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

thumb|300px|Jacobi coordinates for two-body problem; Jacobi coordinates are and with .Differential Equations David Betounes http://books.google.com/books?id=oNvFAzQXBhsC&pg=PA58 0387951407 58; Figure 2.15 2001 Springer thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. See 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,Theory and application of quantum molecular dynamics John Z. H. Zhang http://books.google.com/books?id=b8AzpUPopqQC&pg=PA104 104 9810233884 1999 World Scientific and in celestial mechanics.For example, see Capture Dynamics and Chaotic Motions in Celestial Mechanics Edward Belbruno http://books.google.com/books?id=dK-fl0KrOEIC&pg=PA9 9 0691094802 2004 Princeton University Press An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees.Classical and celestial mechanics Hildeberto Cabral, Florin Diacu http://books.google.com/books?id=q1emz4C4lYQC&pg=PA230 230 Appendix A: Canonical transformations to Jacobi coordinates 0691050228 Princeton University Press 2002 In words, the algorithm is described as follows: Let mj and mk be the masses of two bodies that are replaced by a new body of virtual mass M = mj + mk. The position coordinates xj and xk are replaced by their relative position rjk = xj − xk and by the vector to their center of mass Rjk = (mj qj + mkqk)/(mj + mk). The node in the binary tree corresponding to the virtual body has mj as its right child and mk as its left child. The order of children indicates the relative coordinate points from xk to xj. Repeat the above step for N − 1 bodies, that is, the N − 2 original bodies plus the new virtual body. For the four-body problem the result is:Advanced electromagnetism and vacuum physics Patrick Cornille 102 http://books.google.com/books?id=y8sSFTDkQ20C&pg=PA102 Partition of forces using Jacobi coordinates 9812383670 2003 World Scientific with The vector R is the center of mass of all the bodies:   References