Studying the stability of equilibrium solutions in the planar circular restricted four-body problem
The Newtonian circular restricted four-body problem is considered.We have obtained nonlinear algebraic equations, determining equilibrium solutions in the rotating frame, and found six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we have proved...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2007
|
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/7243 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Studying the stability of equilibrium solutions in the planar circular restricted four-body problem / E.A. Grebenikov, L. Gadomski, A.N. Prokopenya // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 66-82. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-7243 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-72432010-03-29T12:01:16Z Studying the stability of equilibrium solutions in the planar circular restricted four-body problem Grebenikov, E.A. Gadomski, L. Prokopenya, A.N. The Newtonian circular restricted four-body problem is considered.We have obtained nonlinear algebraic equations, determining equilibrium solutions in the rotating frame, and found six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we have proved that the radial equilibrium solutions are unstable while the bisector equilibrium solutions are stable in Liapunov’s sense if the mass parameter μ belongs (0, μ0), where μ0 is 0 a sufficiently small number, and μ ≠ μj , j = 1, 2, 3. We have also proved that for μ = μ1 and μ = μ3 the resonance conditions of the third and the fourth orders, respectively, are fulfilled and for these values of μ the bisector equilibrium are unstable and stable in Liapunov’s sense, respectively. All symbolic and numerical calculations are done with the computer algebra system Mathematica. Розглядається кругова зрiзана проблема ньютонiвської динамiки для чотирьох тiл. Отримано нелiнiйнi алгебраїчнi рiвняння, що визначають рiвноважнi розв’язки вiдносно обертаючої системи вiдлiку, та знайдено шiсть рiвноважних конфiгурацiй системи. При вивченнi стiйкостi рiвноважних розв’язкiв доведено, що радiальнi рiвноважнi розв’язки є нестiйкими, проте бiсекторнi рiвноважнi розв’язки є стiйкими за Ляпуновим, якщо параметр маси μ належить (0, μ0), де μ0 - досить мале число, і μ ≠ μj , j = 1, 2, 3. Також доведено, що для μ = μ1 та μ = μ3 умови резонансу вiдповiдно третього та четвертого порядкiв виконано, iдля цих значень µ бiсекторнi рiвноважнi розв’язки є вiдповiдно нестiйкими та стiйкими за Ляпуновим. Усi символьнi та числовi обчислення виконано за допомогою системи комп’ютерної алгебри "Математика". 2007 Article Studying the stability of equilibrium solutions in the planar circular restricted four-body problem / E.A. Grebenikov, L. Gadomski, A.N. Prokopenya // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 66-82. — Бібліогр.: 17 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/7243 517.9 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The Newtonian circular restricted four-body problem is considered.We have obtained nonlinear algebraic equations, determining equilibrium solutions in the rotating frame, and found six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we have proved that the radial equilibrium solutions are unstable while the bisector equilibrium solutions are stable in Liapunov’s sense if the mass parameter μ belongs (0, μ0), where μ0 is 0 a sufficiently small number, and μ ≠ μj , j = 1, 2, 3. We have also proved that for μ = μ1 and μ = μ3 the resonance conditions of the third and the fourth orders, respectively, are fulfilled and for these values of μ the bisector equilibrium are unstable and stable in Liapunov’s sense, respectively. All symbolic and numerical calculations are done with the computer algebra system Mathematica. |
| format |
Article |
| author |
Grebenikov, E.A. Gadomski, L. Prokopenya, A.N. |
| spellingShingle |
Grebenikov, E.A. Gadomski, L. Prokopenya, A.N. Studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| author_facet |
Grebenikov, E.A. Gadomski, L. Prokopenya, A.N. |
| author_sort |
Grebenikov, E.A. |
| title |
Studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| title_short |
Studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| title_full |
Studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| title_fullStr |
Studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| title_full_unstemmed |
Studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| title_sort |
studying the stability of equilibrium solutions in the planar circular restricted four-body problem |
| publisher |
Інститут математики НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/7243 |
| citation_txt |
Studying the stability of equilibrium solutions in the planar circular restricted four-body problem / E.A. Grebenikov, L. Gadomski, A.N. Prokopenya // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 66-82. — Бібліогр.: 17 назв. — англ. |
| work_keys_str_mv |
AT grebenikovea studyingthestabilityofequilibriumsolutionsintheplanarcircularrestrictedfourbodyproblem AT gadomskil studyingthestabilityofequilibriumsolutionsintheplanarcircularrestrictedfourbodyproblem AT prokopenyaan studyingthestabilityofequilibriumsolutionsintheplanarcircularrestrictedfourbodyproblem |
| first_indexed |
2023-10-18T16:37:03Z |
| last_indexed |
2023-10-18T16:37:03Z |
| _version_ |
1796139450433011712 |