Explicit Equations of Motion for Dynamical Systems with Multiple Constraints

2006 ◽  
Vol 45 (6A) ◽  
pp. 5286-5292
Author(s):  
Po-Chih Chen ◽  
Chia-Ou Chang ◽  
W. T. Chang Chien ◽  
Chan-Shin Chou
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Multiple Constraints ◽  
Explicit Equations Of Motion

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.

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.

Mechanical Systems With Nonideal Constraints: Explicit Equations Without the Use of Generalized Inverses

2004 ◽  
Vol 71 (5) ◽  
pp. 615-621 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Robert E. Kalaba ◽  
Phailaung Phohomsiri
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Generalized Inverses ◽  
Explicit Equations Of Motion

In this paper we obtain the explicit equations of motion for mechanical systems under nonideal constraints without the use of generalized inverses. The new set of equations is shown to be equivalent to that obtained using generalized inverses. Examples demonstrating the use of the general equations are provided.

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.

Direct Linearization of Continuous and Hybrid Dynamical Systems

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado ◽  
Andrew J. Sinclair
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equations ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Hybrid Dynamical Systems ◽  
Linearized Equations ◽  
Direct Linearization ◽  
Taylor Series Expansions ◽  
Nonlinear Equations Of Motion ◽  
And Control

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

scholarly journals A dynamical system analysis of f(R,T) gravity

2016 ◽  
Vol 13 (09) ◽  
pp. 1650108 ◽  
Author(s):  
Behrouz Mirza ◽  
Fatemeh Oboudiat
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Equation Of State ◽  
System Analysis ◽  
Equations Of Motion ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Future Singularity ◽  
First Case ◽  
Stress Energy

We investigate equations of motion and future singularities of [Formula: see text] gravity where [Formula: see text] is the Ricci scalar and [Formula: see text] is the trace of stress-energy tensor. Future singularities for two kinds of equation of state (barotropic perfect fluid and generalized form of equation of state) are studied. While no future singularity is found for the first case, some kind of singularity is found to be possible for the second. We also investigate [Formula: see text] gravity by the method of dynamical systems and obtain some fixed points. Finally, the effect of the Noether symmetry on [Formula: see text] is studied and the consistent form of [Formula: see text] function is found using the symmetry and the conserved charge.

Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints

2000 ◽  
Vol 68 (3) ◽  
pp. 462-467 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Lagrangian Mechanics ◽  
Constrained Motion ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Lagrangian Formulations ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D’Alembert. There are, however, many physical situations where this confining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D’Alembert’s principle, and presents the explicit equations of motion for constrained mechanical systems in which the constraints are nonideal. These equations lead to a simple and new fundamental view of Lagrangian mechanics. They provide a geometrical understanding of constrained motion, and they highlight the simplicity with which Nature seems to operate.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

Sign in / Sign up

Export Citation Format

Share Document