Conserved Quantities of Optimal Continuous-Thrust Trajectories in A Central Gravitational Field

Aimar Negrete; Ossama Abdelkhalik

REVIEW 3 major objections 5 minor 17 references

This paper derives four constants of motion for optimal continuous-thrust trajectories in a central gravitational field, and three are new.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-05 00:11 UTC pith:DTJDSEGL

load-bearing objection A correct derivation of known invariants; the novelty claim is the load-bearing weakness. the 3 major comments →

arxiv 2608.00842 v1 pith:DTJDSEGL submitted 2026-08-01 physics.class-ph cs.SYeess.SY

Conserved Quantities of Optimal Continuous-Thrust Trajectories in A Central Gravitational Field

Aimar Negrete , Ossama Abdelkhalik This is my paper

classification physics.class-ph cs.SYeess.SY

keywords conserved quantitiesNoether's theoremKilling equationsgeneralized Lagrangiancontinuous-thrust trajectory optimizationcentral gravitational fieldminimum-energy optimal control

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper works in the fixed-time, minimum-energy version of the continuous-thrust orbit transfer problem: minimize 1/2 ∫ |u|^2 dt subject to r¨ = -μ r/r^3 + u. It claims that along every optimal trajectory of this problem, the four expressions Φ1–Φ4 in Eq. (61) are constant; Φ4 is the usual conserved Hamiltonian, while Φ1–Φ3 are described as new conserved quantities. The value of having such invariants is that they give analytic checks on numerically optimized trajectories, reveal structure in the optimal control, and reduce the order of the system the same way energy and angular momentum do in classical mechanics. The paper proves the invariance by direct differentiation and by Noether's theorem, and demonstrates it numerically on inclined LEO-to-GEO transfers including a 190-revolution spiral.

Core claim

The paper's claim is that the variational structure of the minimum-energy optimal control problem in an inverse-square field is rich enough to carry integrals of motion beyond energy. Working with a generalized Lagrangian L = ẋ·ů - (μ/r^3) r·u + (1/2) u·u, the authors formulate Killing equations whose solutions are infinitesimal transformations leaving the action invariant. Applying the divergence-invariant form of Noether's theorem yields four independent conserved quantities: three rotational-type invariants that mix position, velocity, control, and control rate (Φ1, Φ2, Φ3), and one energy-type invariant Φ4 equal to the conserved Hamiltonian. The direct proof substitutes the extremal equ

What carries the argument

The machinery is a generalized Lagrangian that turns the optimal control problem into a variational problem without costates, together with the divergence-invariant form of Noether's theorem. The Killing equations are the constraint PDEs enforcing invariance of the action under infinitesimal coordinate and time transformations; their solution gives the generators η and ξ. Each generator plugged into Φ = ∂L/∂q̇·(ξq̇ - η) - ξL + φ yields one conserved quantity. The rotation generators mix position and control (η acts on both r and u), which is why the resulting integrals couple state and control variables.

Load-bearing premise

The derivation assumes the control magnitude is unbounded and the transfer time is fixed with a minimum-energy cost; if a real mission's thrust is bounded, these expressions need not stay constant on arcs where the bound is active.

What would settle it

Solve a minimum-energy transfer with a thrust-magnitude constraint |u| ≤ u_max that has a saturated arc and evaluate Φ1–Φ4 from Eq. (61) along that trajectory: any drift beyond integration error on the constrained arc would show the invariants fail once the unbounded-control assumption is violated.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

Every fixed-time, minimum-energy optimal transfer in an inverse-square field carries four functions that are exactly constant along the extremal, so any candidate optimal trajectory can be checked against them.
In the two-dimensional version of the problem, two of the four invariants vanish identically and the remaining rotational invariant reduces to the planar angular-momentum-type law.
Because the invariants also hold in polar and spherical frames when carried through the generator transformation, they can be written in whatever coordinates a solver uses.
For long, low-thrust spirals, the invariants provide a sensitive numerical test: the paper notes that very tight integrator tolerances are needed to keep the smallest invariant flat, indicating their use as an error diagnostic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

One use the authors do not spell out: the four constants could serve as mission-independent validation metrics for direct optimization software, since a converged solution that slowly drifts in Φ_i has likely not actually reached the minimum-energy extremal.
The rotational generators rotate position and control together; this suggests a generalized 'angular momentum' that couples the steering law to the spacecraft position, and it would be interesting to test whether its presence explains features of minimum-energy transfers such as the structure of the steering profile in multi-revolution spirals.
A natural extension is to add a terminal-cost term or boundary potential: if the symmetry is broken, the same Noether procedure should produce balance equations (rates of change of Φ_i) that could serve as transversality conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A correct derivation of known invariants; the novelty claim is the load-bearing weakness.

