## Archive for the ‘**mechanics**’ Category

## Novel theory addresses centuries-old physics problem

**Hebrew University of Jerusalem Researcher introduces a new approach to the “three-body problem”; predicts its outcome statistics**

**The “three-body problem,”** the term coined for predicting the motion of three gravitating bodies in space, is essential for understanding a variety of astrophysical processes as well as a large class of mechanical problems, and has occupied some of the world’s best physicists, astronomers and mathematicians for over three centuries. Their attempts have led to the discovery of several important fields of science; yet its solution remained a mystery.

At the end of the 17^{th} century, Sir Isaac Newton succeeded in explaining the motion of the planets around the sun through a law of universal gravitation. He also sought to explain the motion of the moon. Since both the earth and the sun determine the motion of the moon, Newton became interested in the problem of predicting the motion of three bodies moving in space under the influence of their mutual gravitational attraction (see attached illustration), a problem that later became known as “the three-body problem”. However, unlike the two-body problem, Newton was unable to obtain a general mathematical solution for it. Indeed, **the three-body problem proved easy to define, yet difficult to solve**.

New research, led by **Prof. Barak Kol** **of the Racah Institute of Physics at the Hebrew University**, adds a step to this scientific journey that began with Newton, touching on **the limits of scientific prediction, and the role of chaos in it**.

The theoretical study presents a novel and exact reduction of the problem, enabled by a re-examination of the basic concepts that underlie previous theories. It allows for a precise prediction of the probability for each of the three bodies to escape the system.

Following Newton and two centuries of fruitful research in the field including by Euler, Lagrange and Jacobi, by the late 19^{th} century the mathematician Poincare discovered that the problem exhibits extreme sensitivity to the bodies’ initial positions and velocities. This sensitivity, which later became known as chaos, has far-reaching implications – it indicates that** there is no deterministic solution in closed-form** to the three-body problem.

In the 20th century, the development of computers made it possible to re-examine the problem with the help of computerized simulations of the bodies’ motion. The simulations showed that under some general assumptions, a three-body system experiences periods of chaotic, or random, motion alternating with periods of regular motion, until finally the system disintegrates into a pair of bodies orbiting their common center of mass and a third one moving away, or escaping, from them.

The chaotic nature implies that not only is a closed-form solution impossible, but also computer simulations cannot provide specific and reliable long-term predictions. However, the availability of large sets of simulations led in 1976 to the idea of seeking a statistical prediction of the system, and in particular, predicting the escape probability of each of the three bodies. In this sense, the original goal, to find a deterministic solution, was found to be wrong, and it was recognized that the **right goal is to find a statistical solution**.

Determining the statistical solution has proven to be no easy task due to three features of this problem: the system presents chaotic motion that alternates with regular motion; it is unbounded and susceptible to disintegration. A year ago,** Dr. Nicholas Stone of the Racah Institute of Physics at the Hebrew University **and his colleagues used a new method of calculation**, **and** for the first time achieved a closed mathematical expression for the statistical solution**. However, this method, like all its predecessor statistical approaches, rests on certain assumptions. Inspired by these results, **Kol **initiated a re-examination of these assumptions.

In order to understand the novelty of the new approach, it is necessary to discuss the notion of “phase space” that underlies all statistical theories in physics. A phase space is nothing but the space of all positions and velocities of the particles that compose a system. For instance, the phase space of a single particle allowed to move on a meter-long track with a velocity of at most two meters per second, is a rectangle, whose width is 1 meter, and whose length is four meters per second (since the velocity can be directed either to the left or to the right).

Normally, physicists identify probability of an event of interest with its associated phase space volume (phase volume, in short). For instance, the probability for the particle to be found in the left half of the track, is associated with the volume of the left half of the phase space rectangle, which is one half of the total volume.

The three-body problem is unbounded and the gravitational force is unlimited in range. This suggests infinite phase space volumes, which would imply **infinite probabilities**. In order to overcome this and related issues, all previous approaches postulated a “strong interaction region” and ignored phase volumes outside of it, such that phase volumes become finite. However, since the gravitational force decreases with distance, but never disappears, this is not an accurate theory, and it introduces a certain arbitrariness into the model.

