Reduced-Order Dynamic Equations

Editor source: Guo Jianyu (2020) "Research on Libration Point Orbit Design and Station-Keeping Strategies Based on the Two-Dominant Invariant Manifold Method"

Website: https://cislunarspace.cn

Definition

Reduced-Order Dynamic Equations are dynamical equations with reduced order obtained through the two-dominant invariant manifold method. By selecting two principal motion directions of a periodic orbit, this method expresses the motion in the third direction as a nonlinear function of the first two, thereby achieving a reduction in the phase space dimensionality.

Theoretical Foundation

Origin

In the Circular Restricted Three-Body Problem (CR3BP), the equations of motion for periodic orbits near libration points can be represented as high-dimensional nonlinear systems. The two-dominant invariant manifold method achieves order reduction through the following steps:

  1. Select two motion directions as principal motions (typically the xx and yy directions)
  2. Use invariant manifold theory to establish the relationship between the third direction (zz direction) and the principal motions
  3. Obtain nonlinear polynomial relations through Legendre polynomial expansion
  4. Obtain the final reduced-order dynamical equations

Mathematical Expression

Let the amplitudes of the principal motion directions be ρ1,ρ2\rho_1, \rho_2; then the amplitude ρ3\rho_3 of the subordinate direction can be expressed as:

ρ3=∑i,jcijρ1iρ2j\rho_3 = \sum_{i,j} c_{ij} \rho_1^i \rho_2^j

where cijc_{ij} are polynomial coefficients determined by the system's dynamical characteristics.

Integration with Lindstedt-Poincare Perturbation Method

The reduced-order dynamic equations can be used in conjunction with the Lindstedt-Poincare perturbation method:

  1. Solve the reduced-order dynamic equations using the Lindstedt-Poincare perturbation method
  2. Obtain third-order approximate analytical solutions for Halo and Lissajous orbits
  3. Use the analytical solutions as initial guesses, then refine through numerical orbit design methods to obtain accurate periodic orbits

Application Value

The main applications of reduced-order dynamic equations include:

  • Orbit design: Providing good initial guesses to reduce the number of numerical iterations
  • Orbit maintenance: Polynomial relations can serve as constraint conditions for real-time orbit corrections
  • Dynamic analysis: Simplifying the analysis of high-dimensional systems and revealing intrinsic laws of orbital motion

Key Elements

Mathematical Definition

Reduced-order dynamic equations are obtained through the two-dominant invariant manifold method by selecting two principal motion directions and expressing the third direction as a nonlinear function, yielding lower-order dynamical equations.

Key Properties

Reduced-order equations preserve the principal dynamical characteristics of periodic orbits while greatly reducing computational complexity. The polynomial coefficients reflect the intrinsic dynamical structure of the system.

Numerical Methods

Solved using Legendre polynomial expansion combined with the Lindstedt-Poincare perturbation method to obtain multi-order approximate analytical solutions.

References

  • Guo Jianyu. Research on Libration Point Orbit Design and Station-Keeping Strategies Based on the Two-Dominant Invariant Manifold Method[D]. Beijing University of Technology, 2020. [in Chinese]
  • Farquhar R W, Kamel A A. Quasi-periodic orbits about the collinear libration points[J]. Celestial Mechanics, 1973, 7(3): 267-289.
  • Richardson D L. Analytic construction of periodic orbits about the collinear points[J]. Celestial Mechanics, 1980, 22(3): 241-253.