About a Classical Gravitational Interaction in a General Non-Inertial Reference Frame: Applications on Celestial Mechanics and Astrodynamics
Abstract
:1. Introduction
2. Mathematical Preliminaries
2.1. A Tensor Operator
- 1.
- is invertible,
- 2.
- ;
- 3.
- ;
- 4.
- ;
- 5.
- , differentiable.
- 6.
- , differentiable.
2.2. Comments and Remarks
2.3. A Vector Differentiation Operator
- 1.
- ;
- 2.
- ;
- 3.
- , differentiable.
- 4.
- ;
- 5.
- ;
- 6.
- ;
- 7.
- ;
- 8.
- ;
- 9.
- (1)
- Solve IVP (12), the inertial problem in the frozen non-inertial reference frame at
- (2)
- Apply to the solution to the IVP (12) to determine the solution to IVP (13).
3. The N-Body Problem in a Non-Inertial Reference Frame: Some First Integrals
The Motion of the Barycenter of N-Body System
4. The Many-Body Motion System in the Barycenter Reference Frame
5. Application on Celestial Mechanics and Astrodynamics
5.1. The Two-Body Problem in Non-Inertial Reference Frames
5.1.1. The Motion of the Barycenter
5.1.2. The Motion Referred to the Barycenter Reference Frame
5.1.3. The Laws of the Motion of Particles in the Non-Inertial Reference Frame
5.1.4. The Gravitational Two-Body Problem in a Non-Inertial Reference Frame: An Exact Closed-Form Solution
- (a)
- if , the closed-form, coordinate-free solution of IVP (116) is given:
- (b)
- If , the closed-form, coordinate-free solution of IVP (116) is given:
- (c)
- For , the closed-form, coordinate-free solution of IVP (116) is given:
- (a)
- If , the closed-form, coordinate-free solution of IVP (116) is given:
- (b)
- If , the closed-form, coordinate-free solution of IVP (116) is given:
- (c)
- For , the closed-form, coordinate-free solution of IVP (116) is given:
5.2. Relative Orbital Motion of Spacecraft: A Closed-Form Coordinate-Free Solution
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
IVP | Initial value problem (IVP). |
Nomenclature
Euclidean three-dimensional vector | |
the norm of Euclidean three-dimensional vector | |
unit real vector associated to vector | |
real Euclidean tensor | |
trace invariant of real Euclidean tensor A | |
Euclidean there-dimensional vectors set | |
time depending real vectorial functions | |
skew–symmetric Euclidean tensor corresponding to the vector | |
real numbers set | |
proper orthogonal real Euclidean tensors set | |
time depending real proper orthogonal Euclidean tensors set | |
skew–symmetric real Euclidean tensors set | |
the set of functions of real variable, with values in . |
Appendix A
Appendix A.1. The Proof of Lemma 1
Appendix A.2. Closed-Form Solution of Kinematic Equation
- If instantaneous angular velocity ω is constant vector, then has the time explicit expression:
- If vector has a regular precession with angular velocity around a fixed axis, expressed like:
- A comprehensive study, together with a closed-form solution to the kinematic equation in the general case, may be found in Ref. [37]
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Condurache, D.; Cojocari, M.; Popa, I. About a Classical Gravitational Interaction in a General Non-Inertial Reference Frame: Applications on Celestial Mechanics and Astrodynamics. Symmetry 2025, 17, 368. https://doi.org/10.3390/sym17030368
Condurache D, Cojocari M, Popa I. About a Classical Gravitational Interaction in a General Non-Inertial Reference Frame: Applications on Celestial Mechanics and Astrodynamics. Symmetry. 2025; 17(3):368. https://doi.org/10.3390/sym17030368
Chicago/Turabian StyleCondurache, Daniel, Mihail Cojocari, and Ionuț Popa. 2025. "About a Classical Gravitational Interaction in a General Non-Inertial Reference Frame: Applications on Celestial Mechanics and Astrodynamics" Symmetry 17, no. 3: 368. https://doi.org/10.3390/sym17030368
APA StyleCondurache, D., Cojocari, M., & Popa, I. (2025). About a Classical Gravitational Interaction in a General Non-Inertial Reference Frame: Applications on Celestial Mechanics and Astrodynamics. Symmetry, 17(3), 368. https://doi.org/10.3390/sym17030368