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]

Jacobi coordinates for two-body problem; Jacobi coordinates are ${\displaystyle {\boldsymbol {R}}={\frac {m_{1}}{M}}{\boldsymbol {x}}_{1}+{\frac {m_{2}}{M}}{\boldsymbol {x}}_{2}}$ and ${\displaystyle {\boldsymbol {r}}={\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{2}}$ with ${\displaystyle M=m_{1}+m_{2}}$.^[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]

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]

${\displaystyle {\boldsymbol {r}}_{j}={\frac {1}{m_{0j}}}\sum _{k=1}^{j}m_{k}{\boldsymbol {x}}_{k}\ -\ {\boldsymbol {x}}_{j+1}\ ,\quad j\in \{1,2,\dots ,N-1\}}$

${\displaystyle {\boldsymbol {r}}_{N}={\frac {1}{m_{0N}}}\sum _{k=1}^{N}m_{k}{\boldsymbol {x}}_{k}\ ,}$

with

${\displaystyle m_{0j}=\sum _{k=1}^{j}\ m_{k}\ .}$

The vector ${\displaystyle {\boldsymbol {r}}_{N}}$ is the center of mass of all the bodies and ${\displaystyle {\boldsymbol {r}}_{1}}$ is the relative coordinate between the particles 1 and 2:

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

This change of coordinates has associated Jacobian equal to ${\displaystyle 1}$.

If one is interested in evaluating a free energy operator in these coordinates, one obtains

${\displaystyle H_{0}=-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\,\partial _{{\boldsymbol {x}}_{j}}^{2}=-{\frac {\hbar ^{2}}{2m_{0N}}}\,\partial _{{\boldsymbol {r}}_{N}}^{2}\!-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N-1}\!\left({\frac {1}{m_{j+1}}}+{\frac {1}{m_{0j}}}\right)\partial _{{\boldsymbol {r}}_{j}}^{2}}$

In the calculations can be useful the following identity

${\displaystyle \sum _{k=j+1}^{N}{\frac {m_{k}}{m_{0k}m_{0k-1}}}={\frac {1}{m_{0j}}}-{\frac {1}{m_{0N}}}}$.

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.