138 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

Hamiltonian System Mechanics on (2,0)-Jet Bundles 

Ibrahim Yousif  I. Abad alrhmana*, Yonnis A. Abu Aashab, Abdulaziz B. M. 

Hamedc 

a,bDepartment of Math and Physics - Faculty of Education, West Kordufan University, Alnhoud City , Sudan 
cDepartment of Mathematics and Statistics, Faculty of Science, Yobe State University , Damaturu, Nigeria 

aEmail: iyibrahimi@gmail.com 
bEmail: sabaya11@gmail.com 

cEmail: aziz.hamed12@gmail.com 

 

Abstract 

The goal of this paper is to present Hamiltonian system Mechanics on (2,0)-jet bundles . In conclusion, some 

differential geometrical and physical results on the related  mechanic systems have been given. 

 Keywords: Jet bundle; holomorphic bundle; complex , Hamiltonian Dynamics. 

1. Introduction  

It is well known that the dynamics of Lagrangian formalisms is characterized by a suitable vector field defined 
on the tangent and cotangent bundles which are phase-spaces of velocities and momentum of a given 

configuration manifold. If 𝓜𝓜 is an m-dimensional configuration manifold [6]. If 𝐇𝐇:𝐓𝐓∗𝓜𝓜⟶𝐑𝐑  is a regular 

Hamiltonian function then there is a unique vector field 𝒁𝒁𝑯𝑯 on cotangent bundle 𝐓𝐓∗𝓜𝓜 such that dynamical 

equations 

𝐢𝐢𝒁𝒁𝐇𝐇𝛟𝛟 = 𝐝𝐝𝐇𝐇                                                        (𝟏𝟏) 

where 𝛟𝛟 is the symplectic form and 𝑯𝑯 stands for Hamiltonian function. The paths of the Hamiltonian vector 

field 𝒁𝒁𝐇𝐇 are the solutions of the Hamiltonian equations shown by 

𝒅𝒅𝒒𝒒𝒊𝒊

𝒅𝒅𝒅𝒅 =
𝝏𝝏𝑯𝑯
𝝏𝝏𝒑𝒑𝒊𝒊

      ,
𝒅𝒅𝒑𝒑𝒊𝒊
𝒅𝒅𝒅𝒅 =

𝝏𝝏𝑯𝑯
𝝏𝝏𝒒𝒒𝒊𝒊                                     (𝟐𝟐) 

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 34, No  1, pp 138-143 

139 
 

where 𝒒𝒒𝒊𝒊  and (𝒒𝒒𝒊𝒊, 𝒑𝒑𝒊𝒊),𝟏𝟏 ≤  𝒊𝒊 ≤  𝒎𝒎, are coordinates of 𝓜𝓜  and 𝐓𝐓∗𝓜𝓜 . The triple (𝐓𝐓∗𝓜𝓜,𝝓𝝓,𝑯𝑯) , is called 

Hamiltonian system on the cotangent bundle 𝐓𝐓∗𝓜𝓜 with symplectic form 𝝓𝝓. Let 𝐓𝐓∗𝓜𝓜 be symplectic manifold 

with closed symplectic form 𝝓𝝓. In this paper related to Hamiltonian equations Hamiltonian system Mechanics 

on (2,0)-jet bundles. 

2. The geometry of holomorphic 𝑱𝑱(𝟐𝟐,𝟎𝟎)𝓜𝓜 bundles  

2.1 Definition  

Let 𝓜𝓜 be a complex manifold, 𝐓𝐓𝐜𝐜𝓜𝓜=  �́�𝐓𝓜𝓜⨁ �́́�𝐓𝓜𝓜, the complexified  tangent bundle of (𝟏𝟏,𝟎𝟎)- and of (𝟎𝟎,𝟏𝟏)-

type vectors, respectively. If (𝐳𝐳𝐢𝐢)𝐢𝐢=𝟏𝟏;𝐧𝐧����� are complex coordinates, then 𝐓𝐓𝐳𝐳́  𝓜𝓜 is spanned by � 𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢
�
𝐢𝐢=𝟏𝟏;𝐧𝐧�����

