4-Walker manifold. Finally, the geometrical-physical results related to complex (paracomplex) mechanical systems are also discussed.

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 indicates the symplectic form. The triple (TQ, , 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 [1-5] 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 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 The canonical forms of the metrics were investigated by Walker [

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 a Walker manifold. Then, and are the set of functions on M_{4}, the set of vector fields on M_{4} and the set of 1-forms on M_{4}, respectively.

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 [

where a, b, c are smooth functions of the coordinates (x_{1}, x_{2}, x_{3}, x_{4}). The parallel null 2-plane D is spanned locally by, where are the abbreviated forms ofi = x_{1}, x_{2}, x_{3}, x_{4}.

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 as follows:

i.e., has the local components

with respect to the natural frame [

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}.

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, , , 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 that, 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 is energy function.

The function induced by and denoted by

is called vertical derivation, where The vertical differentiation is given by

where d is the usual exterior derivation. For the almost complex structure given by (4), the form on Walker manifold M_{4} is the closed 2-form determined by such that

Through a direct computation using (7), the closed 2- form is seen to be as follows:

Then the energy function is found as follows:

Suppose that a curve

be an integral curve of semispray X. According to (1), using (7) and (9) then we find the following equations:

such that the equations calculated in (10) are named complex Euler-Lagrange equations constructed on Walker manifold M_{4} if 2-form is symplectic structure. Thus the triple is named a complex mechanical system on Walker manifold M_{4}.

Now we obtain some corollaries about the equations raised in (10) thinking Remark (p. 387) in [

Corollary 1: In (10), if c = 0 and a = b we find the equations as follows:

By means of Theorem 2 and Theorem 3 (p. 387 and p. 388) in [

Corollary 2: If a and b satisfy the following PDEs:

then the equations in (10) are

and have a solution.

Corollary 3: If the following PDEs hold:

then the equations in (10) have a solution.

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

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:

where

Suppose that a curve

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

In the local coordinates, taking

we find

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} [_{4} if Φ is a symplectic structure.

Then the triple 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 [

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 [

Corollary 5: If a and b satisfy the following PDEs:

then the equations in (21) are

and have a solution.

Corollary 6: If the following PDEs hold:

then the equations in (21) have a solution.

In this section, we produce a paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on a Walker manifold M_{4}.

Let be a proper almost paracomplex structure on a Walker manifold M_{4}, which satisfies

Associated to any Walker metric (3) we consider a proper almost para-Hermitian structure defined by

where, i = x_{1}, x_{2}, x_{3}, x_{4}. denotes the coordinate bases [

If we write as then from (24) we can read off the nonzero components as follows:

i.e., has the local components

with respect to the natural frame.

According to the proper almost paracomplex structure, the 2-form is given by as follows:

By means of, using similar method in Section 3, we get

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 is symplectic structure. Hence the triple is named a paracomplex mechanical system on the Walker manifold M_{4}.

Now, let us consider the proper almost paracomplex structure:

For the proper almost paracomplex structure, the 2-form is given by

By using similar method in Section 4, by means of, we have

Hence, the equations introduced in (27) are named a paracomplex Hamiltonian equations on the Walker manifold M_{4} if the 2-form is symplectic structure.

Then the triple is named a paracomplex Hamiltonian mechanical system on the Walker manifold M_{4}.

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 easily see that the complex (paracomplex) EulerLagrange and Hamilton equations introduced in [4,5] are equivalent those in (10) and (21) ((25) and (27)) if a = b = 0.

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.