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

molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...

s and

chemical reaction A chemical reaction is a process that leads to the chemistry, chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an Gibbs free energy, ...

s, and in

celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

. For example, see An algorithm for generating the Jacobi coordinates for ''N'' bodies may be based upon

binary trees In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theory ...

. In words, the algorithm may be described as follows:

We choose two of the ''N'' bodies with position coordinates x_''j'' and x_''k'' and we replace them with one virtual body at their centre of mass. We define the relative position coordinate r_''jk'' = x_''j'' − x_''k''. We then repeat the process with the ''N'' − 1 bodies consisting of the other ''N'' − 2 plus the new virtual body. After ''N'' − 1 such steps we will have Jacobi coordinates consisting of the relative positions and one coordinate giving the position of the last defined centre of mass.

For the ''N''-body problem the result is: :

\boldsymbol_j= \frac \sum_^j m_k\boldsymbol _k \ - \ \boldsymbol_\ , \quad j \in \

\boldsymbol_N= \frac \sum_^N m_k\boldsymbol _k \ ,

with :

m_ = \sum_^j \ m_k \ .

The vector

\boldsymbol_N

is the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

of all the bodies and

\boldsymbol_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

\boldsymbol_1, \dots, \boldsymbol_

and a center of mass coordinate

\boldsymbol_N

, from iteratively reducing two-body systems within the many-body system. This change of coordinates has associated Jacobian equal to

1

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

H_0=-\sum_^N\frac\, \nabla^2_ = -\frac\,\nabla^2_\!-\frac\sum_^\!\left(\frac+\frac\right)\nabla^2_

In the calculations can be useful the following identity :

\sum_^N \frac=\frac-\frac

References

{{Authority control Molecular vibration Molecular geometry Chemical reactions Hamiltonian mechanics Lagrangian mechanics Coordinate systems Orbits