 and �́́�𝐓𝐳𝐳𝓜𝓜 is 

spanned by � 𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢
�
𝐢𝐢=𝟏𝟏;𝐧𝐧�����

 moreover �́�𝐓𝓜𝓜 is a holomorphic vector bundle 

     let 𝐙𝐙 = (𝐳𝐳𝐢𝐢 ,𝐗𝐗𝐢𝐢 = 𝛈𝛈𝐢𝐢(𝟏𝟏) = 𝐝𝐝𝐳𝐳𝐢𝐢

𝐝𝐝𝐝𝐝
,𝐘𝐘𝐢𝐢 = 𝛈𝛈𝐢𝐢(𝟐𝟐) = 𝐝𝐝𝟐𝟐𝐳𝐳𝐢𝐢

𝐝𝐝𝐝𝐝𝟐𝟐
)  be local complex coordinates in the chart (𝐔𝐔;  𝛙𝛙)  from 

𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜; 

we shall the following notations [1]. 

𝒁𝒁 = �𝒛𝒛𝒊𝒊,𝒙𝒙𝒊𝒊 = 𝜼𝜼𝒊𝒊(𝟏𝟏),𝒚𝒚𝒊𝒊 = 𝜼𝜼𝒊𝒊(𝟐𝟐)� = (𝒛𝒛𝒊𝒊,𝑿𝑿𝒊𝒊,𝒀𝒀𝒊𝒊)                                  (𝟑𝟑) 

2.2 Theorem 

A local basis in �́�𝐓𝐳𝐳�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�is � 𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢

, 𝛛𝛛

𝛛𝛛𝐱𝐱𝐢𝐢𝐢𝐢
, 𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢
�
𝐢𝐢=𝟏𝟏;𝐧𝐧�����

and in �́́�𝐓𝐳𝐳�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜� theirs conjugates � 𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢�

, 𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

, 𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢
�
𝐢𝐢=𝟏𝟏;𝐧𝐧�����

: Due to 

holomorphic changes on 𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜, that is all of   𝛛𝛛�́�𝐳
𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣�
, 𝛛𝛛�́�𝐱

𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣�
, 𝛛𝛛�́�𝐲

𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣�
, 𝛛𝛛�́�𝐱

𝐢𝐢

𝛛𝛛𝐱𝐱𝐣𝐣�
, 𝛛𝛛�́�𝐲

𝐢𝐢

𝛛𝛛𝐱𝐱𝐣𝐣�
, 𝛛𝛛�́�𝐲

𝐢𝐢

𝛛𝛛𝐲𝐲𝐣𝐣�
 j are vanishing, and also theirs 

conjugates, it follows that local bases from �́�𝐓𝐳𝐳�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜� change w.r.t. the transformations  by the rules: 

𝛛𝛛
𝛛𝛛𝐳𝐳𝐣𝐣 =

𝛛𝛛�́�𝐳𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
𝛛𝛛
𝛛𝛛�́�𝐳𝐢𝐢 +

𝛛𝛛�́�𝐱𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
𝛛𝛛
𝛛𝛛�́�𝐱𝐢𝐢 +

𝛛𝛛�́�𝐲𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣

𝛛𝛛
𝛛𝛛�́�𝐲𝐢𝐢
�� 

𝛛𝛛
𝛛𝛛𝐱𝐱𝐣𝐣 =

𝛛𝛛�́�𝐱𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
𝛛𝛛
𝛛𝛛�́�𝐱𝐢𝐢 +

𝛛𝛛�́�𝐲𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
𝛛𝛛
𝛛𝛛�́�𝐲𝐢𝐢 

𝛛𝛛
𝛛𝛛𝐱𝐱𝐣𝐣 =

𝛛𝛛�́�𝐲𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
𝛛𝛛
𝛛𝛛�́�𝐲𝐢𝐢                                         (𝟒𝟒) 

