From the restricted to the full three–body problem

Meyer, Kenneth; Schmidt, Dieter

doi:10.1090/S0002-9947-00-02542-3

From the restricted to the full three–body problem
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by Kenneth R. Meyer and Dieter S. Schmidt PDF

Trans. Amer. Math. Soc. 352 (2000), 2283-2299 Request permission

Abstract:

The three–body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three–body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three–body problem with one small mass is to the first approximation the product of the restricted three–body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three–body problem.

For example, all the non–degenerate periodic solutions, generic bifurcations, Hamiltonian–Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three–body problem. The classic normalization calculations of Deprit and Deprit–Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three–dimensional KAM invariant tori near the Lagrange point in the reduced three–body problem.

References

Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
Dynamical systems. III, 2nd ed., Encyclopaedia of Mathematical Sciences, vol. 3, Springer-Verlag, Berlin, 1993. Mathematical aspects of classical and celestial mechanics; A translation of Current problems in mathematics. Fundamental directions, Vol. 3 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985 [ MR0833508 (87i:58151)]; Translation by A. Iacob; Translation edited by V. I. Arnol′d. MR 1292465
E. A. Belbruno, A new regularization of the restricted three-body problem and an application, Celestial Mech. 25 (1981), no. 4, 397–415. MR 638060, DOI 10.1007/BF01234179
G. D. Birkhoff, The restricted problem of three–bodies, Rend. Circolo Mat. Palermo, 39, 1915, 255–334.
—, Dynamical Systems, Amer. Math. Soc., Providence, RI, 1927.
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Charles Conley, On some new long periodic solutins of the plane restricted three body problem, Comm. Pure Appl. Math. 16 (1963), 449–467. MR 154724, DOI 10.1002/cpa.3160160405
A. Deprit and A. Deprit–Bartholomé, Stability of the Lagrange points, Astron. J., 72, 1967, 173–179.
A. Deprit and J. Henrard, A manifold of periodic solutions, Advan. Astron. Astrophy, 6, 1968, 6–12.
Martin Golubitsky, Jerrold E. Marsden, Ian Stewart, and Michael Dellnitz, The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos (Waterloo, ON, 1992) Fields Inst. Commun., vol. 4, Amer. Math. Soc., Providence, RI, 1995, pp. 81–127. MR 1350545
John D. Hadjidemetriou, The continuation of periodic orbits from the restricted to the general three-body problem, Celestial Mech. 12 (1975), no. 2, 155–174. MR 386389, DOI 10.1007/BF01230209
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4
Kenneth R. Meyer, Comet like periodic orbits in the $N$-body problem, J. Comput. Appl. Math. 52 (1994), no. 1-3, 337–351. Oscillations in nonlinear systems: applications and numerical aspects. MR 1310139, DOI 10.1016/0377-0427(94)90365-4
K. R. Meyer, Generic bifurcation of periodic points, Trans. Amer. Math. Soc. 149 (1970), 95–107. MR 259289, DOI 10.1090/S0002-9947-1970-0259289-X
K. R. Meyer, Generic stability properties of periodic points, Trans. Amer. Math. Soc. 154 (1971), 273–277. MR 271490, DOI 10.1090/S0002-9947-1971-0271490-9
Kenneth R. Meyer, Periodic orbits near infinity in the restricted $N$-body problem, Celestial Mech. 23 (1981), no. 1, 69–81. MR 607227, DOI 10.1007/BF01228545
Kenneth R. Meyer, Periodic solutions of the $N$-body problem, J. Differential Equations 39 (1981), no. 1, 2–38. MR 602429, DOI 10.1016/0022-0396(81)90081-4
Kenneth R. Meyer, Symmetries and integrals in mechanics, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 259–272. MR 0331427
Kenneth R. Meyer and Glen R. Hall, Introduction to Hamiltonian dynamical systems and the $N$-body problem, Applied Mathematical Sciences, vol. 90, Springer-Verlag, New York, 1992. MR 1140006, DOI 10.1007/978-1-4757-4073-8
Kenneth R. Meyer and Julian I. Palmore, A new class of periodic solutions in the restricted three body problem, J. Differential Equations 8 (1970), 264–276. MR 308526, DOI 10.1016/0022-0396(70)90006-9
K. R. Meyer and D. S. Schmidt, Periodic orbits near ${\cal L}_{4}$ for mass ratios near the critical mass ratio of Routh, Celestial Mech. 4 (1971), 99–109. MR 317617, DOI 10.1007/BF01230325
F. R. Moulton, A class of periodic solutions of the problem of three–bodies with applications to lunar theory, Trans. Amer. Math. Soc., 7, 1906, 537–577.
—, A class of periodic orbits of the superior planets, Trans. Amer. Math. Soc., 13, 1912, 96–108.
Julian I. Palmore, Bridges and natural centers in the restricted three body problem, University of Minnesota, Center for Control Sciences, Minneapolis, Minn., 1969. MR 0282570
O. Perron, Neue periodische Lösungen des ebenen Drei und Mehrkörperproblem, Math. Z., 42, 1937, 593–624.
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
D. S. Schmidt, Periodic solutions near a resonant equilibrium of a Hamiltonian system, Celestial Mech. 9 (1974), 81–103. MR 344601, DOI 10.1007/BF01236166
Dieter S. Schmidt, Transformation to versal normal form, Computer aided proofs in analysis (Cincinnati, OH, 1989) IMA Vol. Math. Appl., vol. 28, Springer, New York, 1991, pp. 235–240. MR 1101376, DOI 10.1007/978-1-4613-9092-3_{2}0
Dieter Schmidt, Versal normal form of the Hamiltonian function of the restricted problem of three bodies near $\scr L_4$, J. Comput. Appl. Math. 52 (1994), no. 1-3, 155–176. Oscillations in nonlinear systems: applications and numerical aspects. MR 1310129, DOI 10.1016/0377-0427(94)90355-7
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 187, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme. MR 0502448
A. G. Sokol′skiĭ, On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance, Prikl. Mat. Meh. 41 (1977), no. 1, 24–33 (Russian); English transl., J. Appl. Math. Mech. 41 (1977), no. 1, 20–28. MR 0470991, DOI 10.1016/0021-8928(77)90083-1
E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, 1927.