A geometric algorithm for orbital dynamics based on Lie derivative

doi:10.1007/s10483-026-3377-7

Abstract

Abstract:

Orbital dynamics is a fundamental problem in celestial mechanics, yet its governing equations are characterized by irrational terms and denominator-type nonlinearities. Traditional numerical methods, which are locally discrete, may lead to ambiguities near singularities (e.g., x=y=z=0), thereby limiting the numerical stability and accuracy. To address these challenges, we propose a Lie derivative algorithm that constructs discrete iterative schemes based on the Lie series expansion. Unlike local schemes, this approach discretizes the vector field in a global manner, effectively avoiding singular inconsistencies while ensuring stable long-term integration. Numerical experiments demonstrate that, when compared with a high-accuracy reference solution under uniform step-size settings, the proposed approach not only achieves higher accuracy but also improves computational efficiency by up to 47% in the two-body problem and 67% in the circular restricted three-body problem, relative to classical second-order schemes. These results indicate that the Lie derivative algorithm provides an efficient and practical alternative for high-precision orbital dynamics computations.

Key words: geometric numerical algorithm, Lie derivative, high efficiency, three-body problem

2010 MSC Number:

O316

Yuhan SONG, Shixing LIU, Wenan JIANG, Yongxin GUO. A geometric algorithm for orbital dynamics based on Lie derivative. Applied Mathematics and Mechanics (English Edition), 2026, 47(5): 1157-1176.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3377-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I5/1157

Figures/Tables 18

Fig. 1

Fig. 2

Table 1

Fig. 3

Fig. 4

Fig. 5

Table 2

Relative logarithmic deviation of the global error with respect to the ODE45 reference solution for the two-body 1∶2 resonant orbit with e=0.6 andθ0=0°"

$lg (Δ t)$	Lie	RK2	SE	VI	ODE23
-5.0	-0.67	-0.94	-0.98	-0.95	-0.66
-4.7	-0.71	-1.03	-1.08	-1.03	-0.74
-4.4	-0.82	-1.10	-1.15	-1.11	-0.89
-4.1	-0.91	-1.18	-1.22	-1.18	-1.01
-3.8	-0.99	-1.27	-1.31	-1.27	-1.14
-3.5	-1.09	-1.42	-1.48	-1.43	-1.32
-3.2	-1.25	-1.70	-1.77	-1.70	-1.62
-2.9	-1.58	-2.26	-2.38	-2.28	-2.18
-2.6	-2.66	-4.09	-4.35	-4.14	-3.36

Table 2

Table 3

Fig. 6

Table 4

Table 5

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fg. 12

Fig. A1

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Eccentricity	CPU time/s		Improving efficiency
Eccentricity	Lie	RK2	Improving efficiency
0.1	0.029 7	0.037 4	21%
0.3	0.023 1	0.043 9	47%
0.5	0.024 0	0.038 2	37%
0.7	0.028 2	0.047 0	40%
0.9	0.023 9	0.032 7	27%

Resonance ratio	CPU time/s					Improving efficiency/%
Resonance ratio	Lie	RK2	SE	VI	ODE45	RK2-Lie	SE-Lie	VI-Lie	ODE45-Lie
1∶2	0.009 2	0.013 1	0.009 4	0.120 5	0.014 6	29.8	2.1	92.4	37.0

Initial phase/(°)	CPU time/s		Improving efficiency
Initial phase/(°)	Lie	RK2	Improving efficiency
0	0.006 0	0.011 1	46%
90	0.006 2	0.010 6	42%
180	0.006 2	0.011 0	44%
270	0.005 9	0.010 7	45%

Resonance ratio	CPU time/s		Improving efficiency
Resonance ratio	Lie	RK2	Improving efficiency
1∶1	0.009 5	0.017 3	45%
1∶2	0.007 6	0.012 2	38%
1∶3	0.010 3	0.017 3	40%
1∶4	0.012 9	0.021 8	41%
2∶1	0.005 2	0.015 6	67%
2∶3	0.780 4	1.411 2	45%
3∶1	0.007 4	0.020 6	64%
3∶2	0.003 3	0.007 0	53%
4∶1	0.003 6	0.010 6	65%