An efficient algorithm for equations of motion of molecular 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.

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AN EFFICIENT ALGORITHM FOR THE COMPUTATION OF STABILITY POINTS OF DYNAMICAL SYSTEMS UNDER STEP LOAD

Engineering Computations ◽

10.1108/eb023890 ◽

1992 ◽

Vol 9 (6) ◽

pp. 669-679 ◽

Cited By ~ 3

Author(s):

P. WRIGGERS ◽

C. CARSTENSEN

Keyword(s):

Dynamical Systems ◽

Efficient Algorithm

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Dynamical systems: Approximate Lagrangians and Noether symmetries

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501603 ◽

2019 ◽

Vol 16 (10) ◽

pp. 1950160 ◽

Cited By ~ 1

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].

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A Recursive Householder Transformation for Complex Dynamical Systems With Constraints

Journal of Applied Mechanics ◽

10.1115/1.3125857 ◽

1988 ◽

Vol 55 (3) ◽

pp. 729-734 ◽

Cited By ~ 13

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.

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Reduced conformal geometrodynamics

International Journal of Modern Physics A ◽

10.1142/s0217751x20400230 ◽

2020 ◽

Vol 35 (02n03) ◽

pp. 2040023 ◽

Cited By ~ 1

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.

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Direct Linearization of Continuous and Hybrid Dynamical Systems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.3124092 ◽

2009 ◽

Vol 4 (3) ◽

Cited By ~ 1

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.

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816501085 ◽

2016 ◽

Vol 13 (09) ◽

pp. 1650108 ◽

Cited By ~ 20

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.

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Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2802521 ◽

1999 ◽

Vol 121 (4) ◽

pp. 594-598 ◽

Cited By ~ 1

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.

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Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

Journal of Applied Mechanics ◽

10.1115/1.2900855 ◽

1993 ◽

Vol 60 (3) ◽

pp. 662-668 ◽

Cited By ~ 36

Author(s):

R. E. Kalaba ◽

F. E. Udwadia

Keyword(s):

Dynamical Systems ◽

Closed Form ◽

Programming Problem ◽

Equations Of Motion ◽

Mechanical Systems ◽

Closed Form Solution ◽

Nonholonomic Constraints ◽

Form Solution ◽

Constrained Motion ◽

Constraint Forces

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

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Comparison of Kane’s and Lagrange’s Methods in Analysis of Constrained Dynamical Systems

Robotica ◽

10.1017/s0263574719001899 ◽

2020 ◽

Vol 38 (12) ◽

pp. 2138-2150

Author(s):

Amin Talaeizadeh ◽

Mahmoodreza Forootan ◽

Mehdi Zabihi ◽

Hossein Nejat Pishkenari

Keyword(s):

Dynamical Systems ◽

Equations Of Motion ◽

Computational Cost ◽

Constrained Systems ◽

Robot Dynamics ◽

Energy Error ◽

Kane’S Method ◽

Kane's Method ◽

Constrained Dynamical Systems ◽

Error Energy

SUMMARYDynamic modeling is a fundamental step in analyzing the movement of any mechanical system. Methods for dynamical modeling of constrained systems have been widely developed to improve the accuracy and minimize computational cost during simulations. The necessity to satisfy constraint equations as well as the equations of motion makes it more critical to use numerical techniques that are successful in decreasing the number of computational operations and numerical errors for complex dynamical systems. In this study, performance of a variant of Kane’s method compared to six different techniques based on the Lagrange’s equations is shown. To evaluate the performance of the mentioned methods, snake-like robot dynamics is considered and different aspects such as the number of the most time-consuming computational operations, constraint error, energy error, and CPU time assigned to each method are compared. The simulation results demonstrate the superiority of the variant of Kane’s method concerning the other ones.

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