The new study, recently published in the scientific journal **Celestial Mechanics and Dynamical Astronomy,** focuses on the outgoing **flux of phase-volume**, rather than the phase-volume itself. For instance, consider a volume of gas within a container, a marked gas molecule moving within it, and consider the container wall to have two small holes. In this case, the probability for the molecule’s eventual exit hole would be proportional to the flux of the surrounding gas through each hole.

Since the flux is finite even when the volume is infinite, this **flux-based approach** avoids the artificial problem of infinite probabilities, without ever introducing the artificial strong interaction region.

In order to treat the mix between chaotic and regular motion, the flux-based theory further introduces an unknown quantity, the emissivity. In this way, the statistical prediction exactly factorizes into a closed-form expression, and the emissivity, which is presumably simpler and is left for future study.

The flux-based theory predicts the escape probabilities of each body, under the assumption that the emissivity can be averaged out and ignored. The predictions are different from all previous frameworks, and Prof. Kol emphasizes that “**tests by millions of computer simulations shows strong agreement between theory and simulation**.” The simulations were carried out in collaboration with Viraj Manwadkar from the University of Chicago, Alessandro Trani from the Okinawa Institute in Japan, and Nathan Leigh from University of Concepcion in Chile. This agreement proves that understanding the system requires a paradigm shift and that the new conceptual basis describes the system well.

It turns out, then, that even for the foundations of such an old problem, innovation is possible.

The implications of this study are wide-ranging and is expected to influence both the solution of a variety of astrophysical problems and the understanding of an entire class of problems in mechanics. In astrophysics, it may have application to the mechanism that creates pairs of compact bodies that are the source of gravitational waves, as well as to deepen the understanding of the dynamics within star clusters. In mechanics, the three-body problem is a prototype for a variety of chaotic problems, so progress in it is likely to reflect on additional problems in this important class.

Link to paper: http://old.phys.huji.ac.il/~barak_kol/Kol_3body_CM.pdf

## On modelling bicycle power-meter measurements

**Tomasz Danek, Michael A. Slawinski, Theodore Stanoev**

We combine power-meter measurements with GPS measurements to study the model that accounts for the use of power by a cyclist. The model takes into account the change in elevation and speed along with adverse effects of air, rolling and drivetrain resistance. The focus is on estimating the resistance coefficients using numerical optimization techniques to maintain an agreement between modelled and measured power-meter values, which accounts for the associated uncertainties. The estimation of coefficients is performed for two typical scenarios of road cycling under windless conditions, along a course that is mainly flat as well as a course of near constant inclination. Also, we discuss relations between different combinations of two model parameters, where other quantities are constant, by the implicit function theorem. Using the obtained estimates of resistance coefficients for the two courses, we use the mathematical relations to make inferences on the model and physical conditions. Along with a discussion of results, we provide two appendices. In the first appendix, we illustrate the importance of instantaneous cadence measurements. In the second, we consider the model in constrained optimization using Lagrange multipliers.

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## A Historical Method Approach to Teaching Kepler’s 2nd law

**Wladimir Lyra**

Kepler’s 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained “later on” as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler’s 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the “time=area” statement quantitatively, in the way of Kepler’s equation, M = E – e sin E (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars’ next opposition. This method is partially based on Kepler’s original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.