read the letter

Quick take: the algebra is right and the numerics back it up, but the headline claim — three new conserved quantities — does not survive contact with the literature. The three Phi's are components of r × u_dot − v × u, which are exactly the rotational Noether charges of the Pontryagin Hamiltonian for the minimum-energy problem. That makes them standard first integrals of the primer-vector equations, not new conservation laws. The paper needs to acknowledge this and reposition its contribution.

What is genuinely good: the derivation from the generalized Lagrangian is a coherent alternative route to these integrals. The Killing-equation setup is legitimate, and the direct differentiation proof in Eq. (62) is clean. The appendix's check against the PMP is honest and non-circular. Numerically, the invariants hold to 1e-14 over both a fast and a long spiral transfer, which is what you would want.

Soft spots. First, the novelty claim is not defended; there is no comparison with the extensive primer-vector and optimal-control first-integral literature. Saying "to the best of our knowledge" is not enough. Second, the step "It can be shown that the only consistent solution…" in Section 4.2 is a real gap; a serious referee should ask for the full argument that the Killing equations admit only rotations and time translation. Third, the spherical-frame generators for Phi2 and Phi3 are written down but the corresponding conserved quantities are never explicitly derived or verified; for a paper whose point is to produce invariants, that is unfinished. Fourth, the unbounded-control assumption is stated clearly, so it is not a hidden flaw, but it does limit practical relevance — the quantities will not be conserved on arcs with active thrust constraints.

The paper is worth reading for its method and for a compact derivation of known invariants. It could be a reasonable contribution after major revision, with the novelty claim corrected and the gaps filled. I would send it to peer review, but I would not cite it as a source of new conservation laws.

Referee Report

3 major / 5 minor

Summary. The paper considers the fixed-time, minimum-energy continuous-thrust trajectory optimization problem in a central gravitational field. The authors use a 'generalized Lagrangian' from their prior work, in which the control variables are treated as additional generalized coordinates. They formulate Killing equations for the invariance of the resulting action functional, solve them in the Cartesian case, and apply the divergence-invariant form of Noether's theorem to obtain four conserved quantities Phi1-Phi4 (Eq. 61). They state that Phi1-Phi3 are new and that Phi4 is the Hamiltonian conservation law. They prove directly that Phi3 is invariant along solutions of the optimal-control equations (Eq. 62), state that analogous proofs hold for Phi1 and Phi2, and demonstrate conservation numerically for two LEO-GEO transfers. They also transform the infinitesimal generators to polar and spherical coordinates and obtain the corresponding conserved quantities in those frames.

Significance. If the novelty claim were established, the paper would offer a systematic Noether-type construction of first integrals for an important class of optimal space trajectories, potentially aiding indirect methods by reducing the boundary-value problem. The manuscript's mathematical core is largely sound: the generalized Lagrangian is validated against the Pontryagin Maximum Principle in Appendix 6.1, the invariance of Phi3 is proven by substituting the equations of motion, and the numerical integrations show constancy at the 1e-14 level for all four quantities. The paper also honestly states its scope, namely fixed time, minimum energy, and unbounded control. However, the central claim that Phi1-Phi3 are 'new' is not supported and is in fact likely incorrect: these are exactly the rotational Noether integrals of the PMP Hamiltonian. The paper therefore contributes an alternative derivation and a coordinate-transformation procedure, but the claimed novelty is the weakest point and must be addressed.

major comments (3)