Infer that  𝛛𝛛�́�𝐳
𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
= 𝛛𝛛�́�𝐱𝐢𝐢

𝛛𝛛𝐱𝐱𝐣𝐣
= 𝛛𝛛�́�𝐲𝐢𝐢

𝛛𝛛𝐲𝐲𝐣𝐣
  but in change 𝛛𝛛�́�𝐳

𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
= 𝛛𝛛�́�𝐱𝐢𝐢

𝛛𝛛𝐱𝐱𝐣𝐣
 contain the second order derivatives of  �́�𝐳𝐢𝐢. while 𝛛𝛛�́�𝐱

𝐢𝐢

𝛛𝛛𝐳𝐳𝐣𝐣
 contains 

even the 3-th derivatives of �́�𝐳𝐢𝐢 . 

2.3 Theorem  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 34, No  1, pp 138-143 

140 
 

On 𝑻𝑻𝒄𝒄�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜� the natural complex structure 𝑱𝑱𝟐𝟐 = −𝑰𝑰 acts as follows: 

 𝐽𝐽 � 𝜕𝜕
𝜕𝜕𝑧𝑧𝑗𝑗
� = 𝒊𝒊 𝜕𝜕

𝜕𝜕𝑧𝑧𝑗𝑗
        ,   𝐽𝐽 � 𝜕𝜕

𝜕𝜕𝑥𝑥𝑗𝑗
� = 𝒊𝒊 𝜕𝜕

𝜕𝜕𝑥𝑥𝑗𝑗
    ,    𝐽𝐽 � 𝜕𝜕

𝜕𝜕𝑦𝑦𝑗𝑗
� = 𝒊𝒊 𝜕𝜕

𝜕𝜕𝑦𝑦𝑗𝑗
 

𝑱𝑱 �
𝝏𝝏
𝝏𝝏𝒛𝒛�𝒋𝒋� = −𝒊𝒊

𝝏𝝏
𝝏𝝏𝒛𝒛𝒋𝒋         ,    𝑱𝑱�

𝝏𝝏
𝝏𝝏𝒙𝒙�𝒋𝒋� = −𝒊𝒊

𝝏𝝏
𝝏𝝏𝒙𝒙𝒋𝒋     ,   𝑱𝑱�

𝝏𝝏
𝝏𝝏𝒚𝒚�𝒋𝒋� = −𝒊𝒊

𝝏𝝏
𝝏𝝏𝒚𝒚𝒋𝒋      (𝟓𝟓) 

The dual endomorphism the cotangent space 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�at any point p of manifold 𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜 satifies 𝐉𝐉𝟐𝟐∗ = −𝐈𝐈 

and is defined by  

𝑱𝑱∗(𝒅𝒅𝒛𝒛𝒋𝒋) = 𝒊𝒊𝒅𝒅𝒛𝒛𝒋𝒋         , 𝑱𝑱∗(𝒅𝒅𝒙𝒙𝒋𝒋) = 𝒊𝒊𝒅𝒅𝒙𝒙𝒋𝒋     ,    𝑱𝑱∗(𝒅𝒅𝒚𝒚𝒋𝒋) = 𝒊𝒊𝒅𝒅𝒚𝒚𝒋𝒋 

𝑱𝑱∗(𝒅𝒅𝒛𝒛�𝒋𝒋) = −𝒊𝒊𝒅𝒅𝒛𝒛�𝒋𝒋        , 𝑱𝑱∗(𝒅𝒅𝒙𝒙�𝒋𝒋) = −𝒊𝒊𝒅𝒅𝒙𝒙�𝒋𝒋     ,    𝑱𝑱∗(𝒚𝒚�𝒋𝒋) = −𝒊𝒊𝒅𝒅𝒚𝒚�𝒋𝒋   (𝟔𝟔) 

3. Hamiltonian Dynamical Systems 

In this section, we obtain complex Hamiltonian equations for classical mechanics structured on momentum 

space 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜� that is 2m- dimensional cotangent bundle of an m-dimensional configuration manifold 𝓜𝓜.  