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## Etude des effets non linéaires observés sur les oscillations d’un pendule simple

**Thomas Gibaud, Alain Gibaud**

In this paper we present a study of the non-linear effects of anharmonicity of the potential of the simple pendulum. In a theoretical reminder we highlight that anharmonicity of the potential generates additional harmonics and the non-isochronism of oscillations. These phenomena are all the more important as we move away from the oscillations at small angles, which represent the domain of validity of the harmonic approximation. The measurement is apprehended by means of the acquisition box SYSAM-SP5 coupled with the Latis pro software and the Eurosmart pendulum. We show that only a detailed analysis by fitting the recorded curve can provide sufficient accuracy to describe the quadratic evolution of the period as a function of the amplitude of the oscillations. We we can detect the additional harmonics in the oscillations when the amplitude becomes very high.

read more at https://arxiv.org/ftp/arxiv/papers/1911/1911.11594.pdf

## Nonconservation of Energy and Loss of Determinism

**I. Infinitely Many Balls**

**David Atkinson, Porter Johnson**

An infinite number of elastically colliding balls is considered in a classical, and then in a relativistic setting. Energy and momentum are not necessarily conserved globally, even though each collision does separately conserve them. This result holds in particular when the total mass of all the balls is finite, and even when the spatial extent and temporal duration of the process are also finite. Further, the process is shown to be indeterministic: there is an arbitrary parameter in the general solution that corresponds to the injection of an arbitrary amount of energy (classically), or energy-momentum (relativistically), into the system at the point of accumulation of the locations of the balls. Specific examples are given that illustrate these counter-intuitive results, including one in which all the balls move with the same velocity after every collision has taken place.

Read more at https://arxiv.org/pdf/1908.10458.pdf

**II: Colliding with an Open Set**

An actual infinity of colliding balls can be in a configuration in which the laws of mechanics lead to logical inconsistency. It is argued that one should therefore limit the domain of these laws to a finite, or only a potentially infinite number of elements. With this restriction indeterminism, energy non-conservation and (creatio ex nihilo) no longer occur. A numerical analysis of finite systems of colliding balls is given, and the asymptotic behavior that corresponds to the potentially infinite system is inferred.

Read more at https://arxiv.org/pdf/1908.09865.pdf

## An introduction to the classical three-body problem

**Govind S. Krishnaswami, Himalaya Senapati**

The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of chaos. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical systems.

Read more at https://arxiv.org/pdf/1901.07289.pdf

## Noether’s Theorem and Symmetry

**A.K. Halder, Andronikos Paliathanasis, P.G.L. Leach**

In Noether’s original presentation of her celebrated theorm of 1918 allowance was made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the depenent variable(s), the so-called generalised, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades this dimunition of the power of Noether’s Theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this special issue we emphasise the generality of Noether’s Theorem in its original form and explore the applicability of even more general coefficient functions by alowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables

Read more at https://arxiv.org/pdf/1812.03682.pdf

## Mechanical resonance: 300 years from discovery to the full understanding of its importance

**Jörn Bleck-Neuhaus**

Starting from the observation that the simplest form of forced mechanical oscillation serves as a standard model for analyzing a broad variety of resonance processes in many fields of physics and engineering, the remarkably slow development leading to this insight is reviewed. Forced oscillations and mechanical resonance were already described by Galileo early in the 17th century, even though he misunderstood them. The phenomenon was then completely ignored by Newton but was partly rediscovered in the 18th century, as a purely mathematical surprise, by Euler. Not earlier than in the 19th century did Thomas Young give the first correct description. Until then, forced oscillations were not investigated for the purpose of understanding the motion of a pendulum, or of a mass on a spring, or the acoustic resonance, but in the context of the ocean tides. Thus, in the field of pure mechanics the results by Young had no echo at all. On the other hand, in the 19th century mechanical resonance disasters were observed ever more frequently, e.g. with suspension bridges and steam engines, but were not recognized as such. The equations governing forced mechanical oscillations were then rediscovered in other fields like acoustics and electrodynamics and were later found to play an important role also in quantum mechanics. Only then, in the early 20th century, the importance of the one-dimensional mechanical resonance as a fundamental model process was recognized in various fields, at last in engineering mechanics. There may be various reasons for the enormous time span between the introduction of this simple mechanical phenomenon into science and its due scientific appreciation. One of them can be traced back to the frequently made neglect of friction in the governing equation.

Read more at https://arxiv.org/ftp/arxiv/papers/1811/1811.08353.pdf