Jacobi coordinates

For particles with masses

m_{j}

r_{j}

, Jacobi coordinates consist of the total center of mass and successive relative coordinates

ρ_{j} = r_{j + 1} - (\sum_{l = 1}^{j} m_{l} r_{l}) / (\sum_{l = 1}^{j} m_{l})

. They separate translation from internal motion and diagonalize the kinetic energy with the corresponding reduced masses.

Jacobi coordinates for three particles

For three particles, choose the relative position of particles one and two, the position of their center of mass relative to particle three, and the total center of mass. The kinetic energy diagonalizes with masses

\frac{m _{1} m _{2}}{m _{1} + m _{2}}, \frac{( m _{1} + m _{2} ) m _{3}}{m _{1} + m _{2} + m _{3}}, m_{1} + m_{2} + m_{3} .

a, b, c

and their corresponding reduced masses

α, β, γ

L = α a \times \dot{a} + β b \times \dot{b} + γ c \times \dot{c} .

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Jacobi coordinates are a system of coordinates used in the study of many-body problems in physics, particularly in celestial mechanics and molecular dynamics. They are named after the mathematician Karl Gustav Jacob Jacobi. This coordinate system is especially useful for simplifying the analysis of systems of particles by transforming the coordinates to better exploit the symmetries inherent in the problem. In a typical application, Jacobi coordinates are used to describe the positions of \( n \) particles.