[Abstract and §4.3, Eq. (61)] The novelty claim that Phi1-Phi3 are 'new conserved quantities in this problem' is unsupported and, on the basis of standard optimal-control theory, likely false. For the fixed-time minimum-energy problem, the PMP Hamiltonian is H = 1/2|u|^2 + lambda_r·v + lambda_v·(-mu/r^3 r + u). Rotational symmetry of H implies the Noether integral J = r×lambda_r + v×lambda_v. Using the costate relations lambda_v = -u and lambda_r = dot u (from Eqs. (73)-(75) of the appendix) gives J = r×dot u - v×u, whose components are precisely (Phi1, Phi2, Phi3) up to sign. Thus these are the standard rotational-symmetry integrals of the extended optimal-control phase space, not specialized new conservation laws. The manuscript does not compare with any existing literature on first integrals of the primer-vector equations or Noether-type theorems in optimal control. The authors need to either demonstrate that thes
[§4 (scope) and abstract/title] The title and abstract claim 'optimal continuous-thrust trajectories' without qualification, but the derivation is explicitly for fixed-time, minimum-energy trajectories with an unbounded control magnitude. Equations (27) and the conservation laws hold only when no control constraint is active. On arcs with bounded thrust, the optimality conditions change and Phi1-Phi4 are not expected to be constant. Although Section 4 states this scope in a single sentence, the unqualified framing in the title and abstract will mislead readers, especially because the numerical examples are LEO-GEO transfers of practical interest. The authors should clearly state in the title or abstract that the results apply to the unconstrained minimum-energy problem, and should discuss the constrained-thrust case in the conclusions.
[§4.2, Eqs. (58)-(59)] The step 'It can be shown that the only consistent solution to this equation is...' is a key point in deriving the symmetry generators. No derivation is provided for Eq. (58), and the completeness of the resulting set of generators is asserted rather than proven. For a paper whose main methodological contribution is the Killing-equation route, this gap is important. The final invariants are independently verified by direct differentiation in Eq. (62), so the invariance claim does not rest on this step, but the claimed classification of the admissible symmetries does. The authors should either supply the derivation or, at minimum, clearly state that they do not prove completeness of the symmetry algebra and that the listed generators are only a found set.

minor comments (5)

[§4.2, Eq. (59)] There is a sign inconsistency: the last equation reads eta4 = k2 ux, but the vector form given immediately afterward is [y, -x, uy, -ux]^T, which requires eta4 = -k2 ux. The same sign is used correctly in Eq. (60).
[§4.3, Eq. (62)] The direct proof is shown only for Phi3. The text says similar proofs can be constructed for Phi1 and Phi2; to make the paper self-contained, at least one of those should be included or the proof for all three should be supplied in the appendix.
[Abstract and §1] The abstract contains an extra period after 'system..' and the phrase 'conservative quantities' in the introduction should be 'conserved quantities'.
[§4.4, Eqs. (66)-(68)] There is a cross-reference error: 'a Taylor series of Eq. (71)' should refer to the transformed polar coordinates, which are in Eq. (66). Also, the sign of the generator eta2D is written as [-y,x,-uy,ux]^T in this section, while the earlier 2D generator was [y,-x,uy,-ux]^T; these differ by an overall sign and should be made consistent.
[§5, Figs. 3 and 6] Plotting |Phi| on a logarithmic scale with the signed value given in text is difficult to read. Consider separate linear plots with appropriate scaling, or use a sign-aware visualization, to allow the reader to verify constancy of the signed quantities.

Circularity Check

0 steps flagged

No significant circularity: the generalized Lagrangian is independently validated against Pontryagin's maximum principle, and the conserved quantities are verified by direct differentiation.

full rationale

The central derivation chain is not circular. The generalized Lagrangian in Eq. (24) is taken from the authors' prior work [9], but this is not load-bearing: Appendix 6.1 independently derives the same control differential equations from the Pontryagin Maximum Principle, using the standard primer-vector result, and shows they match Eq. (27). Thus the Lagrangian is externally grounded rather than being an unverified self-citation. The conserved quantities Φ1–Φ4 in Eq. (61) are obtained by solving the Killing equations (35)–(47) and applying Noether's theorem, an external mathematical result; the subsequent direct differentiation check, e.g., Eq. (62), demonstrates invariance using only the equations of motion. No parameter is fitted to the conserved quantities and then 'predicted' back; the numerical examples are demonstrations along propagated optimal trajectories, not fitted predictions. The coordinate-transformation section similarly transforms the symmetry generators and verifies the resulting polar conserved quantity by direct differentiation. The statement that three quantities are 'new' is a novelty claim, not a circularity claim; even if those quantities are standard rotational Noether integrals of the PMP Hamiltonian, that would be a correctness/novelty concern, not a circularity of derivation. Therefore no circular step is present and the appropriate score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities and fits no free parameters. The derivation rests on the generalized Lagrangian (the authors' own construction, justified against PMP) and on standard symmetry theory.

