Nonlinearity and elliptic functions in classical mechanics

Describing the motion of the classical simple pendulum is one of the aims in every undergraduate classical mechanics course. Its analytical solutions are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions this symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ=Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. A discussion of the physical consequences that this symmetry has on the motions of the simple pendulum is presented in this contribution and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. Thus by studying deeper a very familiar mechanical system, it is possible to get an insight to more abstract physical and mathematical concepts. In particular a single solution of pure imaginary time for all allowed values of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

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Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

10.21203/rs.3.rs-929150/v1 ◽

2021 ◽

Author(s):

Wedad Albalawi ◽

Alvaro H. Salas ◽

S. A. El-Tantawy

Keyword(s):

Analytical Solution ◽

Numerical Solution ◽

Numerical Solutions ◽

Classical Mechanics ◽

Elliptic Functions ◽

Approximate Solutions ◽

Electronic Circuits ◽

Chebyshev Collocation ◽

Pendulum Equation ◽

First Time

Abstract In this work, novel semi-analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation on the pivot vertically for arbitrary angles are obtained for the first time. The semi-analytical solution is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical analysis, the Chebyshev collocation numerical method is introduced for analyzing tthe forced damped parametric driven pendulum equation. Moreover, the semi-analytical solution and Chebyshev collocation numerical solution are compared with the Runge-Kutta (RK) numerical solution. Also, the maximum distance error to the obtained approximate solutions is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

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Foundations of Classical Mechanics

10.1017/9781108635639 ◽

2019 ◽

Cited By ~ 1

Author(s):

P. C. Deshmukh

Keyword(s):

Classical Mechanics

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Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

Al-Jabar Jurnal Pendidikan Matematika ◽

10.24042/ajpm.v11i1.5153 ◽

2020 ◽

Vol 11 (1) ◽

pp. 93-100

Author(s):

Vina Apriliani ◽

Ikhsan Maulidi ◽

Budi Azhari

Keyword(s):

Wave Equation ◽

Exact Solutions ◽

Internal Waves ◽

Water Waves ◽

Wave Equations ◽

Elliptic Functions ◽

Expansion Method ◽

Mkdv Equation ◽

Innovative Methods ◽

De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations.

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Deterministic Mechanism of Irreversibility

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v14i3.7759 ◽

2018 ◽

Vol 14 (3) ◽

pp. 5708-5733 ◽

Cited By ~ 1

Author(s):

Vyacheslav Michailovich Somsikov

Keyword(s):

Internal Energy ◽

Hamiltonian Systems ◽

Classical Mechanics ◽

Motion Equation ◽

Holonomic Constraints ◽

Dissipative Processes ◽

Motion Energy ◽

Analytical Review ◽

History Of ◽

Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v4i1.6966 ◽

2014 ◽

Vol 4 (1) ◽

pp. 404-426

Author(s):

Vincze Gy. Szasz A.

Keyword(s):

Harmonic Oscillator ◽

Classical Mechanics ◽

Universal Time ◽

Variation Theory ◽

Quantum Oscillator ◽

Variation Principle ◽

Poisson Brackets ◽

Mathematical Meaning ◽

Damped Harmonic Oscillator ◽

Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

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Sound as a transverse wave

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v13i1.5670 ◽

2017 ◽

Vol 13 (1) ◽

pp. 4522-4534

Author(s):

Armando TomÃ¡s Canero

Keyword(s):

Quantum Mechanics ◽

Gravitational Waves ◽

Longitudinal Wave ◽

Transverse Wave ◽

Sound Propagation ◽

Correspondence Principle ◽

Classical Mechanics ◽

Wave Model ◽

Scientific Experiments

This paper presents sound propagation based on a transverse wave model which does not collide with the interpretation of physical events based on the longitudinal wave model, but responds to the correspondence principle and allows interpreting a significant number of scientific experiments that do not follow the longitudinal wave model. Among the problems that are solved are: the interpretation of the location of nodes and antinodes in a Kundt tube of classical mechanics, the traslation of phonons in the vacuum interparticle of quantum mechanics and gravitational waves in relativistic mechanics.

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