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On some restricted four-body problem

Seminar ECONET ‘2009, 26 - 31 October Bled, Slovenia. On some restricted four-body problem. Dmitry Budzko. Brest State University Bul. Kosmonavtov 21, 224016, Brest, Belarus master_booblik@tut.by.

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On some restricted four-body problem

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  1. Seminar ECONET ‘2009,26 - 31 OctoberBled, Slovenia On some restricted four-body problem Dmitry Budzko Brest State University Bul. Kosmonavtov 21, 224016, Brest, Belarus master_booblik@tut.by

  2. where are momentums, canonically conjugated to coordinates and parameters and . Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  3. Hamiltonian equations: Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  4. The equations, determining equilibrium positions Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  5. Equilibrium positions under , Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  6. Motion of equilibrium positions under , Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  7. Carrying out expansion of the Hamiltonian function in Taylor series in the neighborhood of equilibrium solution we obtain Hamiltonian in the form: Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  8. Linearized equations of perturbed motion Characteristic exponents can be easily found and are given by Necessary and sufficient linear stability condition of the system Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  9. Transformation of the quadratic part of Hamiltonian to normal form Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  10. Linear stability of equilibrium solutions Stability condition of triangular configuration in linear approximation (obtained by Routh in 1875 ) There are exactly eight equilibrium positions for any values of . Theorem 1.The positions of equilibrium of system are unstable in Liapunov’s sense for any parameters from the domain, restricted by inequality. Theorem 2.The equilibrium positions and are stable in linear approximation if parameters posses from the domain shown on following figures. Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  11. Stability boundary (by heavy solid line), third and fourth orderresonance curves (by dashed lines) in the domain (by thin solid line)of the equilibrium position Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  12. The stability boundary of equilibrium position Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  13. Following Birkgoff, let us construct canonical change of variables, for normalizing thus that it will be cancelled. where Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  14. Coefficients can be divided into three independent groups Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  15. Analyzing characteristic exponents, it is easy to observe that in the domain of linear stability there exists only one resonance curve, corresponding to resonance condition Then , and we can not setand some terms in the expansion of the Hamiltonian will survive. where and are arbitrary constants. Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  16. Then using standard canonical transformation where Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  17. Dependencefrom . Theorem.Equilibrium solutions of restricted four-body problem, formulated on the basis of Lagrange’s triangular solutions are unstable in Liapunov’s sense in the case of resonance of third order. Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  18. The case of resonance of fourth order, when Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  19. According to Markeev’s theorem equilibrium position is stable if the following inequality is fulfilled Theorem.Equilibrium solutions of restricted four-body problem, formulated on the basis of Lagrange’s triangular solutions are stable in Liapunov’s sense in the case of resonance of fourth order if . Equilibrium position is unstable under resonance condition under any values of . Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  20. Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  21. Conclusions We found all equilibrium solutions of planar four-body problem, formulated on the basis of Lagrange’s triangular solutions. Equilibrium solutions have been analyzed in linear approximation. Its has been shown that only three of eight equilibrium positions are linearly stable. Stability boundaries for these positions have been constructed. We have constructed the Birghoff’s transformations that normalize third and fourth order terms of Hamiltonian function. The cases of resonance of third and fourth orders were considered, corresponding theorems were formulated. Now according to Arnold-Moser theorem we have missed one case, when . In this case we have a curve, in which points, we have to calculate the value of , and then the stability problem for this model will be completely solved. Dmitry Budzko 26-31 October, Bled, Slovenia 2009

  22. Thank you for your attention!  Dmitry Budzko 26-31 October, Bled, Slovenia 2009

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