Computer simulations, exact trajectories, and the gravitational N-body problem

Hayes, Wayne B.

doi:10.1119/1.1764561

Cited by 8 publications

(5 citation statements)

References 26 publications

(23 reference statements)

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“…The polar coordinate r is related to the cable length s and to the Cartesian coordinate x, according to Figure 5 and Equation (38). In (39), (41) was replaced by (37) after differentiation, wherein the approximation for small values of the angle θ was made. It is important to point out that if, in (37), the approximation for small angles was made first, the information provided by the differential process would be lost.…”

Section: Discussionmentioning

confidence: 99%

“…These details stimulate the interest of many students. Examples of systems with similar effects on students are the elastic pendulum [40] or the three-body problem [41]. These are simple approaches that can present complex or simple dynamic behaviors depending on their state.…”

Section: Discussionmentioning

confidence: 99%

“…It states that only half of the horizontal displacement of point b during Phase II is applied when varying the cable curvature and therefore the angle θ at point B (see Figure 6). Solutions for (39) and (40) results in (41). It provides an expression for both s c and u in Phase II.…”

Section: Discussionmentioning

confidence: 99%

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A Didactic Procedure to Solve the Equation of Steady-Static Response in Suspended Cables

et al. 2020

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Students in the electrical branch of the short-cycle tertiary education program acquire developmental and design skills for low voltage transmission power lines. Aerial power line design requires mathematical tools not covered well enough in the curricula. Designing suspension cables requires the use of a Taylor series and integral calculation to obtain the parabola’s arc length. Moreover, it requires iterative procedures, such as the Newton–Raphson method, to solve the third-order equation of the steady-static response. The aim of this work is to solve the steady-static response equation for suspended cables using simple calculation tools. For this purpose, the influence of the horizontal component of the cable tension on its curvature was decoupled from the cable’s self-weight, which was responsible for the tension’s vertical component. To this end, we analyzed the laying and operation of the suspended cables by defining three phases (i.e., stressing, lifting, and operation). The phenomena that occurred in each phase were analyzed, as was their manifestation in the cable model. Herein, we developed and validated the solution of the steady-static response equation in suspended cables using simple equations supported with intuitive graphics. The best results of the proposed calculation procedure were obtained in conditions of large temperature variations.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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A Didactic Procedure to Solve the Equation of Steady-Static Response in Suspended Cables

et al. 2020

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“…The solution for the two-bodyproblem has been obtained analytically [1]. However, when the number of bodies is more, numerical methods and approximations are needed in general [2][3][4], even for a three-body problem [5][6][7]. This problem hosts interesting physics ranging from the discovery of Neptune [8][9][10][11][12] to non-integrability and chaos [13,14].…”

Section: Introductionmentioning

confidence: 99%

An approximate superposition method to obtain a planet’s orbit

et al. 2021

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We demonstrate that the deviation produced by one celestial object on the Keplerian orbit of another around the Sun is largely independent of the presence of the remaining ones. Hence, to calculate the net deviation of an object from its Keplerian orbit, we superpose the deviations produced by every other object. We show that this method will be useful when dealing with a system containing a large number of objects. As a demonstration, we apply our method to the Solar System, with a particular focus on the orbit of Uranus.

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“…Therefore, this pseudoorbit is actually a good approximation of an exact trajectory for an initial condition close to (−0.5, −0.5) and thus is representative of the system's dynamics. Such backward error analysis through shadowing is also used, e.g., in [14] in the context of galaxy simulations, in [37,36] in the context of fluid mechanics, and in [35] for the rigorous numerical proof of the embedding of symbolic dynamics. Among different numerical shadowing methods, we will focus on the so-called containment method (see [13,15,41]).…”

Section: Introductionmentioning

confidence: 99%

Tinkerbell Is Chaotic

Goldsztejn¹,

Hayes²,

Collins³

2011

SIAM J. Appl. Dyn. Syst.

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Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its hypothesis can be easily verified using interval arithmetic. As an application of this new theorem, shadowing by containment is shown to be applicable to infinite length orbits and is used to provide a computer assisted proof of the presence of chaos in the well-known noninjective Tinkerbell map. A. GOLDSZTEJN, W. HAYES, AND P. COLLINS proof is conceptually very simple since it relies directly on the Poincaré-Miranda theorem. 1 Note that the containment theorem proposed in [21] also does not require differentiability or injectivity: However, its hypotheses are quite difficult to verify. Third, this new containment theorem is extended to bi-infinite orbits, thus reaching the scope of shadowing theorems by refinement like [35].As an application, we show that this containment theorem for bi-infinite orbits can be used to prove that a dynamical system is chaotic in several senses (positive topological entropy, Li-Yorke chaos, and Devaney chaos). Different computed assisted proofs of the presence of chaos were proposed in, e.g., [34,9,42,10,26]. All these approaches are similar in the sense that they rely on some topological fixed point theorems whose hypotheses are translated to sufficient conditions for the presence of chaos. However, the usage of different topological fixed point theorems leads to different statements. The sufficient conditions we obtain here are remarkable by their simplicity, which allows an easy numerical verification by a computer. Interval analysis.Interval analysis is a branch of numerical analysis that was born in the 1960's. It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations (see [27,1,28,17,20] for a survey). Among other applications, interval analysis was used to provide rigorous numerical proofs of mathematical statements (resolution of the 14th Smale problem [38], properties of manifolds [6,7], and properties of chaotic dynamical systems [3,18,4,39,33]).

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Computer simulations, exact trajectories, and the gravitational N-body problem

Cited by 8 publications

References 26 publications

A Didactic Procedure to Solve the Equation of Steady-Static Response in Suspended Cables

A Didactic Procedure to Solve the Equation of Steady-Static Response in Suspended Cables

An approximate superposition method to obtain a planet’s orbit

Tinkerbell Is Chaotic

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