# 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.

# 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.

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.

# 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.

# 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

# 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. W. G. Unruh