Three-Body Lambert Problem

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The Three-Body Lambert Problem (3BLP; also written L3BP — Lambert Three-Body Problem; in the CR3BP model, the CR3BP Lambert problem) is the extension of Lambert's problem to three-body dynamics: in the Circular Restricted Three-Body Problem (CR3BP), given an initial position $\vec r_1$ (at time $t_1$ ), a final position $\vec r_2$ (at time $t_2$ ), and the time of flight $\Delta t=t_2-t_1$ , find the connecting trajectory and the endpoint velocities (Sun et al. 2017).

Because the CR3BP equations of motion are non-integrable and highly sensitive to initial conditions, the three-body Lambert problem has no analytical solution and can only be solved numerically; initial-guess construction is the central difficulty, and simple Newton-Raphson shooting barely converges for long-duration transfers.

Fundamental Differences from the Two-Body Lambert Problem

Aspect	Two-body Lambert	Three-body Lambert
Existence	Guaranteed by Lambert's theorem, at most $2N+1$ solutions	No analytical structure; existence decided numerically
Time-orbit mapping	$\Delta t$ depends only on $a,\,r_1+r_2,\,c$	$\Delta t$ is coupled to orbit shape; Lambert's theorem fails
Multi-valuedness	At most $2N+1$ (by revolution count and short/long way)	May have qualitatively distinct solutions (short arc/long arc/lunar-flyby/multi-revolution/manifold-type)
Conserved quantities	Energy, angular momentum	Jacobi constant $C_J$ (usable as an extra constraint or a free parameter)
Plane	Automatically determined by the two vectors	Two vectors do not uniquely fix the plane (3D transfers are common)
Solution method	One transcendental equation + standard algorithms	Numerical shooting + differential correction, depends on a good initial guess

Initial-Guess Strategies

Solving the three-body Lambert problem is essentially a two-endpoint shooting problem. Common initial-guess strategies:

Two-body solution + homotopic iteration: obtain the two-body Lambert solution $\vec v_1^{(2B)}$ as the initial guess, then introduce a homotopy parameter $\lambda\in[0,1]$ that continuously deforms the dynamics from two-body to CR3BP (homotopy method), with Newton-Raphson convergence at each step. This method is markedly superior to direct shooting for long-duration transfers (Sun et al. 2017).
Genetic algorithm for a reference trajectory: use a GA to globally search a coarse position-time sequence that approximately satisfies the boundary conditions, then refine by homotopic Newton iteration; resolves the poor convergence of long-duration Halo-to-Halo transfers (Sun et al. 2017).
Sukhanov-Prado two-level iteration: in the Hill model, simultaneously correct the initial and final position vectors; converges well but cannot guarantee convergence to the desired solution (because three-body Lambert is multi-valued).
Invariant-manifold patching: for transfers between libration-point orbits, recast the problem as the patching of the unstable manifold of one Halo/Lissajous orbit with the stable manifold of another on a Poincaré section, then refine via three-body Lambert.

Multi-Solution Property

A single set $(\vec r_1,\vec r_2,\Delta t)$ can correspond to multiple qualitatively distinct transfer trajectories: a direct short arc, a long arc via lunar flyby, a multi-revolution transfer around Earth before arrival, or a low-energy transfer exploiting solar perturbation. This is a direct consequence of CR3BP nonlinear dynamics and contrasts with the two-body Lambert bound of $2N+1$ . In practice one usually:

bounds the Jacobi constant $C_J$ to clip the search space to an energetically feasible subset;
fixes the revolution direction (around the Moon / Earth / Sun-Earth $L_1$ );
runs differential correction with multiple initial guesses in parallel and selects the minimum- $\Delta v$ solution.

For Halo orbit computation and libration-point rendezvous, multi-valuedness is both nuisance (filtering required) and opportunity (an extra candidate may save fuel).

Application Notes

Libration-point rendezvous and docking: transfers between two spacecraft in different phases of the same Halo orbit, or between Halos of different amplitudes — a foundational tool for building a cislunar $L_2$ space station (Sun et al. 2017).
Low-energy Earth-Moon transfer: design of WSB and manifold-patching transfers exploiting solar perturbation.
Multi-arc patching: decompose a cislunar transfer into several three-body Lambert arcs, solve each independently, then patch.
No analytical shortcut: every application needs a stable numerical pipeline and a good initial guess — the biggest practical difference from two-body Lambert.

References

Sun Yu, Zhang Jin, Luo Yazhong, 2017, Rendezvous Trajectory Design of Libration Points Based on Three-body Lambert Algorithm (Earth-Moon $L_2$ Halo-to-Halo rendezvous; genetic algorithm + homotopic Newton iteration).
Vallado, 2022, Fundamentals of Astrodynamics and Applications, §7.6 (the two-body Lambert problem as the parent concept and contrast).
Szebehely, 1967, Theory of Orbits: The Restricted Problem of Three Bodies (the CR3BP framework).