Equations of Motion for Classical Dynamical Systems of Variable Mass

1945 ◽  
Vol 68 (5-6) ◽  
pp. 130-141
Author(s):  
Joseph Giarratana
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Variable Mass

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

Variable mass motion in the Hénon–Heiles system

2021 ◽  
pp. 2150150
Author(s):  
Abdullah A. Ansari ◽  
Elbaz I. Abouelmagd
Keyword(s):  
Test Particle ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Stationary Points ◽  
Mass Motion ◽  
Jacobi Integral

In this work, we analyze the motion properties of the test particle, that has a variable mass within the frame of Hénon–Heiles system. We derive the equations of motion of the test particle which varies its mass according to Jean’s law. We also determine the quasi-Jacobi integral which shows the effective variation due to variable mass parameters. Further, we studied the locations of stationary points and their stability, after using Meshcherskii spacetime inverse transformations.

Dynamical systems: Approximate Lagrangians and Noether symmetries

2019 ◽  
Vol 16 (10) ◽  
pp. 1950160 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Kinetic Energy ◽  
Equations Of Motion ◽  
Noether Symmetry ◽  
Noether Symmetries ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Second Order Equations

We determine the approximate Noether point symmetries of the variational principle characterizing second-order equations of motion of a particle in a (finite-dimensional) Riemannian manifold. In particular, the Lagrangian comprises of kinetic energy and a potential [Formula: see text], perturbed to [Formula: see text]. We establish a convenient system of approximate geometric conditions that suffices for the computation of approximate Noether symmetry vectors and moreover, simplifies the problem of the effect of higher orders of the perturbation. The general results are applied to several practical problems of interest and we find extra Noether symmetries at [Formula: see text].

A Recursive Householder Transformation for Complex Dynamical Systems With Constraints

1988 ◽  
Vol 55 (3) ◽  
pp. 729-734 ◽  
Author(s):  
F. M. L. Amirouche ◽  
Tongyi Jia ◽  
Sitki K. Ider
Keyword(s):  
Dynamical Systems ◽  
General Solution ◽  
Equations Of Motion ◽  
New Method ◽  
Systems Dynamics ◽  
Complex Dynamical Systems ◽  
Governing Equations ◽  
Householder Transformation ◽  
Constraint Equations ◽  
Computer Automation

A new method is presented by which equations of motion of complex dynamical systems are reduced when subjected to some constraints. The method developed is used when the governing equations are derived using Kane’s equations with undetermined multipliers. The solution vectors of the constraint equations are determined utilizing the recursive Householder transformation to obtain a Pseudo-Uptriangular matrix. The most general solution in terms of new independent coordinates is then formulated. Methods previously used for handling such systems are discussed and the new method advantages are illustrated. The procedures developed are suitable for computer automation and especially in developing generic programs to study a large class of systems dynamics such as robotics, biosystems, and complex mechanisms.

Reduced conformal geometrodynamics

2020 ◽  
Vol 35 (02n03) ◽  
pp. 2040023 ◽  
Author(s):  
Andrej B. Arbuzov ◽  
Alexander E. Pavlov
Keyword(s):  
Gravitational Field ◽  
Dynamical Systems ◽  
Coordinate System ◽  
Tangent Space ◽  
Equations Of Motion ◽  
Mean Value ◽  
Hamiltonian Reduction ◽  
Square Root ◽  
The Mean ◽  
Static Background

The global time in geometrodynamics is defined in a covariant under diffeomorphisms form. An arbitrary static background metric is taken in the tangent space. The global intrinsic time is identified with the mean value of the logarithm of the square root of the ratio of the metric determinants. The procedures of the Hamiltonian reduction and deparametrization of dynamical systems are implemented. The reduced Hamiltonian equations of motion of gravitational field in semi-geodesic coordinate system are written.

scholarly journals Transient Annular Structures in Exploding Galaxies

1972 ◽  
Vol 44 ◽  
pp. 313-313
Author(s):  
J. L. Sěrsic
Keyword(s):  
Gravitational Field ◽  
Test Particle ◽  
Relative Velocity ◽  
Symmetry Axis ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Physical Space ◽  
Space Time ◽  
High Mass ◽  
New Variables

The explosive events going on in the central parts of some galaxies are related to a very high mass concentration. As an explosion is actually a drastic rearrangement of the concerned masses with energy release, the binding energy of the central core will change and, correspondingly, its effective gravitational mass. A test particle far from the nuclear region, although within the galaxy, will be moving accordingly in a variable-mass Newtonian gravitational field.On the other hand the observations suggest that explosions in galaxies have axial symmetry, so we are concerned with the global properties of the motion of a particle in a variable mass axisymmetric gravitational field. In order to get rid of the mass variation a space-time conformal transformation is made, which, after imposing some not very restrictive conditions, leads to a conservative potential in the new variables. This new potential has additional terms due to the elimination of the variable mass. The equations of motion in the new variables provide the motion of the test particle relative to an expanding or contracting background which depends on the choice of the transformation and the law of the mass variability. The problem is, at this point, formally similar to Hill's. It is possible to write an equation for the relative energy (a generalization of Jacobi's integral) and also to define surfaces of zero relative velocity for the infinitesimal particle. The general topological properties of these surfaces require singular points along the symmetry axis (analogous to the collinear Eulerian points) and also a dense set in a circumference on a plane perpendicular to the symmetry axis (analogous to the Lagrangian points). The latter one is the main feature characterizing the topology of the zero relative velocity surfaces. Even when we lift some of the restrictive conditions, the Lagrangian ring preserves its properties, as for example, the one of being the only region where zero-velocity curves and equi-potentials coincide when the configuration evolves in time (in the transformed space-time).It is easy to understand that the topology of the surfaces is kept when we reverse the transformation and go back to physical space-time. If the dust, gas or stars in the system has definite upper limits for its Jacobian constants, spatial segregation of them will arise, as is the case in radio-galaxies such as NGC 5128, NGC 1316, etc. where ringlike dust structures are observed.

Sign in / Sign up

Export Citation Format

Share Document