Two-Boost Problem in Space Mission Design

• The two-boost problem is defined as finding a Hamiltonian chord on a fixed energy level that connects two phase space points via exactly two instantaneous momentum changes under prescribed energy constraints.
• It leverages advanced tools such as contact geometry, Reeb chord analysis, and Lagrangian Rabinowitz Floer homology to rigorously establish the existence of viable transfer trajectories.
• Analytical techniques including energy–momentum estimates and compact cut-off methods provide explicit energy bounds ensuring that the planned two-boost trajectories remain confined within a bounded region for mission feasibility.

The two-boost problem in space mission design asks, for a specified autonomous Hamiltonian system, whether it is possible to connect two phase space points or two positions, using a trajectory that comprises exactly two instantaneous velocity changes (“boosts”) at prescribed energy. The framework admits natural generalization to dynamical systems such as the planar circular restricted three-body problem, and the central analytical tools are contact geometry, Hamiltonian dynamics, and Lagrangian Rabinowitz Floer homology. Rigorous existence results have been established using Floer-theoretic invariants, under mild decay conditions on the potential, linking physical transfer scenarios to deep symplectic topology (Cieliebak et al., 2024, Wiśniewska, 15 Dec 2025).

1. Precise Formulation of the Two-Boost Problem

Let $Q$ be a configuration manifold (commonly $Q \cong \mathbb{R}^n$ or $\mathbb{R}^2$ minus collision points), and consider the cotangent bundle $T^*Q$ equipped with the canonical one-form $\lambda = p\,dq$ and symplectic form $\omega = d\lambda$. Fix an autonomous Hamiltonian $H: T^*Q \to \mathbb{R}$ and an energy level set $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$.

A "boost" of magnitude $\Delta v$ is defined as an instantaneous change in momentum $p \to p+\Delta p$, subject to $Q \cong \mathbb{R}^n$0. A two-boost trajectory at energy $Q \cong \mathbb{R}^n$1 from $Q \cong \mathbb{R}^n$2 to $Q \cong \mathbb{R}^n$3 consists of:

• An initial Hamiltonian orbit (drift) on $Q \cong \mathbb{R}^n$4 from $Q \cong \mathbb{R}^n$5 to $Q \cong \mathbb{R}^n$6,
• A first boost at time $Q \cong \mathbb{R}^n$7: $Q \cong \mathbb{R}^n$8,
• A second drift,
• A second boost at $Q \cong \mathbb{R}^n$9 into $\mathbb{R}^2$0 with $\mathbb{R}^2$1.

Equivalently, the problem seeks a Hamiltonian chord $\mathbb{R}^2$2 on $\mathbb{R}^2$3, joining $\mathbb{R}^2$4 to $\mathbb{R}^2$5, with exactly two discontinuities in the momentum coordinate.

The core functional encoding such trajectories is the Rabinowitz action: $\mathbb{R}^2$6 Critical points $\mathbb{R}^2$7 with $\mathbb{R}^2$8 yield Hamiltonian chords $\mathbb{R}^2$9 connecting the designated fibers on $T^*Q$0.

2. Model Hamiltonians and Dynamical Systems Classes

The natural physical setting is the planar circular restricted three-body problem, formulated in rotating coordinates. The Hamiltonian is typically:

$T^*Q$1

with $T^*Q$2 representing the gravitational potential of primary masses $T^*Q$3, in polar coordinates given by: $T^*Q$4

Key technical conditions on $T^*Q$5 ensure proper decay/growth at infinity ($T^*Q$6, $T^*Q$7), guaranteeing that $T^*Q$8 is asymptotic to the rotating Kepler problem.

Regularization techniques (Moser, Levi-Civita) exclude collision singularities, so noncompactness is only an issue at spatial infinity. In polar coordinates, the quadratic "Coriolis" form

$T^*Q$9

serves as the model for analytic estimates.

3. Contact Geometry, Reeb Chords, and the Hamiltonian Reformulation

On regular energy hypersurfaces, the Liouville vector field $\lambda = p\,dq$0 is transverse to $\lambda = p\,dq$1, so $\lambda = p\,dq$2 constitutes a contact form. The Reeb vector field $\lambda = p\,dq$3 is specified by $\lambda = p\,dq$4, $\lambda = p\,dq$5. Special Legendrian submanifolds arise as the intersections $\lambda = p\,dq$6.

A Reeb chord from $\lambda = p\,dq$7 to $\lambda = p\,dq$8 is a trajectory $\lambda = p\,dq$9 solving $\omega = d\lambda$0, with endpoints on the respective Legendrians. Time rescaling identifies these objects with Hamiltonian chords as defined above.

