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{{expand Italian|date=December 2011|Campo vettoriale hamiltoniano}}
== not inevitably contact with the dark side of society ==


In [[mathematics]] and [[physics]], a '''Hamiltonian vector field''' on a [[symplectic manifold]] is a [[vector field]], defined for any '''energy function''' or '''Hamiltonian'''. Named after the physicist and mathematician [[William Rowan Hamilton|Sir William Rowan Hamilton]], a Hamiltonian vector field is a geometric manifestation of [[Hamilton's equations]] in [[classical mechanics]]. The [[integral curve]]s of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The [[diffeomorphism]]s of a symplectic manifold arising from the [[flow (mathematics)|flow]] of a Hamiltonian vector field are known as [[canonical transformation]]s in physics and (Hamiltonian) [[symplectomorphism]]s in mathematics.
Very polite authentic forward.<br><br>'not necessarily Yeah, now what lessons have no one to listen to, on this political course, I am afraid that very few people can hear it go in and you see how the belief in question?' Song Ying as casually asked, he wondered, Xie ice age, What concerns should not be concerned about these young people seems to have become obsolete thing.<br><br>'have [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ 激安] faith will have a goal in life, and the feeling [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペードショルダーバッグ新作] of belonging, I think in the spirit of faith, like the o's in the body, as can ignore its existence, but you can [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード 財布 ゴールド] not deny that it is an integral part. 'Xie ice road.<br><br>Song should be also looked back at as strange, it seems appropriate, if the child is not deliberately compliment, [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード ショルダーバッグ] but looked like it a handsome and pious face, he suddenly asked: 'What do you as Interpol, not inevitably contact with the dark side of society, what you believe in? '<br><br>'I believe in human right, evil triumphs.' Xie ice road.<br><br>is still a handsome, but it seemed [http://www.dmwai.com/webalizer/kate-spade-8.html ケイトスペード ハンドバッグ] a bit childish expression, Laosong smiled, patted Kera shoulder and said: 'Paul
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Hamiltonian vector fields can be defined more generally on an arbitrary [[Poisson manifold]]. The [[Lie_bracket_of_vector_fields|Lie bracket]] of two Hamiltonian vector fields corresponding to functions ''f'' and ''g'' on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the
== over sin calm authentic ==
[[Poisson bracket]] of ''f'' and ''g''. 


== Definition ==
............<br><br>'I [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード 財布 値段] did something very simple, like a simple [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペードニューヨーク 財布] scam you, I use your voice to make a word called frequency: ready money, useful to me ...... then this voice message on your regular contact q , sent to your frequently used phone numbers ...... Then, very unexpectedly, actually someone contact me, oh, I did not contact that is linked to your phone. 'over sin road.<br><br>this result is that the people closest to the suspect, I am afraid we must fooled.<br><br>more crucial in the face of terror suspects face, over sin calm authentic: 'and contact me, that thing back on the simple, I told him that someone owes me money, do not pay back the money I'll cut him a leg ...... maybe my sentence credibility is [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード リボン バッグ] not high, but together you face [http://www.dmwai.com/webalizer/kate-spade-15.html kate spade マザーズバッグ] is nosebleed video, is not full of confidence right away? with you to learn, fraud also done like really, like a high success rate. '<br><br>this [http://www.dmwai.com/webalizer/kate-spade-6.html ケイトスペード マザーズバッグ] is through a simple trick, lie to the liar's associates, also brought 'ransom' 肖梦琪 really do not know head over sin is how much longer, so with the loss of tricks can
Suppose that (''M'',''ω'') is a [[symplectic manifold]]. Since the [[symplectic form]] ''ω'' is nondegenerate, it sets up a ''fiberwise-linear'' [[isomorphism]]  
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: <math>\omega:TM\to T^*M, </math>
== but also a sentence sorry ah ==


between the [[tangent bundle]] ''TM'' and the [[cotangent bundle]] ''T*M'', with the inverse
Hand Zuo Yi, even mouth saying:. [http://www.dmwai.com/webalizer/kate-spade-11.html ケイトスペードのバッグ] 'Thank you two, thank you, to again and again few days, I have embarrassed the ministries and agencies of the door.'<br><br>'Cui hall, that there are higher levels to lower levels of thanks.' Wang Shaofeng polite way.<br><br>'got to thank you ...... except thank you, but also a sentence sorry ah, a few hours ago, I have shaken.' Cui director smiled, Wang Shaofeng step ahead to [http://www.dmwai.com/webalizer/kate-spade-0.html ケイトスペード 人気 財布] grab the provincial [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペードバッグセール] secretary, to the leadership of the car door, usually down frankly sitting of the, too, not today, Minister Cui personally opened the back door, please forward two Theravada, they do not dare, or secretary smiled and pushed him hard seat.<br><br>topic nothing else, is curious to find when 璁稿钩绉� After a brief talk, and Cui Ying Italian Chamber smile face froze, leading [http://www.dmwai.com/webalizer/kate-spade-6.html kate spade マザーズバッグ] the flock, half thinking, half luck, they do not fall off the road to [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード ショルダーバッグ] thirty meters of waste wells, abandoned wells orchard from police searches than ten meters in length, attached missed twice, according to the just asking, this guy heard footsteps on the first day, and did not dare to say anything; fainted today is the result a sheep out to hit him awake, see
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: <math>\Omega:T^*M\to TM, \quad \Omega=\omega^{-1}.</math>
== Caoya Jie lead applause ==


