Jacobi coordinates

Jacobi coordinates for two-body problem; Jacobi coordinates are $\boldsymbol{R}=\frac {m_1}{M} \boldsymbol{x}_1 + \frac {m_2}{M} \boldsymbol{x}_2$ and $\boldsymbol{r} = \boldsymbol{x}_1 - \boldsymbol{x}_2$ with $M = m_1+m_2 \$.^[1]

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.^[2]

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_kq_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 N-body problem the result is:^[2]

$\boldsymbol{r_j }= \frac{1}{m_{0j}} \sum_{k=1}^j m_k\boldsymbol {x_k} \ - \ \boldsymbol{x_{j+1}}\ , \quad (j = 1, 2, \dots, N-1)$

$\boldsymbol{r_N }= \frac{1}{m_{0N}} \sum_{k=1}^N m_k\boldsymbol {x_k} \ ,$

with

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

The vector $\boldsymbol{r_N}$ is the center of mass of all the bodies:

The result one is left with is thus a system of N-1 translationally-invariant coordinates $\boldsymbol{r_1}, \dots, \boldsymbol{r_{N-1}}$ and a center of mass coordinate $\boldsymbol{r_N}$, from iteratively treats and reducing two-body systems within the many-body system.

References

1. David Betounes (2001). Differential Equations. Springer. p. 58; Figure 2.15. ISBN 0-387-95140-7.
2. Patrick Cornille (2003). "Partition of forces using Jacobi coordinates". Advanced electromagnetism and vacuum physics. World Scientific. p. 102. ISBN 981-238-367-0.
3. John Z. H. Zhang (1999). Theory and application of quantum molecular dynamics. World Scientific. p. 104. ISBN 981-02-3388-4.
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.
5. 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.