axioms (4)

domain assumption The generalized Lagrangian (24) correctly represents the minimum-energy optimal control problem.
Taken from the authors' prior work [9]; Appendix 6.1 sketches a proof via PMP that the resulting equations match the primer-vector theory.
standard math Noether's theorem and the invariance condition E{L} = -xi_dot L + phi_dot apply to the generalized Lagrangian.
Standard results in Lie group analysis of variational problems; cited to Ref. [15].
domain assumption The optimal control is unbounded in magnitude and continuous in time.
Explicit in Section 4; the derived equations (27) fail if control saturation occurs.
domain assumption The gravitational field is the classical inverse-square central field with parameter mu.
The Lagrangian (24) is built for this potential.

pith-pipeline@v1.3.0-alltime-deepseek · 18262 in / 19834 out tokens · 183721 ms · 2026-08-05T00:11:50.826016+00:00 · methodology

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Cite this review

Pith. "Pith review of Conserved Quantities of Optimal Continuous-Thrust Trajectories in A Central Gravitational Field." pith.science (2026). https://pith.science/paper/DTJDSEGL

@misc{pith2026260800842,
  author       = {Pith},
  title        = {Pith review of: Conserved Quantities of Optimal Continuous-Thrust Trajectories in A Central Gravitational Field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DTJDSEGL}},
  note         = {Machine review of arXiv:2608.00842}
}

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read the original abstract

This paper presents a mathematical derivation for three new conserved quantities in the motion of spacecraft on optimal continuous-thrust trajectories in a central gravitational field. The process presented in this paper is rooted in Noether's theorem that connects the point symmetries of a dynamic system with the associated conservation laws of the system. In the approach presented in this paper, the system's Lagrangian is modified to account for the non-conservative control force. Using this generalized Lagrangian, the action functional to be minimized is written. Then, Killing equations are formulated to find the dynamic symmetries for this system. In this paper a process is laid out for how to solve the Killing equations; Noether's theorem is applied to this solution of the Killing equations to write the conserved quantities of the system. Conserved quantities are presented for both the two-dimensional and the three-dimensional trajectories in several different coordinate frames. Numerical simulations and mathematical proofs are used to demonstrate that the computed quantities are conserved.

Figures

Figures reproduced from arXiv: 2608.00842 by Aimar Negrete, Ossama Abdelkhalik.

**Figure 1.** Figure 1: Optimal LEO to GEO transfer, ∆t ≈ 40 hours the state and control time histories for this transfer. The resulting values were plotted over the transfer time in [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗

**Figure 2.** Figure 2: Control time history for the short time LEO to GEO transfer [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Conserved quantities for the short time LEO to GEO transfer [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Optimal LEO to GEO transfer, ∆t ≈ 46 days [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: Control time history for the long time LEO to GEO transfer [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Conserved quantities for the long time LEO to GEO transfer [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 13 canonical work pages

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MIT Press, 1949

Norbert Wiener.Extrapolation, Interpolation, and Smoothing of Stationary Time Series. MIT Press, 1949. 19 APREPRINT- AUGUST4, 2026

work page 1949

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Princeton University Press, 1957

Richard Bellman.Dynamic Programming. Princeton University Press, 1957

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Pontryagin, V

Lev S. Pontryagin, V . G. Boltyanskii, R. V . Gamkrelidze, and E. F. Mishchenko.The Mathematical Theory of Optimal Processes. Interscience Publishers, 1962

work page 1962

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Arthur E

Jr. Arthur E. Bryson and Yu-Chi Ho.Applied Optimal Control optimization,estimation and control. CRC Press„ 1975

1975

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John T. Betts. Survey of numerical methods for trajectory optimization.Journal of Guidance, Control, and Dynamics, 21(2):193–207, 1998

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Springer, 3rd edition, 2002

Josef Stoer and Roland Bulirsch.Introduction to Numerical Analysis. Springer, 3rd edition, 2002

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Betts.Practical Methods for Optimal Control and Estimation Using Nonlinear Programming

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Anil V Rao. A survey of numerical methods for optimal control.Advances in the Astronautical Sciences, 135(1):497–528, 2009

work page 2009

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Aimar Negrete and Ossama Abdelkhalik. Variational methods of analytical mechanics for trajectory optimization. InAAS/AIAA Space Flight Mechanics Meeting, number AAS 25-416, Kaua’i, Hawaii, January 19-23 2025

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work page 1978

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A group-variational procedure for finding first integrals of dynamical systems.International Journal of Non-Linear Mechanics, 5(2):269–278, 1970

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Conserved Quantities of Optimal Continuous-Thrust Trajectories in A Central Gravitational Field

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it