# Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Jacobi coordinates

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,^{[3]} and in celestial mechanics.^{[4]} An algorithm for generating the Jacobi coordinates for *N* bodies may be based upon binary trees.^{[5]} In words, the algorithm is described as follows:^{[5]}

Let

m_{j}andm_{k}be the masses of two bodies that are replaced by a new body of virtual massM=m_{j}+m_{k}. The position coordinatesx_{j}andx_{k}are replaced by their relative positionr_{jk}=x_{j}−x_{k}and by the vector to their center of massR_{jk}= (m_{j}q_{j}+m_{k}q_{k})/(m_{j}+m_{k}). The node in the binary tree corresponding to the virtual body hasm_{j}as its right child andm_{k}as its left child. The order of children indicates the relative coordinate points fromx_{k}tox_{j}. Repeat the above step forN− 1 bodies, that is, theN− 2 original bodies plus the new virtual body.

For the four-body problem the result is:^{[2]}

with

The vector **R** is the center of mass of all the bodies:

## References[edit]

- ↑ David Betounes (2001).
*Differential Equations*. Springer. p. 58; Figure 2.15. ISBN 0387951407. http://books.google.com/books?id=oNvFAzQXBhsC&pg=PA58. - ↑
^{a}^{b}Patrick Cornille (2003). "Partition of forces using Jacobi coordinates".*Advanced electromagnetism and vacuum physics*. World Scientific. p. 102. ISBN 9812383670. http://books.google.com/books?id=y8sSFTDkQ20C&pg=PA102. - ↑
John Z. H. Zhang (1999).
*Theory and application of quantum molecular dynamics*. World Scientific. p. 104. ISBN 9810233884. http://books.google.com/books?id=b8AzpUPopqQC&pg=PA104. - ↑
For example, see Edward Belbruno (2004).
*Capture Dynamics and Chaotic Motions in Celestial Mechanics*. Princeton University Press. p. 9. ISBN 0691094802. http://books.google.com/books?id=dK-fl0KrOEIC&pg=PA9. - ↑
^{a}^{b}Hildeberto Cabral, Florin Diacu (2002). "Appendix A: Canonical transformations to Jacobi coordinates".*Classical and celestial mechanics*. Princeton University Press. p. 230. ISBN 0691050228. http://books.google.com/books?id=q1emz4C4lYQC&pg=PA230.