The Helmholtz conditions in terms of constants of motion in classical mechanics

1989 ◽  
Vol 30 (9) ◽  
pp. 2054-2061 ◽  
Author(s):  
Francisco Pardo
Keyword(s):  
Classical Mechanics ◽  
Helmholtz Conditions ◽  
Constants Of Motion

Superintegrability in Classical Mechanics: A Contemporary Approach to Bertrand's Theorem

1997 ◽  
Vol 12 (01) ◽  
pp. 271-276 ◽  
Author(s):  
A. L. Salas-Brito ◽  
H. N. Núñez-Yépez ◽  
R. P. Martínez-Y-Romero
Keyword(s):  
Harmonic Oscillator ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Classical System ◽  
Constants Of Motion ◽  
Three Degrees Of Freedom ◽  
Contemporary Approach

Superintegrable Hamiltonians in three degrees of freedom posses more than three functionally independent globally defined and single-valued constants of motion. In this contribution and under the assumption of the existence of only periodic and plane bounded orbits in a classical system we are able to establish the superintegrability of the Hamiltonian. Then, using basic algebraic ideas, we obtain a contemporary proof of Bertrand's theorem. That is, we are able to show that the harmonic oscillator and the Newtonian gravitational potentials are the only 3D potentials whose bounded orbits are all plane and periodic.

Closed orbits and constants of motion in classical mechanics

1992 ◽  
Vol 13 (1) ◽  
pp. 26-31 ◽  
Author(s):  
R P Martinez-y-Romero ◽  
H N Nunez-Yepez ◽  
A L Salas-Brito
Keyword(s):  
Classical Mechanics ◽  
Closed Orbits ◽  
Constants Of Motion

scholarly journals Superintegrability of (2n + 1)-body choreographies, n = 1,2,3,…,∞ on the algebraic lemniscate by Bernoulli (inverse problem of classical mechanics)

2021 ◽  
pp. 2150116
Author(s):  
Alexander V. Turbiner ◽  
Juan Carlos Lopez Vieyra
Keyword(s):  
Inverse Problem ◽  
Angular Momentum ◽  
Total Angular Momentum ◽  
Nearest Neighbor ◽  
Classical Mechanics ◽  
One Dimensional ◽  
Limit Formula ◽  
Constants Of Motion ◽  
Poisson Commute

For one 3-body and two 5-body planar choreographies on the same algebraic lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies [Formula: see text] moving choreographically (without collisions) along given algebraic lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit [Formula: see text] is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic lemniscate and it is characterized by infinitely many constants of motion.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
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.

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

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.

Sound as a transverse wave

2017 ◽  
Vol 13 (1) ◽  
pp. 4522-4534
Author(s):  
Armando Tomás 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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