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n-Body Problem and Choreographies

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The motion of n-point particles of positions \( { x_i(t)\in {\mathbb{R}}^3 } \) and masses \( { m_i > 0 } \), interacting in accordance with Newton's law of gravitation, satisfies thesystem of differentialequations :

$$ -m_i\ddot x_i(t)=G\sum_{j\neq i, j=1}^nm_i m_j \frac{x_i-x_j}{|x_i-x_j|^3}\:,\\ i=1,\ldots,n,\:,\quad t \in {\mathbb{R}}\:. $$
(1)

The n-body problem consists in solving Eq. (1) associated with initial or boundary conditions.

A  simple choreography is a periodic solution to the n-bodyproblem Eq. (1) where the bodies lie on the same curve and exchange their mutual positions aftera fixed time, namely, there exists a function \( {x:{\mathbb{R}}\to{\mathbb{R}} } \) such that

$$ x_i(t)=x(t+(i-1) \tau)\:,\quad i=1,\ldots,n\:,\quad t \in {\mathbb{R}}\:, $$
(2)

where \( { \tau = 2\pi/n } \).

Introduction

The two-body problem can be reduced, by the conservation of the linear momentum, to the one centerKepler problem and can be completely solved either...

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Notes

  1. 1.

    Formula N. 13 quoted by Poincaré is Hamilton equation and covers our class od Dynamical Systems Eq. (3)

  2. 2.

    Highly likely they are not distinct families: this is the recurring phenomenon of “more symmetries than expected” in n-body problems.

Abbreviations

Central configurations:

Are the critical points of the potential constrained on the unitary moment of inertia ellipsoid. Central configurations are associated to particular solutions to the n-body problem: the relative equilibrium and the homographic motions defined in Definition 1.

Choreographical solution:

A choreographical solution of the n-body problem is a solution such that the particles move on the same curve, exchanging their positions after a fixed time. This property can be regarded as a symmetry of the trajectory. This notion finds a natural generalization in that of G-equivariant trajectory defined in Definition 2 for a given group of symmetries-G. The G-equivariant minimization technique consists in seeking action minimizing trajectories among all G-equivariant paths.

Collision and singularities:

When a trajectory can not be extended beyond a certain time b we say that a singularity occurs. Singularities can be collisions if the solution admits a limit configuration as \( { t\to b } \). In such a case we term b a collision instant.

n-Body problem:

The n-body problem is the system of differential equations (1) associated with suitable initial or boundary value data. A solution or trajectory is a doubly differentiable path \( { q(t)=(q_1(t),\dots,q_n(t)) } \) satisfying (1) for all t. The weaker notion of generalized solution is defined in Definition 5 applies to trajectories found by variational methods.

Variational approach:

The variational approach to the n-body problem consists in looking at trajectories as critical points of the action functional defined in (4). Such critical points can be (local) minimizers, or constrained minimizers or mountain pass, or other type.

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  1. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, Italia

    Susanna Terracini

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Terracini, S. (2009). n-Body Problem and Choreographies . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_351

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