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Some minimal characteristics and flat trajectories of a system in the N-body problem

  • Igor Pavlov
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Abstract

For the classical model of a system of N particles with interaction (the N-body problem) we consider some spatial problems of nonlinear programming for finding conditional minima and local extrema for the main characteristics of the system on its phase space. We prove that these conditional minima and local extrema of the system characteristics can only be achieved on flat, in general, not necessarily finite, trajectories of the system in space, (under the condition that the system angular momentum is non-zero). And then we find the basic conditions limiting the values of the integrals of motion: angular momentum and total energy of the system under which these trajectories are flat finite ("circular" or elliptical) trajectories on which the flat structure of the system is preserved in time under the certain restrictions on the values of the system's integrals of motion. And we determine also the values of the system's key characteristics on these flat trajectories. Next, in the Appendix, we consider some additional related results for a more general model of the system.

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Cite

BibTeXRIS
BibTeX
@misc{pavlov2026some,
  title = {{Some minimal characteristics and flat trajectories of a system in the N-body problem}},
  author = {Igor Pavlov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31582},
  eprint = {2610.31582},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31582},
}
RIS
TY  - UNPB
TI  - Some minimal characteristics and flat trajectories of a system in the N-body problem
AU  - Igor Pavlov
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31582
UR  - https://arxiv.org/abs/2610.31582
AB  - For the classical model of a system of N particles with interaction (the N-body problem) we consider some spatial problems of nonlinear programming for finding conditional minima and local extrema for the main characteristics of the system on its phase space. We prove that these conditional minima and local extrema of the system characteristics can only be achieved on flat, in general, not necessarily finite, trajectories of the system in space, (under the condition that the system angular momentum is non-zero). And then we find the basic conditions limiting the values of the integrals of motion: angular momentum and total energy of the system under which these trajectories are flat finite ("circular" or elliptical) trajectories on which the flat structure of the system is preserved in time under the certain restrictions on the values of the system's integrals of motion. And we determine also the values of the system's key characteristics on these flat trajectories. Next, in the Appendix, we consider some additional related results for a more general model of the system.
ER  -