Therefore, [[one-form]]s on a symplectic manifold ''M'' may be identified with [[vector field]]s and every [[differentiable function]] ''H'': ''M'' → '''R''' determines a unique [[vector field]] ''X<sub>H</sub>'', called the '''Hamiltonian vector field''' with the '''Hamiltonian''' ''H'', by requiring that for every vector field ''Y'' on ''M'', the identity
Under the enthusiastic [http://www.dmwai.com/webalizer/kate-spade-14.html ケイトスペード クラッチバッグ] push [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード バッグ ショルダー] his [http://www.dmwai.com/webalizer/kate-spade-11.html ケイトスペードのバッグ] way forward:. 'Well, [http://www.dmwai.com/webalizer/kate-spade-9.html バッグ ケイトスペード] please our only ladies first.'<br><br>'training is over, I want to cut ...... pounds, please supervise me here, thank you.' Li Mei bend one, Caoya Jie lead applause, and the remaining four will join in the fun, and have to Panjie encourage applause.<br><br>'Yu Feng, How about you?' History of Qing Huai ask, this promise is try to figure out a way to student mental state of mind from the target look like Li Mei, obsessed with losing weight.<br><br>'I'm not sure I want to take the CPA exam next month.' Yu Feng direct way, in this atmosphere, it seems nothing can be reservoirs.<br><br>Caoya Jie clearly see the history of the Qing Huai slightly discolored face, I startled the next sin, know why Yu brothers in the unit why frustrated, and nonsense will not say, some people may edge it? And it is clearly in the camp of departure, even Li Mei feel very embarrassed, and the history of the [http://www.dmwai.com/webalizer/kate-spade-8.html ハンドバッグ ケイトスペード] Qing Huai has been good to everyone, this is not personally attack others face it.<br><br>'Oh, ah.' History of Qing Huai little disappointed, but merely
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:<math>\mathrm{d}H(Y) = \omega(X_H,Y)\,</math>
== After the delivery of the vehicle ==


must hold.  
After the delivery of the vehicle, the owner will be waiting in the lounge, small issues that need to wait 30 minutes or so, then this time, in addition to mechanics, but also may have been exposed to it?<br><br>Yu asked the crime: 'No ordinary contact, but before the need to open the car cover normal ...... maybe he monitored areas not covered in your field area, he opened, and soon to [http://www.dmwai.com/webalizer/kate-spade-2.html ケイトスペード 財布 新作] be the hands and feet ...... except you, someone can open it? '<br><br>'hey ... right right right ...... YES YES YES YES ......' suspects excited.<br><br>I sin not say anything, and looked at him, etc. As a result, the suspect hard ineffectively, choke out: 'pick-member ...... Wang Wang Wang ...... Wang.'<br><br>'how contact, talk about the details.' I sin road.<br><br>'general [http://www.dmwai.com/webalizer/kate-spade-7.html ケイトスペード かごバッグ] customers in the door when some guests PDI ...... this is very picky, connected cars, pick-up member must be on the seat, put away the [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード] foot pad, and then [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペードバッグセール] put on the cover, and only then drove to the outside of the workshop If the vehicles waiting [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード リボン バッグ] ...... then have to line up ...... Just north of glass inside can not see. 'suspects excited authentic
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'''Note''': Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.
== psychology and physiology are very normal ==


