Jacobi Coordinates in Few-Body Dynamics

• Jacobi coordinates are a hierarchical system that defines independent relative displacement vectors to separate center-of-mass motion from intrinsic dynamics.
• They simplify the kinetic energy operator in both quantum and classical systems, facilitating the construction of basis functions and integrators.
• While effective for decoupling motion, their complexity increases with particle number and may pose challenges for long-range potential transformations.

Jacobi coordinates are a systematic, hierarchical coordinate system employed in the study of few-body dynamical systems, particularly in quantum mechanics and classical celestial mechanics. They enable the decomposition of multi-particle problems into a set of relative coordinates that efficiently separate center-of-mass motion from internal degrees of freedom. This separation is fundamental to the reduction of the kinetic energy operator and to the derivation of effective equations of motion for interactions among constituents.

1. Definition and Formal Structure

The Jacobi coordinate system provides a recursive scheme for defining $N-1$ independent displacement vectors for an $N$-body system with masses $m_1, \ldots, m_N$ and position vectors $\mathbf{r}_1, \ldots, \mathbf{r}_N$ in Euclidean space. The standard construction is as follows:

• The first Jacobi vector is the relative position between particles 1 and 2:

$\boldsymbol{\xi}_1 = \mathbf{r}_2 - \mathbf{r}_1$

• The second Jacobi vector is the position of the third particle relative to the center of mass of particles 1 and 2:

$$\boldsymbol{\xi}_2 = \mathbf{r}_3 - \frac{m_1\mathbf{r}_1 + m_2\mathbf{r}_2}{m_1 + m_2}$$

• The $k$-th Jacobi vector for $k=2, \ldots, N-1$ generalizes recursively as the position of particle $k+1$ with respect to the center of mass of particles $1,\ldots,k$:

$$\boldsymbol{\xi}_k = \mathbf{r}_{k+1} - \frac{\sum_{i=1}^{k} m_i \mathbf{r}_i}{\sum_{i=1}^k m_i}$$

The total center-of-mass coordinate completes the transformation, yielding a set of $N$ coordinates with a non-singular linear relation to the original positions.

2. Role in Few-Body Quantum and Classical Dynamics

Jacobi coordinates are extensively used to decouple center-of-mass motion in both quantum and classical systems, such as in the analysis of atomic and molecular clusters, nuclear bound states, and hierarchical gravitational systems. By expressing the kinetic energy in terms of Jacobi vectors, the Hamiltonian naturally separates into a center-of-mass term and intrinsic contributions:

$$T = \frac{1}{2 M} \mathbf{P}_{\text{CM}}^2 + \sum_{k=1}^{N-1} \frac{1}{2 \mu_k} \mathbf{p}_k^2$$

where $\mathbf{P}_{\text{CM}}$ is the total momentum, $\mathbf{p}_k$ are canonically conjugate to the Jacobi vectors, $M$ is the total mass, and $\mu_k$ are reduced masses determined by the hierarchical structure. The utility of Jacobi coordinates is especially pronounced in constructing basis functions for solving the Schrödinger equation via the variational principle or hyperspherical harmonics expansion.

3. Hierarchical and Symmetry Properties

The hierarchical definition of Jacobi coordinates reflects the underlying binary structure of organizing the $N$-body system. Different orderings and groupings of particles yield equivalent but distinct Jacobi coordinate sets, which is pertinent in systems featuring indistinguishable particles, permutation symmetry, or the need to exploit particular clustering channels. In systems with underlying symmetries, the selection of Jacobi trees (binary partitions) can be optimized to minimize coupling or maximize the sparsity of interaction terms.

4. Kinetic and Potential Energy Transformation

Transformation to Jacobi coordinates simplifies the kinetic energy operator, reducing the complexity associated with Cartesian coordinates, particularly for systems with translation invariance. The Laplacian decomposes into a sum of independent Laplacians with reduced masses:

$$\nabla^2 = \sum_{i=1}^N \frac{1}{m_i} \nabla^2_{\mathbf{r}_i} = \frac{1}{M} \nabla^2_{\mathbf{R}_{\text{CM}}} + \sum_{k=1}^{N-1} \frac{1}{\mu_k} \nabla^2_{\boldsymbol{\xi}_k}$$

Potential energies expressed as functions of inter-particle distances may become more complicated in Jacobi coordinates, but for certain pairwise or cluster interactions, this coordinate choice remains advantageous.

5. Applications in Computational and Theoretical Methods

Jacobi coordinates are employed in a multitude of theoretical and computational methods:

• Hyperspherical coordinate approaches: Grouping Jacobi vectors into a global hyperradius and set of angular variables facilitates hyperspherical harmonics methods.
• Variational basis construction: Many-body basis functions, such as correlated Gaussians or harmonic oscillator bases, are formulated in Jacobi coordinates to enforce translational invariance.
• Hierarchical integrators and reduction techniques: In celestial mechanics, Jacobi coordinates are used in the reduction of Hamiltonian systems and development of symplectic integrators for hierarchical N-body problems.

6. Significance and Limitations

The primary significance of Jacobi coordinates lies in their ability to isolate intrinsic system properties from collective, non-informative motions. They provide a physically meaningful decomposition for systems where the distinction between internal and global dynamics is critical. However, their usage may become cumbersome for very large $N$ due to the factorial growth of possible partitionings, and for systems with complex long-range potentials, the transformation of the potential may become analytically intractable. In such contexts, alternative coordinate systems or direct use of laboratory-frame coordinates may be preferable.

7. Relation to Modern Research

While Jacobi coordinates are a classical mathematical tool, they continue to influence modern metrological and detection paradigms in physics. For example, recent interferometric studies, such as those exploiting centroid orbiting information in optical fields, utilize the concept of centroids and their geometric trajectories to extract sensitive, robust measurements of perturbations (Fang et al., 12 Jan 2025). This suggests a broader relevance of centroid and coordinate decompositions inspired by Jacobi-type constructions in contemporary precision measurement science. Moreover, the hierarchical and projection principles underlying Jacobi coordinates permeate modern robust centroiding and geometric clustering algorithms.