   Let 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�be the momentum space and  𝐙𝐙 = �𝐳𝐳𝐢𝐢 ,𝐱𝐱𝐢𝐢 = 𝛈𝛈𝐢𝐢(𝟏𝟏),𝐲𝐲𝐢𝐢 = 𝛈𝛈𝐢𝐢(𝟐𝟐)� = (𝐳𝐳𝐢𝐢 ,𝐗𝐗𝐢𝐢,𝐘𝐘𝐢𝐢),   𝟏𝟏 ≤ 𝐢𝐢 ≤ 𝐦𝐦 its 

complex coordinates 

Let almost complex structure 𝐉𝐉∗ and Liouville form 𝛌𝛌 give by 

𝛚𝛚 =
𝟏𝟏
𝟐𝟐

(𝐳𝐳𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐳𝐳�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 + 𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐱𝐱�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 + 𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢 + 𝐲𝐲�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢)                    (𝟕𝟕) 

𝛌𝛌 = (𝐉𝐉∗𝛚𝛚) =
𝟏𝟏
𝟐𝟐𝐉𝐉

∗(𝐳𝐳𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐳𝐳�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 + 𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐱𝐱�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 + 𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢 + 𝐲𝐲�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢) 

or  

𝛌𝛌 = (𝐉𝐉∗𝛚𝛚) =
𝟏𝟏
𝟐𝟐

(𝐳𝐳𝐢𝐢𝐉𝐉∗(𝐝𝐝𝐳𝐳�𝐢𝐢) + 𝐳𝐳�𝐢𝐢𝐉𝐉∗(𝐝𝐝𝐳𝐳𝐢𝐢) + 𝐱𝐱𝐢𝐢𝐉𝐉∗(𝐝𝐝𝐱𝐱�𝐢𝐢) + 𝐱𝐱�𝐢𝐢𝐉𝐉∗(𝐝𝐝𝐱𝐱𝐢𝐢) + 𝐲𝐲𝐢𝐢𝐉𝐉∗(𝐝𝐝𝐲𝐲�𝐢𝐢) + 𝐲𝐲�𝐢𝐢𝐉𝐉∗(𝐝𝐝𝐲𝐲𝐢𝐢) 

𝛌𝛌 =
𝟏𝟏
𝟐𝟐

(−𝐢𝐢𝐳𝐳𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐢𝐢𝐳𝐳�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 − 𝐢𝐢𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐢𝐢𝐱𝐱�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 − 𝐢𝐢𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢 + 𝐢𝐢𝐲𝐲�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢) 

or  

𝛌𝛌 =
𝟏𝟏
𝟐𝟐𝐢𝐢

(−𝐳𝐳𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐳𝐳�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 − 𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐱𝐱�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 − 𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢 + 𝐲𝐲�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢))                 (𝟖𝟖) 

𝐬𝐬𝐮𝐮𝐜𝐜𝐜𝐜 𝐭𝐭𝐜𝐜𝐭𝐭𝐭𝐭 𝛚𝛚 complex 1-form on 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�.  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 34, No  1, pp 138-143 

141 
 

      If    𝛟𝛟 = −𝐝𝐝𝛌𝛌    is  closed  Kahlerian form, then 𝛟𝛟 is also a symplectic structure on 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�. 

𝛟𝛟 = −𝐝𝐝𝛌𝛌= −𝐝𝐝(
𝟏𝟏
𝟐𝟐𝐢𝐢

(−𝐳𝐳𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐳𝐳�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 − 𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐱𝐱�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 − 𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢 + 𝐲𝐲�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢)) 

𝛟𝛟 = −𝐝𝐝𝛌𝛌 = −𝐢𝐢𝐝𝐝(−𝐳𝐳𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐳𝐳�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢)− 𝐢𝐢𝐝𝐝(−𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐱𝐱�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢)− 𝐢𝐢𝐝𝐝(−𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢 + 𝐲𝐲�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢) 

