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Global Regularization of the Spatial Elliptic Restricted Three-body Problem: Quaternionic, Hamiltonian Formulation

Kenshiro Oguri

arXiv:2610.13969v1math.DScs.NAmath.NA

TL;DR

Near gravitational collisions, numerical integration can become unreliable, and existing WA formulations do not explicitly incorporate arbitrary external forces. This paper recasts global spatial ER3BP regularization in quaternionic Hamiltonian form; its analysis establishes regularity properties of the transformation and flow, while physical momenta remain unrecoverable at collision epochs.

  • Problem

    Near gravitational collisions, ODE accuracy degrades and adaptive solvers may fail to converge reasonably, while existing WA formulations do not explicitly incorporate arbitrary external forces.

    Source: §1 · pp. 1, 3

  • Method

    The paper reformulates the WA global spatial ER3BP transformation through quaternionic mappings and Hamiltonian mechanics to derive canonical momenta and incorporate external acceleration.

    Source: §1 · pp. 3–4

  • Results

    The WA coordinate map is regular off a set corresponding to physical infinity, and dΓ/ds remains zero at collision states in the regularized flow.

    Source: §4.5 · p. 15, §5.3 · p. 20

  • Takeaways & Limitations

    The Hamiltonian formulation supports analysis of the WA transformation and regularized dynamics while accommodating external acceleration through canonical generalized forces.

    Source: §1 · pp. 3–4

    The regularized flow can propagate smoothly through collisions, but physical momenta cannot be recovered at collision epochs.

    Source: §4.5 · p. 16

Abstract

from arXiv · show

Any objects in space may encounter singularity due to the inverse-square law of the gravitational force, hindering the analysis and propagation of particle motion in space. Regularization in celestial mechanics removes such gravitational singularities in equations of motion by transforming the time and coordinates. This paper addresses a class of regularization methods known as global regularization for restricted three-body problems (R3BPs), which simultaneously removes the two singularities (e.g., Earth and Moon in cislunar space). It enables analysis, propagation, and prediction of particle motion throughout the domain of R3BPs, which is in stark contrast to regularizing each singularity individually. In particular, this paper develops a formulation for global regularization of the spatial elliptic R3BP (ER3BP) under external perturbations. The presented formulation leverages mathematical tools from quaternionic (hyper-complex) algebra and Hamiltonian mechanics. This approach enables us to (i) elegantly handle the four-dimensional coordinates arising from the dimension-lifting operation for spatial regularization, (ii) systematically derive the coordinate-momentum transformation while preserving the canonical structure and incorporating perturbing forces, and (iii) provide geometric insights for analyzing the regularity of the transformation, resulting Hamiltonian, and dynamical flow. Numerical examples demonstrate the validity of the mathematical results as well as significant numerical benefits of the regularization formulation in perturbed Earth-Moon ER3BP.

1. Introduction.

Gravitational singularities can undermine numerical integration, motivating regularization that handles both primaries in restricted three-body problems. This paper develops a quaternionic, Hamiltonian formulation of global regularization for the spatial ER3BP, including external perturbations.

1. Introduction.

  • Near a gravitational singularity, ODE accuracy degrades and adaptive solvers may fail to converge within reasonable computational time or step size.The issue arises as distance to a gravitational body approaches zero under the inverse-square force law.Source: §1 · p. 1
  • Global regularization simultaneously removes collisions with both primaries, unlike local methods that remove one singularity at a time.Earlier global ER3BP formulations extended to elliptic motion were limited to planar problems.Source: §1 · pp. 2–3
  • Existing WA formulations globally regularize the spatial ER3BP, but use Lagrangian mechanics and do not explicitly incorporate arbitrary external forces.The paper offers a quaternionic, Hamiltonian alternative for analyzing the transformation, Hamiltonian, and differential equations.Source: §1 · p. 3
  • The quaternionic approach presents regularizing coordinates as conformal-like mappings and streamlines their derivation.The KS transformation is called “conformal-like” because it is not strictly conformal.Source: §1 · p. 3
  • Hamiltonian mechanics systematically derives the associated momentum maps while preserving canonical structure and incorporating external accelerations through generalized forces.Examples include solar radiation pressure, third-body perturbations, and continuous thrusting.Source: §1 · p. 4

2. Hyper-complex numbers.

The section introduces quaternions as the paper’s hyper-complex representation and explains how they relate to real four- and three-dimensional vectors.

2. Hyper-complex numbers.

  • A quaternion h belongs to H and has three imaginary components and one real component; the paper’s quaternions are generally not unit-norm.The three imaginary units are i, j, and k.Source: §2 · p. 4
  • Quaternion multiplication is generally noncommutative, so left and right division must be distinguished.The conjugate of a product reverses the factor order.Source: §2 · p. 4
  • The maps quat and vecH identify real four-vectors with quaternions, while vecIH identifies pure-imaginary quaternions with real three-vectors.The paper defines these maps to clarify the relation between real vectors and quaternion representations.Source: §2 · p. 5

3. Elliptic Restricted Three-body Problem (ER3BP).

The ER3BP is expressed in a pulsating-rotating frame, shifted coordinates, and an extended Hamiltonian phase space. A time transformation preserves the canonical formulation but does not by itself remove collision singularities.

3.1. Formulation in the Pulsating-Rotating Frame.

  • The pulsating-rotating frame tracks the primary line and orbital angular momentum, with time variation in frame rotation and primary separation captured through ωz(t).The frame uses a non-dimensional formulation with the instantaneous primary separation as characteristic length.Source: §3.1 · pp. 5–6

3.2. ER3BP Hamiltonian in Shifted Coordinates.

  • Shifted coordinates place the primaries symmetrically, define q as the shifted particle position, and include a frame-rotation term in the associated momentum p.For the non-autonomous Hamiltonian, the phase space is extended with q0 = t and its conjugate momentum p0 = −H; generalized forces represent external acceleration.Source: §3.1 · p. 6, §3.2 · pp. 6–7

3.3. Time-transformed ER3BP Hamiltonian.

  • Time regularization introduces fictitious time and a scaled Hamiltonian, with generalized forces scaled to preserve the canonical system, but collision singularities remain.The transformed equations still contain derivatives involving division by the distances to the primaries.Source: §3.3 · pp. 7–8

4. Spatial Extension of Birkhoff’s Coordinates.

The section constructs the spatial Waldvogel–Arenstorf (WA) coordinates through quaternionic mappings and derives their canonical momentum transformations. The cotangent-lift formulation preserves canonical structure without requiring quaternion differentiability.

4.1. Waldvogel-Arenstorf Coordinates.

  • The WA map extends Birkhoff’s planar construction to the spatial problem by combining quaternionic Möbius maps with a Kustaanheimo–Stiefel map.The spatial coordinates are lifted from R3 to R4; the KS map replaces the planar Levi-Civita map.Source: §4.1 · pp. 8–9

4.2. Canonical Momenta in Waldvogel-Arenstorf Coordinates.

  • The cotangent-lift approach derives canonical momenta without quaternion differentiability, avoiding complications from noncommutative quaternion multiplication.The formulation explicitly preserves the canonical structure.Source: §4.2 · p. 9

4.2.1. Quaternionic Cotangent Lift.

  • Quaternionic cotangent lift obtains the momentum map for a smooth coordinate map by solving the canonical relation using the map’s differential and adjoint.The method is based on preserving the canonical one-form.Source: §4.2.1 · p. 10

4.2.2. Quaternionic Waldvogel-Arenstorf Momenta Derivation.

  • The WA momentum derivation applies the cotangent lift to its component maps; the KS momentum map returns physical-space momenta when its input satisfies the admissibility condition.The admissible set requires Re(q ip) = 0.Source: §4.2.2 · p. 11

4.3. Condition for Physical Projection.

This section characterizes which WA states project to physical coordinates and momenta. The admissible-state condition is equivalent to the input and output both representing physical three-dimensional states.

4.3. Condition for Physical Projection.

  • An arbitrary WA state may project to a momentum that is not purely imaginary, so not every quaternionic momentum represents a physical three-dimensional momentum.The condition is needed to ensure the intermediate KS momentum lies in AKS.Source: §4.3 · pp. 12–13
  • A WA state belongs to AWA exactly when its projection returns coordinates and momenta in the pure-imaginary quaternion subspace representing physical space.The condition is equivalent to the intermediate KS state satisfying its admissibility constraint.Source: §4.3 · pp. 13–14
  • The physical projection maps AWA states to physical coordinate-momentum pairs, and the inverse lifting map maps those pairs back into AWA.The paper names these maps physical projection and lifting map, respectively.Source: §4.3 · p. 14

4.4. Lifting Map: Physical to WA coordinates.

The lifting map constructs WA variables from a physical state by inverting the component transformations. Its construction is non-unique, but every lifted state satisfies the admissibility condition.

4.4. Lifting Map: Physical to WA coordinates.

  • The inverse component maps construct WA coordinates and momenta from a physical state, and the resulting lifted state automatically lies in AWA.The coordinate lift includes a choice of q(2) satisfying q(1) = q(2)iq(2).Source: §4.4 · p. 14
  • The lifting map is not unique: different choices of q(2) represent the same physical state with different WA states.The paper identifies this non-uniqueness with the Hopf fibration and parameterizes the choices by θ ∈ [0, 2π).Source: §4.4 · pp. 14–15

4.5. Regularity of the WA Transformation.

The WA coordinate map is smooth away from the real line in quaternion space, which corresponds to infinity in physical space. The section also gives a vector-matrix expression for the transformation.

4.5. Regularity of the WA Transformation.

  • The WA coordinate transformation is C∞ outside S0, the real line in quaternion space, and S0 maps to infinity in physical space.Thus, the coordinate-map exception lies at physical infinity.Source: §4.5 · p. 15
  • At collisions, the WA momentum map is singular and physical momentum cannot be recovered, although the regularized Hamiltonian flow can pass smoothly through collision.At collision, dt/ds = 0, making conversion to physical-time velocity singular.Source: §4.5 · p. 16

4.6. Vector-Matrix Expression.

  • The vector-matrix form expresses physical coordinates and momenta directly in terms of WA variables, facilitating the partial derivatives needed for generalized forces.The quaternionic expression is inconvenient for taking those partial derivatives.Source: §4.6 · p. 16
  • The notation ϱ and χ(±) compactly represents distance from the real axis and squared distances from the two primaries, respectively.The real-axis distance also corresponds to distance from physical infinity.Source: §4.6 · p. 16

5. Regularized Hamiltonian.

The WA transformation yields a Hamiltonian in regularized coordinates whose Hamiltonian and equations of motion are regular on the stated domain. Its flow preserves admissibility under specified conditions, while numerical use has practical boundary cases and the framework extends to some perturbed-primary systems.

5. Regularized Hamiltonian.

  • Applying the WA mapping to the time-transformed ER3BP Hamiltonian gives an explicit Hamiltonian K in the WA state (z, ξ).The original time-transformed Hamiltonian still has singularities before the WA mapping is applied.Source: §5 · p. 17

5.1. Regularity of the Hamiltonian.

  • Theorem 5.2 establishes that the WA-transformed Hamiltonian is C∞ for practical trajectories, with possible loss of regularity only on S0, corresponding to escape to infinity.The paper therefore treats the Hamiltonian as regular for trajectories of practical interest.Source: §5.1 · p. 18

5.2. Hamilton’s Equation.

  • Theorem 5.3 establishes regularity of the WA-space equations of motion, including generalized-force terms that remain smooth at primary collisions.The force term is regular on the domain because the collision singularities in the momentum map are removed after multiplication by ηWA.Source: §5.2 · pp. 18–19

5.3. Admissibility of the Dynamical Flow.

  • The flow conserves Γ, preserving AWA when the initial state lies in AWA and the z-trajectory remains away from S0.The paper extends Γ conservation to collision states using continuity of the regularized vector field.Source: §5.3 · pp. 19–21

5.4. Numerical Implementation.

  • Numerical propagation uses s as the independent variable and should stop when ϱ(z) becomes small, signaling approach to escape at infinity.Although the regularized state propagates smoothly through collisions, the inverse momentum map is undefined at an exact primary collision.Source: §5.4 · pp. 21–22

5.5. Directions for Further Extensions.

  • The framework extends to primaries on perturbed non-Keplerian orbits when they remain in a fixed co-orbital plane, R12(T) is smooth, and their angular momentum vector remains nonzero.The paper includes perturbations from other gravitational bodies in the same orbital plane as an example.Source: §5.5 · p. 22

6. Numerical Examples.

The Earth–Moon simulations validate agreement between regularized and Cartesian propagation away from collisions, then demonstrate successful regularized propagation through primary encounters. In collision cases, Cartesian integration can become inaccurate or fail, while regularization maintains stable steps and achieves substantial speedups.

6.1. Physical System Definition.

  • The Earth–Moon study models external solar radiation pressure with a standard cannon-ball model and uses ode89 with 1 × 10^-12 tolerances.The reported nondimensional SRP acceleration is 3 × 10^-5, approximately corresponding to a small satellite with area-to-mass ratio 0.012 m^2/kg and reflectivity ratio 0.5 at 1 AU.Source: §6 · p. 22, §6.1 · p. 22

6.2. Validation.

  • Validation uses a spatial synodic resonant orbit across the Earth–Moon system, which is not periodic under the perturbed ER3BP model.The paper notes that it is periodic in the unperturbed CR3BP; initial conditions are tabulated in Cartesian and regularization coordinates.Source: Figure 1 · p. 23, §6.2 · p. 24, Table 2 · p. 24
  • The bottom-left panel shows position and velocity difference histories between the regularized and Cartesian trajectories, indicating agreement to the numerical integration tolerance (1×10−12) in the non-dimensional unit.The regularized trajectory respects Γ = 0 and Re(q) = Re(p) = 0 within numerical noise. Hamiltonian remains constant up to numerical integration error, while fictitious time s increases monotonically with physical time t.Source: §6.2 · p. 24

6.3. Trajectories with Collisions.

  • The regularized propagator completes integrations through primary collisions, while Cartesian propagation can diverge or fail, supporting the claimed regularity of the formulation.The collision trajectories also pass through the primaries in position space, validating the regular-to-Cartesian coordinate transformation.Source: §6.3 · pp. 25–26
  • Regularization achieved >1450× speedup compared with Cartesian propagation; when encountering the Moon, the Cartesian step size shrank to 2.5 × 10^-21 nondimensional time units.The regularized step size remained roughly constant throughout propagation.Source: §6.3 · pp. 25–26
  • For the SRP-perturbed Earth-collision case, Cartesian ode89 cannot meet the requested tolerance, while regularized steps remain stable.The requested relative and absolute tolerances are 1 × 10^-12; the Cartesian step falls below 2 × 10^-24.Source: §6.3 · pp. 26–27, Figure 3 · p. 27
  • >2200× speedup over Cartesian propagation is reported for the SRP-perturbed ER3BP Moon-encounter case.The comparison is attributed to larger regularized independent-variable steps throughout propagation.Source: §6.3 · p. 27

7. Conclusions.

The paper develops a quaternionic, Hamiltonian formulation that globally regularizes both singularities of the spatial ER3BP, including under external acceleration. It proves regularity properties and reports numerical validation and advantages in Earth–Moon simulations.

7. Conclusions.

  • The formulation simultaneously regularizes both singularities in the spatial elliptic restricted three-body problem.It treats the transformation as three conformal-like mappings in quaternionic space and uses Hamiltonian mechanics to derive it systematically while accommodating external acceleration.Source: §7 · p. 28
  • The paper proves regularity of the transformation, Hamiltonian, and equations of motion, with the regularized dynamics singular only at physical infinity.The latter condition has implications for numerical implementation.Source: §7 · p. 28
  • Earth–Moon numerical simulations demonstrate the mathematical results and numerical advantages over the conventional Cartesian representation.Source: §7 · p. 28

Appendix A. Proof of Lemma 4.5.

The proof derives the coordinate and momentum maps by manipulating the quaternionic transformations and their cotangent lifts. It then substitutes the resulting identities to obtain the stated momentum map.

Appendix A. Proof of Lemma 4.5.

  • The proof derives the coordinate map by substituting the inverse quaternion expression into the preceding transformation.It uses a quaternion identity to simplify the inverse factor before establishing the map.Source: Appendix A · p. 28
  • The momentum map is obtained by combining the three cotangent lifts and applying the associated quaternion identities.The proof uses the intermediate expressions for q(2), q(1), and p(2) to transform the product.Source: Appendix A · p. 29
  • Substituting the inverse of the final transformation into the derived product yields the momentum map in equation (4.29b).Source: Appendix A · p. 29

Appendix B. Proof of Lemma 4.8.

The proof simplifies the coordinate and momentum maps using quaternionic inverse identities, then verifies that the momentum map has zero real part. These steps establish the expressions and constraint used in Lemma 4.8.

Appendix B. Proof of Lemma 4.8.

  • The coordinate map is obtained by substituting the inverse identity for z − z̄ into the quaternionic transformation and expanding.The proof uses the relation (z − z̄)^-1 = −(z − z̄)/(4ϱ^2).Source: Appendix B · p. 29
  • The momentum map is reduced using the identities for (z − i)^-1 and (z − z̄)^-1, then expanded component-wise.The expansion collects coefficients of ξ to form the matrix M.Source: Appendix B · pp. 29–30
  • The proof verifies that the momentum map’s real component vanishes using quaternion-product identities and cyclic permutation inside the real-part operator.Source: Appendix B · p. 30

Appendix C. Proof of Theorem 5.1.

Appendix C proves Theorem 5.1 by deriving equations (5.2) and (5.3) through substitutions and quaternionic identities.

Appendix C. Proof of Theorem 5.1.

  • Equation (4.37) directly yields the first and third terms of equation (5.3).Source: Appendix C · p. 30
  • The rotation term in equation (5.3) follows from q2p1 − q1p2 = Re(kqp), the momentum map, and coordinate-map identities.Source: Appendix C · p. 30
  • Substituting (C.2), (C.3), and (C.5) into (5.1) gives equation (5.2).Source: Appendix C · p. 30
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