scholarly journals On a Linearly Damped 2 Body Problem

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Body Problem ◽  
Singular Potential ◽  
Blow Up ◽  
Atomic Model ◽  
The Sun ◽  
Moving Particle ◽  
New Phenomena

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in nite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover some formal calculations suggest the existence of special orbits with an asymptotically spiraling convergence to the center.

scholarly journals On a Linearly Damped 2 Body Problem

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Finite Time ◽  
Body Problem ◽  
Singular Potential ◽  
Blow Up ◽  
Fast Convergence ◽  
Atomic Model ◽  
The Sun ◽  
Bounded Solutions ◽  
Moving Particle ◽  
New Phenomena

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center.

scholarly journals On Some Damped 2 Body Problems

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Singular Potential ◽  
Blow Up ◽  
Central Charge ◽  
The Sun ◽  
Bounded Solutions ◽  
High Scale ◽  
Electrical Devices ◽  
Moving Particle ◽  
New Phenomena ◽  
Dissipative Model

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center. A related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology.

scholarly journals On Some Damped 2 Body Problems

2020 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Singular Potential ◽  
Blow Up ◽  
Central Charge ◽  
The Sun ◽  
Bounded Solutions ◽  
High Scale ◽  
Electrical Devices ◽  
Moving Particle ◽  
New Phenomena ◽  
Dissipative Model

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center. A related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology.

scholarly journals On Some Damped 2 Body Problems

2021 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Singular Potential ◽  
Central Charge ◽  
The Sun ◽  
Initial State ◽  
High Scale ◽  
All Solutions ◽  
Electrical Devices ◽  
New Phenomena ◽  
Formal Calculation ◽  
Dissipative Model

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear which were investigated thoroughly by R. Ortega and his co-authors between 2014 and 2017, in particular all solutions are bounded and tend to $0$ for $t$ large, some of them with asymptotically spiraling exponentially fast convergence to the center. We provide explicit estimates for the bounds in the general case that we refine under specific restrictions on the initial state, and we give a formal calculation which could be used to determine practically some special asymptotically spiraling orbits. Besides, a related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology

scholarly journals Symmetries in central-force problems

2003 ◽  
pp. 47-52 ◽  
Author(s):  
V. Mioc ◽  
M. Barbosu
Keyword(s):  
Vector Fields ◽  
Body Problem ◽  
Blow Up ◽  
Central Force ◽  
Polar Coordinates ◽  
Global Flow ◽  
Two Body Problem ◽  
Concrete Problem ◽  
Central Force Problems ◽  
Force Problem

The two-body problem in central fields (reducible to a central-force problem) models a lot of concrete astronomical situations. The corresponding vector fields (in Cartesian and polar coordinates, extended via collision-blow-up and infinity-blow-up transformations) exhibit nice symmetries that form eight-element Abelian groups endowed with an idempotent structure. All these groups are isomorphic, which is not a trivial result, given the different structures of the corresponding phase spaces. Each of these groups contains seven four-element subgroups isomorphic to Klein?s group. These symmetries are of much help in understanding various characteristics of the global flow of the general problem or of a concrete problem at hand, and are essential in searching for periodic orbits.

scholarly journals Canonical Variables of The Second Kind and The Reduction of The N-Body Problem

1999 ◽  
Vol 172 ◽  
pp. 269-280
Author(s):  
John G. Bryant
Keyword(s):  
Hamiltonian System ◽  
Body Problem ◽  
Blow Up ◽  
Kepler Problem ◽  
Geometrical Optics ◽  
Canonical Variables ◽  
N Body Problem ◽  
Homogeneous Reduction ◽  
New Variables

AbstractWe introduce a new kind of canonical variables that prove very useful when the order of a Hamiltonian system can be reduced by one, as in the case of isoenergetic reduction, and of what we call homogeneous reduction. The Kepler Problem, Geometrical Optics and McGehee Blow-up are discussed as examples. Finally we carry out the isoenergetic reduction of the general N-Body Problem using the new variables, and briefly discuss its application to the problem of collision.

Sign in / Sign up

Export Citation Format

Share Document