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 DOI: 10.9790/5728-1303046871 www.iosrjournals.org 68 | Page 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 Para- Complex. 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                ,    , be para complex manifold .In local coordinates system on a neighborhood V of  .We define the vector fields by