TY - JOUR

T1 - Two Methods for Defining Jacobi Coordinates in the Planetary Problem

AU - Микрюков, Денис Викторович

PY - 2026/3/1

Y1 - 2026/3/1

N2 - Abstract: Two methods for defining Jacobi coordinates in the planetary problem are considered and compared. The first method is classical, the second is proposed for the first time. The methods differ in the specification of an auxiliary vector required for nondegeneracy of the transformation of the initial absolute coordinates (this vector is usually supplied with a zero index). In the classical version, this vector specifies the absolute position of the barycenter of the system, whereas in the one proposed in this work, it specifies the absolute position of the central star. When considering each method, expressions for canonically conjugate momenta are derived. As a result of a detailed comparative analysis performed on the basis of the Hamiltonian formalism of mechanics, it is shown that after the reduction of the center of mass, both methods lead to the same system of equations of planetary motion. It is remarkable that the representation of potential energy, and along with it the representation of the disturbing function, turn out to be invariant in Jacobi coordinates with respect to the definitions under consideration. Formulas are given that are convenient for the practical application of perturbation theory methods.

AB - Abstract: Two methods for defining Jacobi coordinates in the planetary problem are considered and compared. The first method is classical, the second is proposed for the first time. The methods differ in the specification of an auxiliary vector required for nondegeneracy of the transformation of the initial absolute coordinates (this vector is usually supplied with a zero index). In the classical version, this vector specifies the absolute position of the barycenter of the system, whereas in the one proposed in this work, it specifies the absolute position of the central star. When considering each method, expressions for canonically conjugate momenta are derived. As a result of a detailed comparative analysis performed on the basis of the Hamiltonian formalism of mechanics, it is shown that after the reduction of the center of mass, both methods lead to the same system of equations of planetary motion. It is remarkable that the representation of potential energy, and along with it the representation of the disturbing function, turn out to be invariant in Jacobi coordinates with respect to the definitions under consideration. Formulas are given that are convenient for the practical application of perturbation theory methods.

KW - Hamiltonian

KW - Jacobi coordinates

KW - heliocentric coordinates

KW - long-term dynamics of planetary systems

KW - osculating elements

KW - perturbation function

KW - planetary N-body problem

UR - https://www.mendeley.com/catalogue/6f3163dd-ec55-3b7b-8dce-e1d406b1369e/

U2 - 10.1134/S1063454125700840

DO - 10.1134/S1063454125700840

M3 - Article

VL - 59

SP - 115

EP - 124

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -