The mathematical heart of the paper lies in a systematic method for extending dimension-raising point transformations into canonical transformations—the special class of coordinate changes that preserve Hamilton’s equations and the symplectic structure of phase space. The authors begin with Hamilton’s principle, requiring the action integral to be stationary along physical trajectories in both the original and the transformed coordinate systems. Because the new coordinates are redundant—four coordinates describe a three-dimensional position—they must satisfy a constraint, and the researchers handle this by introducing a Lagrange multiplier directly into the canonical structure. This yields explicit formulas for the transformed momenta and the new Hamiltonian, and it generalizes earlier work by Ferrándiz and Sansaturio while allowing for time-dependent transformations and constraints.

The particular family of projective transformations considered takes a position vector r and decomposes it as r = u raised to the power n, multiplied by q raised to the power m, times a vector q constrained to unit length. The parameters n and m act as adjustable knobs that generate an entire family of canonical coordinate systems, with the well-known Burdet–Ferrándiz (BF) transformation recovered for specific values. After analyzing the properties of each member of the family, the authors settle on a preferred transformation with n = m = -1, meaning r = (1/u) times the unit vector along q. This choice, they argue, subtly improves on BF at the configuration level and significantly at the momentum level, and it places attitude dynamics and angular momentum at the center of the formulation rather than at its margins.

Several technical virtues of the preferred choice stand out. First, radial distance becomes r = 1/u, a function only of the scalar coordinate u, so central forces are isolated entirely within the one-dimensional radial subsystem. Second, the six coordinates describing the rotational part of the motion directly characterize the attitude kinematics of the local vertical local horizontal (LVLH) frame—the rotating orthonormal basis used constantly in spacecraft operations. Third, rotational and radial motion decouple completely, and the rotational subsystem is linear for any central force whatsoever, not merely for gravity. Finally, and perhaps most strikingly, the transformation fully linearizes both Kepler and Manev dynamics without any need to carefully impose constraint conditions; the two extra integrals of motion built into the redundant formulation may be used when convenient but are never required for linearity itself.

Coordinate changes alone do not finish the job. The key additional step is a reparameterization of time: replacing physical time t with a new evolution parameter s defined by dt = r squared times ds, or alternatively a parameter tau defined by dt = (r squared divided by angular momentum) times d tau. For Kepler orbits, tau corresponds to the true anomaly up to an additive constant—a physically meaningful quantity long used by orbital analysts. Under either parameterization, the equations for the rotational coordinates (q, p) become those of a special orthogonal rotation, solvable in closed form via the Rodrigues rotation formula or equivalently via matrix exponentials. The radial equations, meanwhile, become those of a perturbed linear harmonic oscillator precisely when the central potential is of Manev type, V = -k1/r – k2/(2 r squared), with Kepler gravity appearing as the special case k2 = 0.

The Manev potential deserves particular attention. Introduced originally as a classical approximation to certain relativistic corrections, it preserves the conic orbits and integrability of the Kepler problem while adding perihelion precession—the slow rotation of an orbit’s closest approach point that also afflicts Mercury in Einstein’s general relativity. That the same projective machinery linearizes Manev dynamics as readily as Kepler dynamics suggests the framework extends naturally beyond the idealized inverse-square law. In the transformed system, the radial oscillator has a frequency determined by the difference between the squared angular momentum and the Manev parameter k2, and closed-form solutions follow immediately from standard oscillator theory.

Practical consequences for astrodynamics flow directly from these formal results. Because the linearized system admits closed-form solutions, closed-form state transition matrices—the sensitivity matrices that map small changes in initial conditions to changes in the final state—become readily available. State transition matrices are workhorses of modern mission design: they underpin orbit determination, uncertainty propagation, station-keeping, and targeting algorithms. Obtaining them in closed form, rather than integrating the variational equations numerically, promises both computational savings and improved long-term accuracy for orbit propagation, a goal that earlier linearization schemes such as the Kustaanheimo–Stiefel (KS) quaternionic transformation and the BF transformation have long pursued.

The authors demonstrate the reach of the method with a worked example of considerable practical importance: the J2-perturbed Kepler problem, often called the main satellite problem. Here the perturbation arises from the oblateness of the central body, such as Earth, whose equatorial bulge modifies the gravitational potential at leading order. Under the projective transformation, the perturbing potential becomes a polynomial expression in the new coordinates, and the resulting perturbation forces enter the linearized equations as explicit, tractable terms. This is exactly the sort of structure that perturbation theorists and designers of symplectic integrators prize, since polynomial perturbations of a linear oscillator can be handled by well-developed analytical and numerical machinery. Notably, the transformation preserves the form of angular momentum across the entire family of projective coordinates, and the constraint function and Lagrange multiplier turn out to be integrals of motion even in the presence of arbitrary, possibly nonconservative, perturbing forces—a robustness the authors verify with Poisson-bracket calculations.

The historical lineage of this work is rich. Regularization techniques—coordinate and time transformations that tame the singularity at collision—date back over a century, with contributions from Burdet, Vitins, Silver, Schumacher, Bond, Cid and colleagues, and a modern renaissance by Roa, Majji, Baù and collaborators. The KS transformation, rooted in spinor algebra and now commonly interpreted through quaternions, remains the most celebrated Hamiltonian linearization of the Kepler problem. Moser’s stereographic-projection approach offers yet another route. What the new paper adds is a projective alternative developed from first principles via Hamilton’s principle, unifying and extending the Burdet–Ferrándiz line of work while clarifying which choices of transformation parameters yield linear dynamics and why. The derivation that n must equal -1 for Kepler–Manev linearization is given explicitly, removing guesswork from the construction.

The authors point toward several promising future directions: quantitative comparisons with the KS transformation in terms of computational efficiency and long-term propagation stability, the derivation of action–angle coordinates for use in symplectic perturbation methods, and exploitation of the combined linearity and Hamiltonian structure for symplectic integration or even the quantization of classical central-force dynamics. For a field in which spacecraft navigation, debris tracking, and interplanetary mission design all hinge on accurately propagating orbits over long spans of time, a transformation that renders the underlying dynamics linear while preserving the full Hamiltonian structure is more than a mathematical curiosity—it is a practical instrument. If the promised efficiency gains materialize, these projective coordinates may soon find their way from the pages of a dynamics journal into the flight software of real missions.

Subject of Research: Projective canonical transformations that linearize central-force orbital dynamics, including Kepler and Manev problems, in a Hamiltonian framework.

Article Title: Projective transformations for linearized and regularized central-force dynamics: Hamiltonian formulation

Article References: Peterson, J. T. A., Majji, M., & Junkins, J. L. (2026). Projective transformations for linearized and regularized central-force dynamics: Hamiltonian formulation. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 54. https://doi.org/10.1007/s10569-026-10304-3

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10304-3

Keywords: celestial mechanics, Hamiltonian dynamics, canonical transformations, projective coordinates, Kepler problem, Manev potential, regularization, linearization, state transition matrices, J2 perturbation, orbital mechanics, symplectic structure

The mathematical heart of the new system is the Hori-Deprit method, a canonical perturbation technique based on Lie transformations that was developed in the 1960s by Gen-ichiro Hori and André Deprit. The method works on a Hamiltonian, the energy function that governs orbital motion, expanded in powers of a small parameter representing the strength of a perturbing force. Through a sequence of recursive canonical transformations, the algorithm systematically strips away the short periodic oscillations in the satellite’s motion, leaving behind a simplified, averaged Hamiltonian that captures only the long-term, secular evolution of the orbit. The approach rose to fame when Deprit and Rom used it in 1970 to produce an automated third-order solution to the main problem of artificial satellite theory, eliminating the small-eccentricity singularities that had plagued earlier analytical solutions.

What the new framework adds is a fully automated, open pipeline for carrying out such calculations at modern scale. The symcelmech package represents Hamiltonians as Poisson series, multivariate Fourier series whose coefficients are themselves Laurent series in the orbital variables, and it manipulates them within a closed algebraic domain so that differentiation, averaging and integration never produce unmanageable new mathematical objects. A dedicated Poisson bracket engine, equipped with memoization that caches previously computed brackets and exploits antisymmetry to avoid redundant work, drives the recursive Lie transformation. Because high-order analytical theories are often defeated not by physics but by expression swell, the explosive growth of symbolic terms, the authors devoted considerable effort to controlling that growth, including targeted rational simplification, selective trigonometric expansion on a single chosen variable, and eccentricity truncation that caps the number of terms entering each stage of the recursion.

The automation extends to the most computationally delicate step: integration. Maxima’s general-purpose symbolic integrator relies on heuristic searches that can buckle when asked to integrate expressions containing thousands of trigonometric terms. The researchers therefore wrote specialized routines that exploit the structure of Poisson series. Averaging is performed through a divide-and-conquer routine that classifies each term as secular or periodic in linear time, while quadrature for the generating functions is carried out by mapping trigonometric terms into the complex exponential domain, where integration becomes simple division by the angular frequency. For perturbations naturally expressed in the true anomaly, such as third-body gravity and solar radiation pressure, the framework offers closed-form averaging routes based on Hansen coefficients or on a change of variable to the eccentric anomaly, preserving exact eccentricity dependence rather than relying on truncated series.

Once the algebra is complete, the analytical expressions are exported as optimized text files and parsed by CelestialMechanics.jl, which converts them into fast Julia functions ready for numerical evaluation. The Julia toolkit assembles full-fidelity force models, including zonal harmonics up to degree 18, tesseral and sectorial gravity terms, third-body perturbations and solar radiation pressure with shadow models, drawing ephemerides from NASA’s SPICE library and gravity fields from standard planetary data files. It supports three distinct propagation modes, Cartesian Cowell integration, integration of Hamilton’s equations in Delaunay variables, and integration of Lagrange’s planetary equations in classical orbital elements, all feeding a unified post-processing pipeline that converts states into comparable orbital elements and generates scientific visualizations. For long-duration runs, trajectory data are streamed to disk to avoid exhausting memory.

The authors validated the pipeline in two increasingly demanding applications. In the first, they used symcelmech to normalize the Hamiltonian of the main J2 problem, the dominant perturbation caused by Earth’s equatorial bulge, through two successive Lie transformations that removed both the fast orbital angle and the argument of perigee. The entire symbolic computation, including a second-order generating function with thousands of terms, ran in a fraction of a second on an ordinary laptop. Predictions of the secular precession rates of perigee and ascending node derived from the doubly averaged Hamiltonian were then checked against two independent numerical propagations of the full, non-averaged J2 dynamics, agreeing to within roughly a part in a thousand, exactly consistent with the expected size of the neglected third-order contributions.

The second application targeted frozen orbits around the Moon, special orbits whose eccentricity and argument of perilune remain constant over time, making them ideal for stable low-altitude observation missions. The team built a high-fidelity lunar gravity model including zonal harmonics from J2 through J9 together with the leading tesseral and sectorial coefficients, and normalized the Hamiltonian in the Moon’s rotating body-fixed frame. Solving the frozen orbit conditions across a grid of semi-major axes and inclinations revealed distinct families of equilibria: bands of low-eccentricity frozen orbits at intermediate inclinations, a polar family with eccentricities below 0.01, and narrow strips of higher-eccentricity solutions near the equator driven by the odd zonal harmonics. Phase portraits confirmed the expected libration structure, with stable equilibria surrounded by closed energy contours and separatrices dividing librating from circulating orbits.

A representative near-polar frozen orbit was then propagated for 800 days, roughly nine thousand lunar revolutions, using an eighth-order symplectic integrator with a fixed one-minute step. The argument of perilune librated around 270 degrees with an amplitude of about 70 degrees, confirming that the orbit remained trapped in the libration island predicted by the analytical theory, while the periapsis altitude stayed confined between 165 and 185 kilometers. Crucially, the Jacobi constant, an exact integral of motion in the rotating frame, was conserved to about five parts in one hundred million with no secular drift, demonstrating that the long-term evolution observed in the simulation reflected genuine physics rather than numerical artifacts. The authors also documented a known limitation honestly: near-circular frozen orbits expose a coordinate singularity in Delaunay variables that will require reformulating the transformation in Poincare variables, a development already partially implemented.

Beyond the immediate results, the significance of the work lies in its openness and its architecture. The ecosystem of analytical celestial mechanics has long been dominated by proprietary mission analysis suites whose internal algorithms cannot be audited or extended by the community, and by aging research codes tied to obsolete languages. By releasing both symcelmech and CelestialMechanics.jl as open-source projects, the researchers hope to democratize access to sophisticated perturbation methods and invite collaborative refinement. Planned extensions include the elimination of the parallax in polar-nodal variables to tame memory demands at third order, computation of Birkhoff normal forms for stability analysis, non-singular Poincare formulations for near-circular orbits, and integration with the SciML ecosystem to explore physics-informed neural networks and machine-learned surrogate models for long-term orbit prediction. For an era of rapidly multiplying lunar and planetary missions, the framework offers a rigorous, transparent and computationally efficient foundation for designing the orbits of the future.

Subject of Research: An open-source hybrid symbolic-numerical framework automating Hori-Deprit perturbation theories for artificial satellite dynamics and space mission design.

Article Title: A hybrid symbolic-numerical framework for artificial satellite theory and dynamics using Maxima and Julia

Article References: de Oliveira Paes, G., Berton, L., & de Moraes, R. V. (2026). A hybrid symbolic-numerical framework for artificial satellite theory and dynamics using Maxima and Julia. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 56. https://doi.org/10.1007/s10569-026-10331-0

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10331-0

Keywords: celestial mechanics, artificial satellite theory, Hori-Deprit method, Lie transformations, computer algebra, Maxima, Julia, Poisson series, frozen orbits, lunar orbits, Hamiltonian dynamics, orbital perturbation theory