Hamiltonian mechanics with Two Almost para-Complex Structures on Symplectic Geometry

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 3 Ver. IV (May - June 2017), PP 68-71 www.iosrjournals.org

Hamiltonian mechanics with Two Almost para-Complex Structures on Symplectic Geometry Ibrahim Yousif .I. Abad Alrhman Department of Mathematics - Faculty of Education West Kordufan University- Alnhoud City-Sudan

Abstract: In this study, we concluded the Hamiltonian equations on

, being a model. Finally introduce, some geometrical and physical results on the related mechanic systems have been discussed.

Key words: Symplectic manifold, Almost para-complex structure, holomorphic structures, Hamiltonian Dynamics...

I. Introduction The study of differential geometry is one of the most important branches of Geometry due to its direct correlation with many engineering and physical sciences and other practical fields There are also a large number of studies on this subject, for example [1] where the researcher to find the equations of Hamiltonian Dynamics .

Its is some important work for examples [2][3][4][5] In this paper we will study Hamiltonian mechanics .using Almost Para-Complex Where this paper contains three sections ,its detailed description is as follows . The first section is presented with an Introduction, the second section gives some principles and concepts in Almost ParaComplex. The three section we derive the Hamiltonian equations. Then presented formulas equations of Hamiltonian Dynamical Systems. Finally, we presented the results and the conclusions. provided this paper is with list of references that have been used in this paper.

II. Almost Para-Complex Let be a para-complex manifold of para-complex dimension n and denote by the manifold considered as a real 2n-dimensional manifold with the induced almost para-complex structure .

Definition 2.1. [6]A para-complex Riemannian metric on is a covariant symmetric 2-tensor field , which is non-degenerate at each point of M and satisfies , the relation (1) is equivalent to Definition 2.2 Let be configuration manifold of real dimension 2m ,A tensor field on Is called almost para -complex manifold .such that .

Definition 2.3[3] An almost para -complex structure on amanifold is a differentiable map on the tangent bundle of such that preserves each fiber ,A manifold with affixed almost para -complex structure is called an almost para -complex manifold Definition 2.4 Let para complex manifold . In local holomorphic coordinates one can define the tangent space and and and cotangent space and and .respectively .then I ,as denoted

And

Definition 2.5 Let neighborhood V of

, , be para complex manifold .In local coordinates system on a .We define the vector fields by

DOI: 10.9790/5728-1303046871

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Hamiltonian mechanics with Two Almost para-Complex Structures on Symplectic Geometry

And the dual convector fields

Definition 2.6 she holomorphic structures on cotangent space for Hamilton equations, of cotangent space of manifold satisfying Theorem 2.7 Suppose that

be a real coordinate system on , and bases cotangent space

stands for the dual endomorphism

). Then the bases tangent space

If is defined as a para complex manifold then Theorem 2.8 the dual endomorphism of cotangent space bases cotangent space ,and is defined by

of manifold

satisfying

and

Proof

Definition 2.9 Almost para –Hermition structure is differentiable manifolds almost product structure and pseudo –Riemannian metric , compatible in the sense that

endowed with an

Definition 2.10 [4] Almost para –Hermition structure is said to be a para- Kahlerian manifold if closed .Also .by means of geometric structure one may show that is para- Kahlerian manifold ( defined by Lemma 2.11[9] If (i) (ii) (iii)

and

is

form be respectively then

III. Hamiltonian Mechanics In this part we will derive the Hamiltonian Dynamical Systems through previous theorems and definitions as follows :Definition 3.1 [5]. A Hamiltonian function for a Hamiltonian vector field X on is a smooth function such that Definition 3. 2[4]. A Hamiltonian system is a triple is a function, called the Hamiltonian function.

where

is a Symplectic manifold and

Suppose that an almost real structure, a Liouville form and 1-form on respectively. Then we have DOI: 10.9790/5728-1303046871

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are shown by

,

and

,

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Hamiltonian mechanics with Two Almost para-Complex Structures on Symplectic Geometry

And

We substitute equation (4) in equation (8) we get

differential of

It is known that if

is a closed 2- form on

If Hamiltonian vector field

, then

is also a symplectic structure on

.

associated with Hamiltonian energy H is given by

Calculates a value

So we find that

Moreover, the differential of Hamiltonian energy is written as follows:

Suppose that a curve is an integral curve of the Hamiltonian vector field

, i.e.,

In the local coordinates, if it is considered to be we obtain

Taking the equation (14) = the equation (12)

By comparing the two sides of the equation we get the

Thus Hamilton's equations are

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Hamiltonian mechanics with Two Almost para-Complex Structures on Symplectic Geometry

Hence the triple

is shown to be a Hamiltonian mechanical system which are deduced by means of

an almost para-complex structure

and using of basis

on the distributions

IV. Conclusions The equations of Hamiltonian equations (15) with two Almost para- Complex Structures.

References [1].

[2].

[3].

[4].

[5].

[6].

[7].

[8].

[9].

M. Tekkoyun, On Para-Euler Lagrange and Para- Hamiltonian Equations, Physics Letters A, Vol. 340, (2005) 7-11 M.Tekkoyun, M. Sari., Bi-para-mechanical systems on the bi-Lagrangian manifold, Physical B-Condensed Matter, Vol. 405, Issue 10, 2390-2393, (2010).

Mehmet Tekkoyun-Murat Sari -Constrained Para complex Mechanical Systems-arXiv:0902.4120v1 [math.DS] 24 Feb 2009 Mehmet Tekkoyun-Lagrangian and Hamiltonian Mechanical Systems on Para-Quaternionic Kahler Manifolds-arXiv:1001.3490v1 [math-ph] 20 Jan 2010 Mehmet Tekkoyun -Mechanics Systems on Para-Kahlerian Manifolds of Constant J-Sectional Curvature-arXiv:0902.3569v1 [mathph] June 11, 2013 Cristian Ida, Alexandru Ionescu and Adelina Manea, A note on para-holomorphic Riemannian Einstein manifolds, arXiv:1507.01114v2 [math.DG] 19 May 2016.

V. Cruceanu, P. Fortuny, P. M. Gadea, A survey on paracomplex geometry, Rocky Mountain J. Math. 26 (1996), 83–115.

C. Ida, A. Ionescu, On a metric holomorphic connection in complex Lie groups. BSG Proceedings 21, (2014), 74–83. loring .W.Tu , S.Axler , K.A.Ribet . An Introduction to Manifolds ,Springer,

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