4. Lagrangian Rabinowitz Floer Homology Construction and Existence Results

Lagrangian Rabinowitz Floer homology (LRFH) is built from the action functional $\omega = d\lambda$1 over the space $\omega = d\lambda$2. Critical points correspond to valid two-boost trajectories.

The chain complex $\omega = d\lambda$3 is generated by critical points, graded via Maslov index, with an action filtration. The boundary operator $\omega = d\lambda$4 counts rigid Floer trajectories, solutions of

$\omega = d\lambda$5

with specified asymptotics.

By analytic continuation (Floer theory invariance under compact perturbations), the positive part $\omega = d\lambda$6 is isomorphic to the ordinary homology of the based path space $\omega = d\lambda$7: $\omega = d\lambda$8 This is nontrivial in degree $\omega = d\lambda$9, confirming existence of a positive action chord (i.e., a two-boost solution).

Existence theorem (Cieliebak et al., 2024, Wiśniewska, 15 Dec 2025): If $H: T^*Q \to \mathbb{R}$0 satisfies the decay conditions and $H: T^*Q \to \mathbb{R}$1 are within a ball of radius $H: T^*Q \to \mathbb{R}$2, then for

$H: T^*Q \to \mathbb{R}$3

there exists a Hamiltonian chord at energy $H: T^*Q \to \mathbb{R}$4 joining $H: T^*Q \to \mathbb{R}$5 and $H: T^*Q \to \mathbb{R}$6. This result is optimal within the technical framework imposed by the behavior of $H: T^*Q \to \mathbb{R}$7 at infinity.

5. Boundedness of Reeb Chords and Analytical Techniques at Infinity

A central difficulty arises from noncompactness at infinity, especially ensuring boundedness of the critical chord sets (Reeb chord moduli). The key geometric fact is that at sufficiently large energy $H: T^*Q \to \mathbb{R}$8, flow outside a suitable compact set is hyperbolic, precluding return of a chord that leaves that set.

Analytical handles include:

• Energy–momentum estimate: For $H: T^*Q \to \mathbb{R}$9 with $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$0,

$\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$1

implying bounds on $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$2, enforcing non-return from infinity.

• No radial maxima at infinity (Lemma 2.2): The second Poisson bracket,

$\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$3

ensures that a chord cannot attain its radial maximum outside $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$4 without contradiction.

• Compact cut-off technique: Define compactly supported perturbation $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$5 so that $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$6 within $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$7 and $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$8 outside a compact set, reducing the moduli problems to compact analysis.

These estimates guarantee all critical chords lie in a bounded region, permitting standard transversality and gluing arguments of Floer theory.

6. Physical Interpretation and Mission Design Implications

The analytic existence result rigorously validates the classical two-impulse mission design (e.g., Hohmann transfer and its generalizations), even in the context of two gravitational primaries with more complex dynamics. For any two positions $\Sigma_c = \{(q, p) \in T^*Q \mid H(q, p) = c\}$9 within a specified ball, a sufficiently high energy $\Delta v$0 ensures the existence of a transfer orbit connecting $\Delta v$1 and $\Delta v$2 with one boost at launch and one at arrival, coasting otherwise under the gravitational and Coriolis forces.

Required energy bounds are explicit: $\Delta v$3 where $\Delta v$4 corresponds to decay estimates of $\Delta v$5. This guarantees the trajectory remains in the hyperbolic regime at spatial infinity; thus, the system's geometric and spectral invariants translate directly into mission feasibility constraints (Wiśniewska, 15 Dec 2025).

7. Floer-Theoretic Spectral Invariants and Boost Energy Relations

The spectral invariant associated with the shortest action chord is given by

$\Delta v$6

Nontriviality of $\Delta v$7 is equivalent to $\Delta v$8, guaranteeing existence of a valid chord within energy $\Delta v$9. In the quadratic model $p \to p+\Delta p$0, one finds $p \to p+\Delta p$1, so any $p \to p+\Delta p$2 admits a solution. For perturbed models $p \to p+\Delta p$3, penalization and continuation methods extend the existence to all $p \to p+\Delta p$4 as detailed above.

These results clarify the interplay between system geometry, Hamiltonian dynamics, and the minimum energy requirements for two-impulse trajectory existence, grounding traditional mission planning within a rigorous symplectic and Floer-theoretic framework (Cieliebak et al., 2024, Wiśniewska, 15 Dec 2025).