scholarly journals Classical n-body system in geometrical and volume variables: I. Three-body case

2021 ◽  
pp. 2150140
Author(s):  
A. M. Escobar-Ruiz ◽  
R. Linares ◽  
Alexander V. Turbiner ◽  
Willard Miller
Keyword(s):  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Symmetric Polynomial ◽  
Body System ◽  
Generalized Coordinates ◽  
New Variables ◽  
Three Body ◽  
Mass Dependent

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

THE LAGRANGIAN DERIVATION OF KANE’S EQUATIONS

2007 ◽  
Vol 31 (4) ◽  
pp. 407-420 ◽  
Author(s):  
Kourosh Parsa
Keyword(s):  
Kinetic Energy ◽  
Partial Derivative ◽  
Equations Of Motion ◽  
Forward Kinematics ◽  
Body System ◽  
Generalized Coordinates ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Multi Body ◽  
Velocity Level

The Lagrangian approach to the development of dynamics equations for a multi-body system, constrained or otherwise, requires solving the forward kinematics of the system at velocity level in order to derive the kinetic energy of the system. The kinetic-energy expression should then be differentiated multiple times to derive the equations of motion of the system. Among these differentiations, the partial derivative of kinetic energy with respect to the system generalized coordinates is specially cumbersome. In this paper, we will derive this partial derivative using a novel kinematic relation for the partial derivative of angular velocity with respect to the system generalized coordinates. It will be shown that, as a result of the use of this relation, the equations of motion of the system are directly derived in the form of Kane’s equations.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Oblateness Effects on Solar Sail in the Restricted Three–body Problem

2020 ◽  
pp. 24-34
Author(s):  
Fatma M. Elmalky ◽  
M. N. Ismail ◽  
Ghada F. Mohamedien
Keyword(s):  
Equations Of Motion ◽  
Solar Sail ◽  
Special Importance ◽  
Systems Of Equations ◽  
Body System ◽  
Three Body Problem ◽  
Laplace Transformations ◽  
Space Dynamics ◽  
Three Body ◽  
Good Agreement

In the present work, the equations of motion of the solar sail are derived in the restricted three–body system. The dimensionless coordinates are used to obtain the solution of the problem. The Laplace transformations are used to solve these systems of equations to obtain the components of the solar sail acceleration. The motion about L2, L4 and its stability are studied under obalteness effects. The results obtained are in good agreement with previous results in this field. It is remarked that this model has special importance in space-dynamics to enabling spacecraft to do some maneuvers depends on the solar sail acceleration.

scholarly journals Nonintegrability of the problem of a spherical top rolling on a vibrating plane

2020 ◽  
Vol 30 (4) ◽  
pp. 628-644
Author(s):  
A.A. Kilin ◽  
E.N. Pivovarova
Keyword(s):  
Equilibrium Point ◽  
Horizontal Plane ◽  
Degrees Of Freedom ◽  
Symmetry Axis ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Rolling Motion ◽  
Lagrange Top ◽  
Unstable Equilibrium Point

This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.

scholarly journals MATHEMATICAL MODEL OF THE DYNAMICS OF A SIX-MASS SYSTEM OF A PARALLEL STRUCTURE MANIPULATOR MECHANISM

2021 ◽  
pp. 32-37
Author(s):  
V. V. Dyashkin-Titov ◽  
N. S. Vorob’eva ◽  
V. V. Zhoga
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Kinetic Energy ◽  
Quadratic Form ◽  
Degrees Of Freedom ◽  
Parallel Structure ◽  
Mass System ◽  
Generalized Coordinates ◽  
Comparative Results

The paper is devoted to the construction of a mathematical model of the dynamics of a parallel structure manipulator with three controlled degrees of freedom, based on the reduction of the kinetic energy of the manipulator to a quadratic form relative to three independent generalized coordinates, comparative results of mathematical modeling are presented.

Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part II—System Equations

1990 ◽  
Vol 112 (2) ◽  
pp. 215-224 ◽  
Author(s):  
S. Nagarajan ◽  
D. A. Turcic
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Elastic Response ◽  
Mutual Dependence ◽  
Generalized Coordinates ◽  
Lagrange’S Equation ◽  
Elastic Mechanism

The first step in the derivation of the equations of motion for general elastic mechanism systems was described in Part I of this work. The equations were derived at the elemental level using Lagrange’s equation and the generalized coordinates were both the rigid body degrees of freedom, and the elastic degrees of freedom of element ‘e’. Each rigid body degree of freedom gave rise to a scalar equation of motion, and the elastic degrees of freedom of element e gave rise to a vector equation of motion. Since both the rigid body degrees of freedom and elastic degrees of freedom are considered as generalized coordinates, the equations derived take into account the mutual dependence between the rigid body and elastic motions. This is important for mechanisms that are built using lightweight and flexible members and which operate at high speeds. A schematic diagram of how the equations of motion are obtained in this work is shown in Fig. 1 in Part I. The transformation step in the figure refers to the rotational transformation of the nodal elastic displacements (which were measured in the element coordinate system), so that they are measured in terms of the reference coordinate system. This transformation is necessary in order to ensure compatibility of the displacement, velocity and acceleration of the degrees of freedom that are common to two or more links during the assembly of the equations of motion. This final set of equations after assembly are obtained in closed form, and, given external torques and forces, can be solved for the rigid body and elastic response simultaneously taking into account the mutual dependence between the two responses.

Mathematics and physics preliminaries: of hills and plains and other things

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Calculus Of Variations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Space Research ◽  
Lagrange Equations ◽  
Virtual Displacement ◽  
Generalized Coordinates ◽  
Flat Region ◽  
Fixed End ◽  
Mathematics And Physics

The link between mathematics and physics is explained, and how the concepts “coordinates,” “generalized coordinates,” “time,” and “space” have evolved, starting with Galileo. It is also shown that “degrees of freedom” is a slippery but crucial idea. The important developments in “space research”, from Pythagoras to Riemann, are sketched. This is followed by the motivations for finding a flat region of “space”, and for Riemann’s invariant interval. A careful explanation of the three ways of taking an infinitesimal step (actual, virtual, and imperfect) is given. The programme of the Calculus of Variations is described and how this requires a virtual variation of a whole path, a path taken between fixed end-states. This then culminates in the Euler-Lagrange Equations or the Lagrange Equations of Motion. Along the way, the ideas virtual displacement and extremum are explained.

scholarly journals Motion of the Infinitesimal Variable Mass in the Generalized Circular Restricted Three-Body Problem under the Effect of Asteroids Belt

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Ferdaous Bouaziz-Kellil
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Variable Mass ◽  
Three Body Problem ◽  
Common Center ◽  
The Stability ◽  
Effects Of Radiation ◽  
Three Body

The present paper deals with the study of the motion’s properties of the infinitesimal variable mass body moving in the same orbital plan as two massive bodies (considered as primaries). It is assumed that the massive bodies have radiating effects, have oblate shapes, and are moving in circular orbits around their common center of mass. Using the procedures established by Singh and Abouelmagd, we determined the equations of motion of the infinitesimal body for which we assumed that under the effects of radiation and oblateness of the primaries, its mass varies following Jean’s law. We evaluated analytically and numerically the locations of equilibrium points and examined the stability of these equilibrium points. Finally, we found that all the points are unstable.

scholarly journals Study of Approximations for Electron?Atom Direct Reactions

1983 ◽  
Vol 36 (5) ◽  
pp. 665 ◽  
Author(s):  
IE McCarthy ◽  
AT Stelbovics
Keyword(s):  
Experimental Data ◽  
Electron Scattering ◽  
Experimental Error ◽  
Degrees Of Freedom ◽  
Body System ◽  
Many Body ◽  
Coupled Channels ◽  
Direct Reactions ◽  
Many Body Systems ◽  
Three Body

The electron-hydrogen system is a true three-body system which provides an excellent test for theories of reactions in many-body systems that approximately involve only three-body degrees of freedom. The coupled-channels optical approximation reproduces experimental data in most cases within experimental error. The approximation may be extended to a larger space of coupled channels by various approximations which are tested with the example of 54�42 e V electron scattering on the Is, 2s and 2p space for hydrogen, extended by the addition of 3s and 3p channels. Channels outside this five-state space are treated by including the corresponding polarization potentials.

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