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

Lagrange’s periodic solution with three bodies at vertices of equilateral triangles. The constant ratios of separations
are functions of the mass ratios alone

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

Kepler’s Laws without Calculus

W. G. Unruh
Newton in the Principia used the inverse squared force law (and Galileo’s idea of compound motion) to derive Kepler’s laws. As usual for him, the proof is a purely geometric proof, using no calculus. Maxwell [2] gave a very different proof. In the 1960’s, Feynman[3] gave a geometric proof very similar to Maxwell’s. Finally, Vogt[4] in the American Journal of Physics also carried out a derivation which started out with the energy conservation equation and the angular momentum conservation to again present a geometric proof. In all of these cases, the derivation that the orbit actually is an ellipse was somewhat torturous, difficult to follow, and non obvious. In the following, following initially the Maxwell-Feynman’s approach, the derivation that the orbit is an ellipse is simplified about as much as possible.

Read more at https://arxiv.org/pdf/1803.06770.pdf

Leonardo da Vinci’s studies of friction

Sketches from two different pages in Leonardo’s notebooks: (a, b) from Codex Atlanticus, Biblioteca Ambrosiana, Milan (CA folio 532r c. 1506-8), and (c) from Codex Arundel, British Library, London (Arundel folio 41r c. 1500-05)

Sketches from two different pages in Leonardo’s notebooks: (a, b) from Codex
Atlanticus, Biblioteca Ambrosiana, Milan (CA folio 532r c. 1506-8), and (c) from Codex Arundel, British Library, London (Arundel folio 41r c. 1500-05)

Ian M. Hutchings
Based on a detailed study of Leonardo da Vinci’s notebooks, this review examines the development of his understanding of the laws of friction and their application. His work on friction originated in studies of the rotational resistance of axles and the mechanics of screw threads.
He pursued the topic for more than 20 years, incorporating his empirical knowledge of friction into models for several mechanical systems. Diagrams which have been assumed to represent his experimental apparatus are misleading, but his work was undoubtedly based on experimental measurements and probably largely involved lubricated contacts.
Although his work had no influence on the development of the subject over the succeeding centuries, Leonardo da Vinci holds a unique position as a pioneer in tribology.
Read more at http://www.ifm.eng.cam.ac.uk/uploads/Hutchings_Leonardo_Friction_2016_v2.pdf

Read also: Study reveals Leonardo da Vinci’s ‘irrelevant’ scribbles mark the spot where he first recorded the laws of friction