== Examples ==
Talk about anything. 'Dong Shao Jun Shan Xiaozhao, sitting two bedside, and [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ 激安] Sun Yi thought of something, jumped up, grabbed Dong Shao Jun, chin, cheeks squeezed asked suspiciously:' I see what changes you kid '?<br><br>'what changed?' Dong Shaojun puzzled.<br><br>'is before that fried it?'<br><br>'how could not it?'<br><br>'how I feel there has changed?'<br><br>'did not change it? psychology and physiology [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード] are very normal, do not believe you put a portion of a sheet-check erectile function.'<br><br>ah Peipei Pei, SUN Yi Dong Shaojun straight to push aside, this stock since addicted excrement research, talk like drained. Sun Yi push, Wu Guangyu but it is with a smile and asked the puzzled out, straight forward:? 'Biscuits, I say a lot of your courage ah, how dare mess on more than cheap dinner Tau also to people up and down the drug, this matter [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード リボン バッグ] poke out, get off your kid for two years. '<br><br>'As police, speculation and hearsay alone judgment, contrary to your work [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード リボン バッグ] ethic.' Dong Shao Jun is not [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード バッグ ショルダー] red in the face
Suppose that ''M'' is a 2''n''-dimensional symplectic manifold. Then locally, one may choose [[canonical coordinates]] (''q''<sup>1</sup>, ..., ''q<sup>n</sup>'', ''p''<sub>1</sub>, ..., ''p<sub>n</sub>'') on ''M'', in which the symplectic form is expressed as
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:<math>\omega=\sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,</math>
 
 
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where d denotes the [[exterior derivative]] and ∧ denotes the [[exterior product]]. Then the Hamiltonian vector field with Hamiltonian ''H'' takes the form
 
 
  <li>[http://desertbreezenv.com/cgi-bin/guestbook/guestbook.cgi http://desertbreezenv.com/cgi-bin/guestbook/guestbook.cgi]</li>
:<math>\Chi_H=\left( \frac{\partial H}{\partial p_i},  
 
- \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,</math>
  <li>[http://coinauction.org/cgi-bin/auction/logout.cgi http://coinauction.org/cgi-bin/auction/logout.cgi]</li>
 
 
where ''Ω'' is a 2''n'' by 2''n'' square matrix
</ul>
 
:<math>\Omega =
\begin{bmatrix}
0 & I_n \\
-I_n & 0 \\
\end{bmatrix},</math>
and
:<math> \mathrm{d}H=\begin{bmatrix} \frac{\partial H}{\partial q^i} \\
\frac{\partial H}{\partial p_i} \end{bmatrix}.</math>
Suppose that ''M'' = '''R'''<sup>2''n''</sup> is the 2''n''-dimensional [[symplectic vector space]] with (global) canonical coordinates.  
 
* If ''H'' = ''p<sub>i</sub>'' then <math>X_H=\partial/\partial q^i; </math>  
* if ''H'' = ''q<sup>i</sup>'' then  <math>X_H=-\partial/\partial p^i; </math>
* if <math>H=1/2\sum (p_i)^2</math> then  <math>X_H=\sum p_i\partial/\partial q^i; </math>
* if <math>H=1/2\sum a_{ij} q^i q^j, a_{ij}=a_{ji} </math> then  <math>X_H=-\sum a_{ij} q_i\partial/\partial p^j. </math>
 
== Properties ==
 
* The assignment ''f'' ↦ ''X<sub>f</sub>'' is [[linear map|linear]], so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
 
* Suppose that (''q''<sup>1</sup>, ..., ''q<sup>n</sup>'', ''p''<sub>1</sub>, ..., ''p<sub>n</sub>'') are canonical coordinates on ''M'' (see above). Then a curve γ(''t'')=''(q(t),p(t))'' is an [[integral curve]] of the Hamiltonian vector field ''X<sub>H</sub>'' if and only if it is a solution of the [[Hamilton's equations]]:
 
:<math>\dot{q}^i = \frac {\partial H}{\partial p_i}</math>
:<math>\dot{p}_i = - \frac {\partial H}{\partial q^i}.</math>
 
* The Hamiltonian ''H'' is constant along the integral curves, because <math><dH, \dot{\gamma}> = \omega(X_H(\gamma),X_H(\gamma)) = 0</math>. That is, ''H''(γ(''t'')) is actually independent of ''t''. This property corresponds to the [[conservation of energy]] in [[Hamiltonian mechanics]].
 
* More generally, if two functions ''F'' and ''H'' have a zero [[Poisson bracket]] (cf. below), then ''F'' is constant along the integral curves of ''H'', and similarly, ''H'' is constant along the integral curves of ''F''. This fact is the abstract mathematical principle behind [[Noether's theorem]].
 