𝛟𝛟 = −𝐝𝐝𝛌𝛌 = −𝐢𝐢(𝐝𝐝𝐳𝐳�𝐢𝐢⋀𝐝𝐝𝐳𝐳𝐢𝐢)− 𝐢𝐢(𝐝𝐝𝐱𝐱�𝐢𝐢⋀𝐝𝐝𝐱𝐱𝐢𝐢)− 𝐢𝐢(𝐝𝐝𝐲𝐲�𝐢𝐢⋀𝐝𝐝𝐲𝐲𝐢𝐢)                         (𝟗𝟗) 

Let 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜� be momentum space with closed Kaehlerian form 𝛟𝛟 .Consider that Hamiltonian vector  field 

 𝐙𝐙𝐇𝐇 associated Hamiltonian energy 𝐇𝐇 is given by 

𝐙𝐙 = 𝐙𝐙𝐇𝐇 = 𝐙𝐙𝐢𝐢
𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 + 𝐙𝐙�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

+ 𝐗𝐗𝐢𝐢 𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 + 𝐗𝐗�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

+ 𝐘𝐘𝐢𝐢
𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 + 𝐘𝐘�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

          𝟏𝟏 ≤ 𝐢𝐢 ≤ 𝐦𝐦 

From the isomorphism given in, we calculate by 

𝐢𝐢𝐙𝐙𝐇𝐇 𝛟𝛟 = 𝐢𝐢𝐙𝐙𝐇𝐇(−𝐝𝐝𝛌𝛌)

= �𝐙𝐙𝐢𝐢
𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 + 𝐙𝐙�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

+ 𝐗𝐗𝐢𝐢 𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 + 𝐗𝐗�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

+ 𝐘𝐘𝐢𝐢
𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 + 𝐘𝐘�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

� �−𝐢𝐢(𝐝𝐝𝐳𝐳�𝐢𝐢⋀𝐝𝐝𝐳𝐳𝐢𝐢)− 𝐢𝐢(𝐝𝐝𝐱𝐱�𝐢𝐢⋀𝐝𝐝𝐱𝐱𝐢𝐢)

− 𝐢𝐢(𝐝𝐝𝐲𝐲�𝐢𝐢⋀𝐝𝐝𝐲𝐲𝐢𝐢)� 

𝐢𝐢𝐙𝐙𝐇𝐇  𝛟𝛟 = 𝐢𝐢𝐙𝐙�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 + 𝐢𝐢𝐙𝐙𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐢𝐢𝐗𝐗�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 + 𝐢𝐢𝐗𝐗𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐢𝐢𝐘𝐘�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢 + 𝐢𝐢𝐘𝐘𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢                     (𝟏𝟏𝟎𝟎) 

On the other hand, we obtain as 

𝐝𝐝𝐇𝐇 =
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳𝐢𝐢 𝐝𝐝𝐳𝐳

𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳�𝐢𝐢

𝐝𝐝𝐳𝐳�𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱𝐢𝐢 𝐝𝐝𝐱𝐱

𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱�𝐢𝐢

𝐝𝐝𝐱𝐱�𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲𝐢𝐢𝐝𝐝𝐲𝐲

𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲�𝐢𝐢

𝐝𝐝𝐲𝐲�𝐢𝐢            (𝟏𝟏𝟏𝟏) 

the differential of Hamiltonian energy. From  𝐢𝐢𝐙𝐙𝐇𝐇𝛟𝛟 =  𝐝𝐝𝐇𝐇, we find as 

𝐢𝐢𝐙𝐙𝐇𝐇𝛟𝛟 =  𝐝𝐝𝐇𝐇 = 𝐙𝐙�𝐢𝐢𝐝𝐝𝐳𝐳𝐢𝐢 + 𝐢𝐢𝐙𝐙𝐢𝐢𝐝𝐝𝐳𝐳�𝐢𝐢 + 𝐢𝐢𝐗𝐗�𝐢𝐢𝐝𝐝𝐱𝐱𝐢𝐢 + 𝐢𝐢𝐗𝐗𝐢𝐢𝐝𝐝𝐱𝐱�𝐢𝐢 + 𝐢𝐢𝐘𝐘�𝐢𝐢𝐝𝐝𝐲𝐲𝐢𝐢 + 𝐢𝐢𝐘𝐘𝐢𝐢𝐝𝐝𝐲𝐲�𝐢𝐢

