Jacobi coordinates

Jacobian coordinates illustrated for four bodies. The virtual masses are shown in light blue. The Jacobian coordinates are r ₁ , r ₂ , r ₃ and R.

The Jacobian coordinates are a system of generalized coordinates for n-body systems in physics . They are used in particular in celestial mechanics and the consideration of polyatomic molecules and chemical reactions .

Jacobian coordinates for N particles

The algorithm for obtaining the Jacobian coordinates can be described as follows:
One considers two of the particles and calculates their center of gravity , their total mass and the relative position to each other . The two particles are now replaced by a new virtual particle with mass in place . The relative distance is thereby the first Jacobi coordinate represents: . This is repeated for the other particles as well as the new virtual particle. After such steps, the Jacobian coordinates are obtained as and from the last step. ${\displaystyle N}$ ${\displaystyle {\vec {R}}=(m_{1}{\vec {r}}_{1}+m_{2}{\vec {r}}_{2})/(m_{1}+m_{2})}$ ${\displaystyle m_{12}=m_{1}+m_{2}}$ ${\displaystyle {\vec {r}}_{12}={\vec {r}}_{1}-{\vec {r}}_{2}}$ ${\displaystyle m_{12}}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle {\vec {R}}_{1}={\vec {r}}_{12}}$ ${\displaystyle N-2}$ ${\displaystyle N-1}$ ${\displaystyle {\vec {R}}_{i}}$ ${\displaystyle {\vec {R}}_{N}={\vec {R}}}$

The Jacobi coordinates result in formulas

${\displaystyle {\vec {R}}_{1}={\vec {r}}_{1}-{\vec {r}}_{2}\;,\qquad {\vec {R}}_{j}={\frac {1}{m_{0j}}}\sum \limits _{k=1}^{j}m_{k}{\vec {r}}_{k}-{\vec {r}}_{j+1}\qquad {\text{und}}\qquad {\vec {R}}_{N}={\frac {1}{m_{0}}}\sum \limits _{k=1}^{N}m_{k}{\vec {r}}_{k}}$

With

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

Where is the total mass of the system. The last Jacobian coordinate corresponds to the center of gravity of the system. The associated speeds are calculated as ${\displaystyle m_{0N}=M}$ ${\displaystyle {\vec {R}}_{N}}$

${\displaystyle {\vec {W}}_{j}={\frac {\mathrm {d} {\vec {R}}_{j}}{\mathrm {d} t}}}$

to

${\displaystyle {\vec {W}}_{1}={\vec {v}}_{1}-{\vec {v}}_{2}\;,\qquad {\vec {W}}_{j}={\frac {1}{m_{0j}}}\sum \limits _{k=1}^{j}m_{k}{\vec {v}}_{k}-{\vec {v}}_{j+1}\qquad {\text{und}}\qquad {\vec {W}}_{N}={\frac {1}{m_{0N}}}\sum \limits _{k=1}^{N}m_{k}{\vec {v}}_{k}\;.}$

use

In celestial mechanics, the Jacobi coordinates enable the Hamilton function of a planetary system to be split into a Keplerian and an interaction part. Wisdom and Holman used these in 1991 to construct a high-speed symplectic integrator, which was widely used in the implementation called Swift by Levison and Duncan.

Individual evidence

1. ^ John ZH Zhang, Theory and application of quantum molecular dynamics , World Scientific 1999, p. 104.
2. ↑ ^{a b} Patrick Cornille: Advanced electromagnetism and vacuum physics . World Scientific, 2003, ISBN 981-238-367-0 , Partition of forces using Jacobi coordinates, p. 102 ( limited preview in Google Book search).
3. ^ J. Wisdom, MJ Holman: Symplectic maps for the n-body problem . The Astronomical Journal 102, 1991, pp. 1528-1538, doi: 10.1086 / 115978 .
4. ^ HF Levison, MJ Duncan: The long-term dynamical behavior of short-period comets . Icarus 108, 1994, pp. 18-36, doi: 10.1006 / icar.1994.1039 .