*The [[symplectic form]] ω is preserved by the Hamiltonian flow. Equivalently, the [[Lie derivative]] <math>\mathcal{L}_{X_H} \omega= 0</math>
 
==Poisson bracket==
The notion of a Hamiltonian vector field leads to a [[skew-symmetric]], bilinear operation on the differentiable functions on a symplectic manifold ''M'', the '''[[Poisson bracket]]''', defined by the formula
   
:<math>\{f,g\} = \omega(X_g, X_f)= dg(X_f) = \mathcal{L}_{X_f} g</math>
 
where <math>\mathcal{L}_X</math> denotes the [[Lie derivative]] along a vector field ''X''. Moreover, one can check that the following identity holds:
 
: <math> X_{\{f,g\}}= [X_f,X_g], </math> 
 
where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians ''f'' and ''g''. As a consequence (a proof at [[Poisson bracket]]), the Poisson bracket satisfies the [[Jacobi identity]]
 
: <math> \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0, </math>
 
which means that the vector space of differentiable functions on ''M'', endowed with the Poisson bracket, has the structure of a [[Lie algebra]] over '''R''', and the assignment ''f'' ↦ ''X<sub>f</sub>'' is a [[Lie algebra homomorphism]], whose [[kernel (linear algebra)|kernel]] consists of the locally constant functions (constant functions if ''M'' is connected).
 
== References ==
* {{cite book|last=Abraham|first=Ralph|authorlink=Ralph Abraham|coauthors=[[Jerrold E. Marsden|Marsden, Jerrold E.]]|title=Foundations of Mechanics|publisher=Benjamin-Cummings|location=London|year=1978|isbn=0-8053-1012-X {{Please check ISBN|reason=Check digit (X) does not correspond to calculated figure.}}}}''See section 3.2''.
* {{cite book|last=Arnol'd|first=V.I.|authorlink=Vladimir Arnold|title=Mathematical Methods of Classical Mechanics|publisher=Springer |location=Berlin etc|year=1997|isbn=0-387-96890-3}}
* {{cite book|last=Frankel|first=Theodore|title=The Geometry of Physics|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-38753-1}}
* {{cite book|last=McDuff|first=Dusa|coauthors=Salamon, D.|authorlink=Dusa McDuff|title=Introduction to Symplectic Topology|series=Oxford Mathematical Monographs|year=1998|isbn=0-19-850451-9}}
 
[[Category:Symplectic geometry]]
[[Category:Hamiltonian mechanics]]

Latest revision as of 19:35, 13 April 2014

not inevitably contact with the dark side of society

Very polite authentic forward.

'not necessarily Yeah, now what lessons have no one to listen to, on this political course, I am afraid that very few people can hear it go in and you see how the belief in question?' Song Ying as casually asked, he wondered, Xie ice age, What concerns should not be concerned about these young people seems to have become obsolete thing.

'have ケイトスペード バッグ 激安 faith will have a goal in life, and the feeling ケイトスペードショルダーバッグ新作 of belonging, I think in the spirit of faith, like the o's in the body, as can ignore its existence, but you can ケイトスペード 財布 ゴールド not deny that it is an integral part. 'Xie ice road.

Song should be also looked back at as strange, it seems appropriate, if the child is not deliberately compliment, ケイトスペード ショルダーバッグ but looked like it a handsome and pious face, he suddenly asked: 'What do you as Interpol, not inevitably contact with the dark side of society, what you believe in? '

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this result is that the people closest to the suspect, I am afraid we must fooled.

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but also a sentence sorry ah

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'Cui hall, that there are higher levels to lower levels of thanks.' Wang Shaofeng polite way.

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'Yu Feng, How about you?' History of Qing Huai ask, this promise is try to figure out a way to student mental state of mind from the target look like Li Mei, obsessed with losing weight.

'I'm not sure I want to take the CPA exam next month.' Yu Feng direct way, in this atmosphere, it seems nothing can be reservoirs.

Caoya Jie clearly see the history of the Qing Huai slightly discolored face, I startled the next sin, know why Yu brothers in the unit why frustrated, and nonsense will not say, some people may edge it? And it is clearly in the camp of departure, even Li Mei feel very embarrassed, and the history of the ハンドバッグ ケイトスペード Qing Huai has been good to everyone, this is not personally attack others face it.

'Oh, ah.' History of Qing Huai little disappointed, but merely 相关的主题文章：

After the delivery of the vehicle

After the delivery of the vehicle, the owner will be waiting in the lounge, small issues that need to wait 30 minutes or so, then this time, in addition to mechanics, but also may have been exposed to it?

Yu asked the crime: 'No ordinary contact, but before the need to open the car cover normal ...... maybe he monitored areas not covered in your field area, he opened, and soon to ケイトスペード 財布 新作 be the hands and feet ...... except you, someone can open it? '

'hey ... right right right ...... YES YES YES YES ......' suspects excited.

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psychology and physiology are very normal

Talk about anything. 'Dong Shao Jun Shan Xiaozhao, sitting two bedside, and ケイトスペード バッグ 激安 Sun Yi thought of something, jumped up, grabbed Dong Shao Jun, chin, cheeks squeezed asked suspiciously:' I see what changes you kid '?

'what changed?' Dong Shaojun puzzled.

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