Five new methods of celestial mechanics

Alexander Bruno; Alexander Bruno

doi:10.3934/math.2020340

AIMS Mathematics

2020, Volume 5, Issue 5: 5309-5319. doi: 10.3934/math.2020340

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Research article

Five new methods of celestial mechanics

Alexander Bruno ^,

Keldysh Institute of Applied Mathematics of RAS, Miusskaya sq. 4, Moscow, 125047, Russia

Received: 06 September 2019 Accepted: 12 June 2020 Published: 22 June 2020
MSC : 37J40, 37J45

The last volume of the book "Les méthods nouvelles de la Mécanique céleste" by Poincaré ^[28] was published more than 120 years ago. Since then, the following methods have arisen. 1. Method of normal forms, allowing to study regular perturbations near a stationary solution, near a periodic solution and so on. 2. Method of truncated systems, which are found with a help of the Newton polyhedrons, allowing to study singular perturbations. 3. Method of generating families of periodic solutions (regular and singular). 4. Method of generalized problems, allowing bodies with negative masses. 5. Computation of a net of families of periodic solutions as a "skeleton" of a part of the phase space.
- Hamiltonian system,
- normal form,
- truncated Hamiltonian,
- family of periodic solutions,
- generated family,
- negative mass,
- skeleton
Citation: Alexander Bruno. Five new methods of celestial mechanics[J]. AIMS Mathematics, 2020, 5(5): 5309-5319. doi: 10.3934/math.2020340

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Abstract

The last volume of the book "Les méthods nouvelles de la Mécanique céleste" by Poincaré ^[28] was published more than 120 years ago. Since then, the following methods have arisen. 1. Method of normal forms, allowing to study regular perturbations near a stationary solution, near a periodic solution and so on. 2. Method of truncated systems, which are found with a help of the Newton polyhedrons, allowing to study singular perturbations. 3. Method of generating families of periodic solutions (regular and singular). 4. Method of generalized problems, allowing bodies with negative masses. 5. Computation of a net of families of periodic solutions as a "skeleton" of a part of the phase space.

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