Birkhoff-Gustavson Normal Form

Author: Tianjiang Talk
Edited from: Qiao et al. (2025) "Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points"
Website: https://cislunarspace.cn

Definition

The Birkhoff-Gustavson Normal Form (B-G Normal Form) is a canonical transformation method that expands a Hamiltonian system near an equilibrium point into a diagonalized polynomial form. It was proposed by Birkhoff (1927) and later applied by Gustavson (1966) to stellar system problems in celestial mechanics.

In the Circular Restricted Three-Body Problem (CR3BP), the Birkhoff-Gustavson Normal Form performs Legendre expansion and Lie transformation on the Hamiltonian function near a libration point. Higher-order nonlinear terms are progressively eliminated, ultimately yielding a separable, diagonalized Hamiltonian expression that gives the system integrability under small perturbations.

Mathematical Background

From CR3BP to Polynomial Hamiltonian

The CR3BP Hamiltonian, after coordinate translation and normalization near a libration point, can be expanded into a sequence of homogeneous polynomials:

H = \sum_{n \geq 2} H_n = H_2 + H_3 + H_4 + \cdots

where $H_n$ is an $n$ -th order homogeneous polynomial. Through Legendre expansion, the nonlinear terms $(1-\mu)/r_1$ and $\mu/r_2$ can be transformed into polynomial form:

\frac{1}{\sqrt{(x-A)^2+(y-B)^2+(z-C)^2}} = \frac{1}{D}\sum_{n=0}^{\infty}\left(\frac{\rho}{D}\right)^n P_n\left(\frac{Ax+By+Cz}{D\rho}\right)

where $P_n$ is the $n$ -th order Legendre polynomial.

Diagonalization of the Linear Term $H_2$

In the neighborhood of a libration point, the linearized Hamiltonian $H_2$ corresponds to a saddle × center × center dynamics structure:

H_2 = \lambda q_1 p_1 + \frac{\omega_p}{2}(q_2^2 + p_2^2) + \frac{\omega_\nu}{2}(q_3^2 + p_3^2)

where:

$\lambda$ : hyperbolic characteristic frequency (unstable direction)
$\omega_p$ , $\omega_\nu$ : characteristic frequencies of the two center modes

Through a real linear symplectic transformation matrix $C$ (satisfying $C^TJC = J$ ), the original coordinates can be mapped to the diagonalized basis.

Gustavson's Contribution

Gustavson (1966) proved that by normalizing the Hamiltonian to infinite order, additional integrals (beyond the Hamiltonian itself) can be obtained. This method is called the "indirect method," using Lie series to transform the Hamiltonian into Birkhoff-Gustavson Normal Form, thereby directly identifying the integrals.

Lie Transformation Process

The construction of the normal form is achieved via Lie transformation. For an $n$ -th order generating function $G_n$ , the transformed Hamiltonian is:

\hat{H}_n = H_n + \{H_2, G_n\}

By choosing appropriate generating functions $G_n$ , non-resonant terms can be progressively eliminated while resonant terms are retained (resonant terms are critical for understanding the bifurcation of Halo orbit families). Specifically:

For third-order terms: use $G_3$ to eliminate terms with $i_1 \neq j_1$ (preserving the hyperbolic part $q_1 p_1$ )
For higher-order terms: use $G_n$ progressively for elimination

There is a trade-off between normalization precision and computational cost. Qiao et al. (2025) noted that when the expansion order $N$ exceeds 13, error reduction slows (limited by 15 significant digits of double-precision floating point), recommending $N=15$ .

Relationship with Other Methods

Method	Precision	Integrability	Limitations
Linearization ( $H_2$ )	Low	Exactly integrable	Only applicable in the immediate vicinity of libration points
B-G Normal Form (low-medium order)	Medium	Approximately integrable	Resonant terms eliminated; large errors for high-amplitude orbits
Complete B-G Normal Form ( $N \to \infty$ )	High	Integrable	Computational cost grows exponentially
Central Manifold Theory	—	Semi-integrable	Only handles center directions

Qiao et al. (2025) built on this foundation by introducing central manifold theory, decoupling the hyperbolic unstable direction from the central manifold to form a more complete parameterization system.

Applications

In Qiao et al. (2025), Birkhoff-Gustavson Normal Form combined with central manifold theory is used to:

Decompose the CR3BP six-dimensional phase space into hyperbolic directions ( $q_1, p_1$ ) and central manifold directions ( $I_2, \theta_2, I_3, \theta_3$ )
Establish a bijective correspondence from Cartesian coordinates to characteristic parameters
Establish a libration point orbit distribution map via Poincare sections

Central Manifold
Action-Angle Variables
Circular Restricted Three-Body Problem (CR3BP)
Lie Transformation
Hamiltonian System
Poincare Section

References

Birkhoff G D. Dynamical systems[M]. American Mathematical Society, 1927.
Gustavson F G. On constructing formal integrals of a Hamiltonian system near an equilibrium point[J]. Astronomical Journal, 1966, 71: 670.
Jorba A, Masdemont J. Dynamics in the center manifold of the collinear points of the restricted three body problem[J]. Phys D, 1999, 132(1-2): 189-213.
Qiao C, Long X, Yang L, et al. Orbital parameter characterization and objects cataloging for Earth-Moon collinear libration points[J]. Chinese Journal of Aeronautics, 2025. doi: 10.1016/j.cja.2025.103869.