Discrete Newtonian gravitation and the three-body problem

Greenspan, Donald

doi:10.1007/bf00712693

Cited by 18 publications

(4 citation statements)

References 6 publications

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“…The fourth-order symplectic method, which is applied to equation ( 2), G: Greenspan's energy conserving method (Greenspan 1974;LaBudde & Greenspan 1974), which is second-order accurate and is applied to equation (2), d-CRGNBP: The d-CRGNBP, which is given by equations ( 48) and ( 50), and is second-order accurate.…”

Section: Si4mentioning

confidence: 99%

d'Alembert-type scheme with a chain regularization for N-body problem

Minesaki

2014

Preprint

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We design an accurate orbital integration scheme for the general N-body problem preserving all the conserved quantities but the angular momentum. This scheme is based on the chain concept (Mikkola & Aarseth 1993) and is regarded as an extension of a d'Alembert-type scheme (Betsch 2005) for constrained Hamiltonian systems. It also coincides with the discrete-time general three-body problem (Minesaki 2013a) for particle number N = 3. Although the proposed scheme is only second-order accurate, it can accurately reproduce some periodic orbits, which generic geometric numerical integrators cannot do.

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Section: Si4mentioning

confidence: 99%

d'Alembert-type scheme with a chain regularization for N-body problem

Minesaki

2014

Preprint

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“…Since the general three-body problem (G3BP) has six degrees of freedom, we need the values of 11 independent conserved quantities to establish an orbit uniquely. However, only six conserved quantities are known: the energy, two components of the position of the mass center, two components of the linear momentum, and one component of the angular momentum (Greenspan 1974). Therefore, monitoring these conserved quantities alone is not enough to confirm whether a numerical integration reproduces the orbits of the G3BP.…”

Section: Introductionmentioning

confidence: 98%

“…Therefore, monitoring these conserved quantities alone is not enough to confirm whether a numerical integration reproduces the orbits of the G3BP. Although Greenspan's energy-momentum integration (Greenspan 1974) preserves all six known conserved quantities, it does not reproduce the orbits of the Lagrange triangle solutions. We can assume that this integration does not conserve all five unknown quantities.…”

Section: Introductionmentioning

confidence: 99%

Lagrange Solutions to the Discrete-Time General Three-Body Problem

Minesaki

2013

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There is no known integrator that yields exact orbits for the general three-body problem (G3BP). It is difficult to verify whether a numerical procedure yields the correct solutions to the G3BP because doing so requires knowledge of all 11 conserved quantities, whereas only six are known. Without tracking all of the conserved quantities, it is possible to show that the discrete general three-body problem (d-G3BP) yields the correct orbits corresponding to Lagrange solutions of the G3BP. We show that the d-G3BP yields the correct solutions to the G3BP for two special cases: the equilateral triangle and collinear configurations. For the triangular solution, we use the fact that the solution to the three-body case is a superposition of the solutions to the three two-body cases, and we show that the three bodies maintain the same relative distances at all times. To obtain the collinear solution, we assume a specific permutation of the three bodies arranged along a straight rotating line, and we show that the d-G3BP maintains the same distance ratio between two bodies as in the G3BP. Proving that the d-G3BP solutions for these cases are equivalent to those of the G3BP makes it likely that the d-G3BP and G3BP solutions are equivalent in other cases. To our knowledge, this is the first work that proves the equivalence of the discrete solutions and the Lagrange orbits.

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“…A survey of the methods for solving the N-body problem is given, among others, in [1] and [9], From the point of view of the solution accuracy obtained, the most often used numerical methods are the Gragg-Bulirsch-Stoer method based on a rational approximation [3,5], the Everhart method [4], and the Taylor-Steffensen method [15], which uses the Taylor series for the right-hand side functions occurring in the differential equations and recursive formulas for coefficients of this series. Some special methods conserving and using constants (integrals) of the motion should also be mentioned (see e. g. [6][7][8][9][10][11][12][13]).…”

Section: Introductionmentioning

confidence: 99%