=
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳𝐢𝐢 𝐝𝐝𝐳𝐳

𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳�𝐢𝐢

𝐝𝐝𝐳𝐳�𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱𝐢𝐢𝐝𝐝𝐱𝐱

𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱�𝐢𝐢

𝐝𝐝𝐱𝐱�𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲𝐢𝐢 𝐝𝐝𝐲𝐲

𝐢𝐢 +
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲�𝐢𝐢

𝐝𝐝𝐲𝐲�𝐢𝐢     (𝟏𝟏𝟐𝟐) 

Or 

𝐙𝐙𝐇𝐇 =
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

+
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

+
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

       (𝟏𝟏𝟑𝟑)  

𝟏𝟏 ≤ 𝐢𝐢 ≤ 𝐦𝐦 

Let {𝐙𝐙 = (𝐳𝐳𝐢𝐢 ,𝐳𝐳�𝐢𝐢 ,𝐱𝐱𝐢𝐢 ,𝐱𝐱�𝐢𝐢 ,𝐲𝐲𝐢𝐢 ,𝐲𝐲�𝐢𝐢) ∶ 𝟏𝟏 ≤ 𝐢𝐢 ≤ 𝐦𝐦}  be the complex coordinates in the momentum space. Suppose that 

the curve 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 34, No  1, pp 138-143 

142 
 

𝛂𝛂: 𝐈𝐈 ⊂  𝐂𝐂 → 𝐓𝐓𝓜𝓜 

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

𝐙𝐙𝐇𝐇�𝛂𝛂(𝐭𝐭)�= �̇�𝛂     , 𝐭𝐭 ∈  𝐈𝐈. 

In the local coordinates we have 

𝛂𝛂(𝐭𝐭) =  �𝐳𝐳𝐢𝐢(𝐭𝐭),𝐳𝐳�𝐢𝐢(𝐭𝐭),𝐱𝐱𝐢𝐢(𝐭𝐭),𝐱𝐱�𝐢𝐢(𝐭𝐭),𝐲𝐲𝐢𝐢(𝐭𝐭),𝐲𝐲�𝐢𝐢(𝐭𝐭)�, 

And  

�̇�𝛂(𝐭𝐭) =
𝐝𝐝𝐳𝐳𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 +

𝐝𝐝𝐳𝐳�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

+
𝐝𝐝𝐱𝐱𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 +

𝐝𝐝𝐱𝐱�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

+
𝐝𝐝𝐲𝐲𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 +

𝐝𝐝𝐲𝐲�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

               (𝟏𝟏𝟒𝟒)  

   the Hamiltonian vector field on momentum space 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜� with closed Kaehlerian form 𝛟𝛟.Now, from 

𝐙𝐙𝐇𝐇�𝛂𝛂(𝐭𝐭)�= �̇�𝛂,  

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

+
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

+
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

=
𝐝𝐝𝐳𝐳𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 +

𝐝𝐝𝐳𝐳�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

+
𝐝𝐝𝐱𝐱𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 +

𝐝𝐝𝐱𝐱�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

+
𝐝𝐝𝐲𝐲𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 +

𝐝𝐝𝐲𝐲�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

 

then we infer the following equations 

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢 =

𝐝𝐝𝐳𝐳𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐳𝐳𝐢𝐢     →

𝐝𝐝𝐳𝐳𝐢𝐢

𝐝𝐝𝐭𝐭 =
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳�𝐢𝐢

 

