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# 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} and *m*_{k} be the masses of two bodies that are replaced by a new body of virtual mass *M* = *m*_{j} + *m*_{k}. The position coordinates **x**_{j} and **x**_{k} are replaced by their relative position **r**_{jk} = **x**_{j} − **x**_{k} and by the vector to their center of mass **R**_{jk} = (*m*_{j} *q*_{j} + *m*_{k}*q*_{k})/(*m*_{j} + *m*_{k}). The node in the binary tree corresponding to the virtual body has *m*_{j} as its right child and *m*_{k} as its left child. The order of children indicates the relative coordinate points from **x**_{k} to **x**_{j}. 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:[2]

\( \boldsymbol{r_1 = x_1 - x_2} \ , \)

\( \boldsymbol{r_j }= \frac{1}{m_{0j}} \sum_{k=1}^j m_k\boldsymbol {x_k} \ - \ \boldsymbol{x_{j+1}}\ , \)

with

\( m_{0j} = \sum_{k=1}^j \ m_k \ . \)

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

\( \boldsymbol R = \frac{1}{m_0} \sum_{k=1}^N\ m_k \boldsymbol{x_k} \ ; m_0 = \sum_{k=1}^N\ m_k \ . \)

The result one is left with is thus a system of N translationally-invariant coordinates and a reduced mass, from iteratively treats and reducing two-body systems within the many-body system.

References

David Betounes (2001). Differential Equations. Springer. p. 58; Figure 2.15. ISBN 0-387-95140-7.

Patrick Cornille (2003). "Partition of forces using Jacobi coordinates". Advanced electromagnetism and vacuum physics. World Scientific. p. 102. ISBN 981-238-367-0.

John Z. H. Zhang (1999). Theory and application of quantum molecular dynamics. World Scientific. p. 104. ISBN 981-02-3388-4.

For example, see Edward Belbruno (2004). Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press. p. 9. ISBN 0-691-09480-2.

Hildeberto Cabral, Florin Diacu (2002). "Appendix A: Canonical transformations to Jacobi coordinates". Classical and celestial mechanics. Princeton University Press. p. 230. ISBN 0-691-05022-8.

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