The use of integrals in numerical integrations of theN-body problem

Nacozy, P. E.

doi:10.1007/bf00649193

Cited by 89 publications

(28 citation statements)

References 4 publications

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“…Methods based on other principles of stabilization have been investigated in the references Stiefel and Scheifele (1971), Baumgarte (1972aBaumgarte ( , b, 1974, Baumgarte and Stiefel (1973), Hochfeld (1957), Nacozy (1971), and Sigrist (1974). All these methods use first integrals of the dynamical system at hand.…”

Section: Introductionmentioning

confidence: 98%

Stabilization by manipulation of the Hamiltonian

Baumgarte

Stiefel

1974

Celestial Mechanics

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Numerical integration of unstable differential equations should be avoided since a numerical error during the nth step produces erroneous initial values for the next step and thus deteriorates the subsequent integration in an unstable manner. A method is offered to stabilize the equations of motion corresponding to a given Hamiltonian H by transforming H into a new Hamiltonian H* which is equivalent to the Hamiltonian of a harmonic oscillator. In contrast to other methods of stabilization the realm of canonical mechanics is thus not abandoned. Perturbations are discussed and as examples the Keplerian motion and the motion of a gyroscope are presented.

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Section: Introductionmentioning

confidence: 98%

Stabilization by manipulation of the Hamiltonian

Baumgarte

Stiefel

1974

Celestial Mechanics

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“…Although his approach succeeded in the case of truly unordered orbits such as a general 25-body problem (Nacozy 1971), it failed to follow almost ordered orbits such as in the solar system (Huang & Innanen 1983;Hairer et al 1999). Around a couple of decades later, Murison (1989) demonstrated the effectiveness of the same idea in the restricted three-body problems where he used the constancy of the Jacobi constant.…”

Section: Historical Notementioning

confidence: 96%

“…In orbital mechanics, which we believe to be the first field where manifold corrections were applied, the original idea was invented by Nacozy (1971). He noticed that the total energy of a classic N -body system is a constant of integration.…”

Section: Historical Notementioning

confidence: 99%

Simple, Regular, and Efficient Numerical Integration of Rotational Motion

Fukushima

2008

The Astronomical Journal

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Among various formulations to integrate rotational motions numerically, we recommend the integration of the combination of Euler parameters and the angular velocity components referred to the body-fixed reference frame with the renormalization of the Euler parameters at every integration step. This is because the formulation proposed is not only simple and regular, but also stable and precise in studying rotational motions numerically. Its efficiency is not degraded even for almost uniform rotations if their nominal uniform part is separated by Encke's method.

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“…1 In short, they are an extension of Nacozy's method of manifold correction (Nacozy 1971) using the concept of the integral invariant relation (Szebehely & Bettis 1972). These methods numerically integrate not only the position and velocity but also some quantities to be conserved in the unperturbed case, such as the Kepler energy, the orbital angular momentum vector, and/or the Laplace integral.…”

Section: Introductionmentioning

confidence: 99%

Efficient Orbit Integration by Linear Transformation for Consistency of Kepler Energy, Full Laplace Integral, and Angular Momentum Vector

Fukushima

2004

Astron J

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By adopting a general linear transformation as the method of manifold correction, we modify our dual scaling method to integrate quasi-Keplerian orbits numerically. The new method adjusts the integrated position and velocity at each integration step in order to exactly satisfy the relations for the Kepler energy, angular momentum vector, and the full Laplace vector. In the case of no perturbation, the integration errors in all the orbital elements except the mean longitude at the epoch, which grows linearly with time, are reduced to the level of the machine epsilon throughout the integration. For perturbed orbits, the integration errors in position are smaller than with the previous methods of manifold correction. Since its wide applicability is unchanged and the cost of additional computation is similarly negligible, we recommend the new method as the best of our methods of manifold correction.

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The use of integrals in numerical integrations of theN-body problem

Cited by 89 publications

References 4 publications

Stabilization by manipulation of the Hamiltonian

Stabilization by manipulation of the Hamiltonian

Simple, Regular, and Efficient Numerical Integration of Rotational Motion

Efficient Orbit Integration by Linear Transformation for Consistency of Kepler Energy, Full Laplace Integral, and Angular Momentum Vector

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