−
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳𝐢𝐢

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

=
𝐝𝐝𝐳𝐳�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐳𝐳�𝐢𝐢

→
𝐝𝐝𝐳𝐳�𝐢𝐢
𝐝𝐝𝐭𝐭 = −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐳𝐳𝐢𝐢 

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢 =

𝐝𝐝𝐱𝐱𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐱𝐱𝐢𝐢  →

𝐝𝐝𝐱𝐱𝐢𝐢

𝐝𝐝𝐭𝐭 =
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱�𝐢𝐢

 

−
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱𝐢𝐢

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

=
𝐝𝐝𝐱𝐱�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐱𝐱�𝐢𝐢

→
𝐝𝐝𝐱𝐱�𝐢𝐢
𝐝𝐝𝐭𝐭 = −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐱𝐱𝐢𝐢 

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲�𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 =

𝐝𝐝𝐲𝐲𝐢𝐢

𝐝𝐝𝐭𝐭
𝛛𝛛
𝛛𝛛𝐲𝐲𝐢𝐢 →

𝐝𝐝𝐲𝐲𝐢𝐢

𝐝𝐝𝐭𝐭 =
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲�𝐢𝐢

 

−
𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲𝐢𝐢

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

=
𝐝𝐝𝐲𝐲�𝐢𝐢
𝐝𝐝𝐭𝐭

𝛛𝛛
𝛛𝛛𝐲𝐲�𝐢𝐢

→
𝐝𝐝𝐲𝐲�𝐢𝐢
𝐝𝐝𝐭𝐭 = −

𝟏𝟏
𝐢𝐢
𝛛𝛛𝐇𝐇
𝛛𝛛𝐲𝐲𝐢𝐢 

which are called complex Hamiltonian equations on momentum space 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�. we have the complex 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 34, No  1, pp 138-143 

143 
 

Hamiltonian equations given by 

dz�i

dt =
1
i
∂H
∂zi

      ,           
dzi
dt = −

1
i
∂H
∂z�i     

d𝑥𝑥i

dt =
1
i
∂H
∂�̅�𝑥i

               ,      
d�̅�𝑥i
dt = −

1
i
∂H
∂𝑥𝑥i 

d𝑦𝑦i

dt =
1
i
∂H
∂𝑦𝑦�i

               ,      
d𝑦𝑦�i
dt = −

1
i
∂H
∂𝑦𝑦i                                    (15) 

Thus, by complex Hamiltonian equations ,we may call the equations obtained in (15) on 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�. Then the 

quartet (𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�,𝛟𝛟𝐇𝐇 ,𝐙𝐙𝐇𝐇) is named mechanical system with  

4. Conclusions 

The solutions of the Hamiltonian equations determined by (15) on the mechanical system 

(𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�,𝛟𝛟𝐋𝐋 ,𝐙𝐙𝐇𝐇)are the paths of vector field 𝐙𝐙𝐇𝐇 on 𝐓𝐓𝐜𝐜∗�𝐉𝐉(𝟐𝟐,𝟎𝟎)𝓜𝓜�. 

References  

[1] Violeta, Zalutchi, (2010),The geometry of (2; 0)-jet bundles , University "Transilvania of Brasov, of 

Brasov,Faculty of Mathematics and Informatics,311-320. 

[2] loring .W.Tu , S.Axler , K.A.Ribet (2009) , An Introduction to Manifolds ,Springer 

[3] A. Manea, (2010),A decomposition of the bundle of second order jets on a complex manifold, Analle St. 

Univ. "Al. I. Cuza", Iasi, LVI, 151-162. 

[4] W. Stoll, P.-M. Wong, (2000) ,On holomorphic jet bunldes, preprint, arxiv:math/ 0003226v1/. 

[5] Liz Lane-Harvard, Melissa Swager (2010)  -Hamiltonian Systems and Chaos Overview- 

[6] Mehmet Tekkoyun , (2009) ,Lagrangian and Hamiltonian Dynamics on Para-Kahlerian Space Form 

rXiv:0902.4522v1 [math.DS] 26.