Lindstedt-Poincare Method (Lindstedt-Poincare Method)

Author: Tianjiang Shuo

Reference: 钱霙婧(2014) "Research on Autonomous Navigation and Orbit Keeping of Spacecraft on Quasi-Periodic Orbits in Cislunar Space"

Website: https://cislunarspace.cn

Definition

The Lindstedt-Poincare method is a perturbation analysis method for solving periodic solutions of nonlinear vibration systems, independently proposed by Lindstedt (1883) and Poincare (1892). This method eliminates secular terms by introducing scaled time (stretched time), obtaining uniformly valid expansions of periodic solutions.

In libration point orbit research, the Lindstedt-Poincare method is used to derive analytical approximations of Halo orbits, Lissajous orbits, and Lyapunov orbits, providing high-quality initial guesses for numerical computation.

Method Principles

Difficulties with Traditional Perturbation Methods

For nonlinear vibration equations:

x¨+ω02x+ϵf(x,x˙)=0\ddot{x} + \omega_0^2 x + \epsilon f(x, \dot{x}) = 0

Traditional perturbation methods assume a solution of the form:

x(t)=x0(t)+ϵx1(t)+ϵ2x2(t)+⋯x(t) = x_0(t) + \epsilon x_1(t) + \epsilon^2 x_2(t) + \cdots

Substituting into the equation produces secular terms, i.e., terms that grow linearly with time, destroying the periodic solution assumption.

Core Idea of Lindstedt-Poincare Method

The Lindstedt-Poincare method eliminates secular terms through time scaling transformation:

τ=ωt\tau = \omega t

Where ω\omega is an undetermined amplitude-dependent frequency. Rewriting the equation in terms of τ\tau and selecting an appropriate expansion for ω\omega eliminates secular terms.

Algorithm Steps

  1. Scaled transformation: Let τ=ωt\tau = \omega t, replacing time variable with τ\tau
  2. Frequency expansion: Expand frequency as ω=ω0+ϵω1+ϵ2ω2+⋯\omega = \omega_0 + \epsilon \omega_1 + \epsilon^2 \omega_2 + \cdots
  3. Solution expansion: Expand solution as x=x0+ϵx1+ϵ2x2+⋯x = x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots
  4. Solve order by order: Solve by orders of ϵ\epsilon, selecting appropriate ωi\omega_i at each step to eliminate secular terms

Application in Libration Point Orbit Research

Analytical Solutions for Halo Orbits

Farquhar and Kamel (1973) used the Lindstedt-Poincare method to derive third-order and fourth-order approximate solutions for Halo orbits near the Earth-Moon L2 point.

Analytical Solutions for Lyapunov Orbits

Richardson (1980) derived analytical solutions for libration point Lyapunov orbits, widely used as initial guesses for Halo and Lissajous orbits.

Solution Accuracy

Analytical solutions from the Lindstedt-Poincare method have limited accuracy, typically used as initial guesses for numerical computation:

OrderAccuracyApplicable Scenario
First order~10⁻³Qualitative analysis
Second order~10⁻⁵Initial guess
Third order~10⁻⁷High-precision initial guess
Fourth order~10⁻⁹Refined initial guess

Relationship with Multiple Shooting Method

The Lindstedt-Poincare method and multiple shooting method represent two levels of periodic orbit solving:

MethodTypeAccuracyComputational Efficiency
Lindstedt-PoincareAnalyticalMediumHigh (closed-form solution)
Multiple ShootingNumericalHighLower (requires iteration)

Typical workflow:

  1. Use Lindstedt-Poincare method to obtain analytical solution as initial guess
  2. Use multiple shooting method for numerical refinement

Limitations

  1. Convergence: For large-amplitude orbits, higher-order terms may diverge
  2. Applicability: Mainly suitable for weakly nonlinear systems
  3. Computational complexity: Derivation of high-order expansions is tedious

References

  • Lindstedt A. Uber die Integration einer für die Storungstheorie wichtigen Differentialgleichung[J]. Astronomische Nachrichten, 1883.
  • Farquhar R W, Kamel A A. Quasi-periodic orbits about the translunar libration point[J]. Celestial Mechanics, 1973.
  • Richardson D L. A halo orbit solution[J]. Celestial Mechanics, 1980.
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Contributors: Cron Job, Ou Yang Jiahong