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[[File:Two-body Jacobi coordinates.JPG|thumb|300px|Jacobi coordinates for two-body problem; Jacobi coordinates are <math>\boldsymbol{R}=\frac {m_1}{M} \boldsymbol{x}_1 + \frac {m_2}{M} \boldsymbol{x}_2 </math> and <math>\boldsymbol{r} = \boldsymbol{x}_1 - \boldsymbol{x}_2 </math> with <math>M = m_1+m_2 \ </math>.<ref name=Betounes>{{cite book |title=Differential Equations |author=David Betounes |url=http://books.google.com/books?id=oNvFAzQXBhsC&pg=PA58 |isbn=0-387-95140-7 |page=58; Figure 2.15 |year=2001 |publisher=Springer}}</ref> ]]
If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>
[[File:Four-body Jacobi coordinates.JPG|thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, '''r'''<sub>3</sub> and the center of mass '''R'''. See Cornille.<ref name=Cornille/>]]
In the theory of many-particle systems, '''Jacobi coordinates''' often are used to simplify the mathematical formulation.  These coordinates are particularly common in treating polyatomic molecules and chemical reactions,<ref name= Zhang>


{{cite book |title=Theory and application of quantum molecular dynamics |author= John Z. H. Zhang |url=http://books.google.com/books?id=b8AzpUPopqQC&pg=PA104 |page=104 |isbn=981-02-3388-4 |year=1999 |publisher=World Scientific}}
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</ref> and in celestial mechanics.<ref name=Belbruno>
 
For example, see {{cite book |title=Capture Dynamics and Chaotic Motions in Celestial Mechanics |author= Edward Belbruno |url=http://books.google.com/books?id=dK-fl0KrOEIC&pg=PA9 |page=9 |isbn=0-691-09480-2 |year=2004 |publisher=Princeton University Press}}
 
</ref> An algorithm for generating the Jacobi coordinates for ''N'' bodies may be based upon [[binary trees]].<ref name=Cabral>
 
{{cite book |title=Classical and celestial mechanics |author= Hildeberto Cabral, Florin Diacu |url=http://books.google.com/books?id=q1emz4C4lYQC&pg=PA230 |page=230 |chapter=Appendix A: Canonical transformations to Jacobi coordinates |isbn=0-691-05022-8 |publisher=Princeton University Press |year=2002}}
 
</ref> In words, the algorithm is described as follows:<ref name=Cabral/>
<blockquote>Let ''m''<sub>''j''</sub> and ''m''<sub>''k''</sub> be the masses of two bodies that are replaced by a new body of virtual mass ''M'' = ''m''<sub>''j''</sub> + ''m''<sub>''k''</sub>. The position coordinates '''x'''<sub>''j''</sub> and '''x'''<sub>''k''</sub> are replaced by their relative position  '''r'''<sub>''jk''</sub> = '''x'''<sub>''j''</sub>&nbsp;&minus;&nbsp;'''x'''<sub>''k''</sub> and by the vector to their center of mass '''R'''<sub>''jk''</sub> = (''m''<sub>''j''</sub> ''q''<sub>''j''</sub> + ''m''<sub>''k''</sub>''q''<sub>''k''</sub>)/(''m''<sub>''j''</sub> +  ''m''<sub>''k''</sub>). The node in the binary tree corresponding to the virtual body has ''m''<sub>''j''</sub> as its right child and ''m''<sub>''k''</sub> as its left child. The order of children indicates the relative coordinate points from '''x'''<sub>''k''</sub> to '''x'''<sub>''j''</sub>. Repeat the above step for ''N''&nbsp;&minus;&nbsp;1 bodies, that is, the ''N''&nbsp;&minus;&nbsp;2 original bodies plus the new virtual body. </blockquote>
 
For the four-body problem the result is:<ref name=Cornille>{{cite book |title=Advanced electromagnetism and vacuum physics |author=Patrick Cornille |page=102 |url=http://books.google.com/books?id=y8sSFTDkQ20C&pg=PA102 |chapter=Partition of forces using Jacobi coordinates  |isbn=981-238-367-0 |year=2003 |publisher=World Scientific}}</ref>
:<math>\boldsymbol{r_1 = x_1 - x_2} \ , </math>
:<math>\boldsymbol{r_j }= \frac{1}{m_{0j}} \sum_{k=1}^j m_k\boldsymbol {x_k} \ - \ \boldsymbol{x_{j+1}}\ , </math>
 
with
 
:<math>m_{0j} = \sum_{k=1}^j \ m_k \ . </math>
 
The vector '''R''' is the center of mass of all the bodies:
 
:<math>\boldsymbol R = \frac{1}{m_0} \sum_{k=1}^N\ m_k \boldsymbol{x_k} \ ; </math> &ensp; <math>m_0 = \sum_{k=1}^N\ m_k \ . </math>
 
 
==References==
<references/>
 
[[Category:Molecular vibration]]
[[Category:Molecular geometry]]
[[Category:Chemical reactions]]
[[Category:Hamiltonian mechanics]]
[[Category:Lagrangian mechanics]]
[[Category:Coordinate systems]]
[[Category:Orbits]]

Latest revision as of 14:55, 1 November 2014

If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts.

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