scholarly journals Normal forms for elements of o(p, q) and Hamiltonians with integrals linear in momenta

1992 ◽  
Vol 33 (4) ◽  
pp. 486-507 ◽  
Author(s):  
Gerard Thompson
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Lie Algebra ◽  
Normal Forms ◽  
First Integral ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Hamiltonian Dynamical Systems ◽  
Matrix Normal

AbstractWe solve the problem of finding a simultaneous matrix normal form for an element of the Lie algebra o(p, q) and the underlying indefinite inner product. The results are used to determine several classes of classical Hamiltonian dynamical systems which possess a first integral linear in the momentum variables.

On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

1995 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Normal Form ◽  
Linear Part ◽  
Normal Form Theory ◽  
Mathieu’S Equation ◽  
Form Theory ◽  
Hamiltonian Dynamical Systems ◽  
Time Invariant ◽  
Time Periodic

Abstract In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

FORMAL DECOMPOSITION METHOD AND PARAMETRIC NORMAL FORMS

2010 ◽  
Vol 20 (11) ◽  
pp. 3487-3515 ◽  
Author(s):  
MAJID GAZOR ◽  
PEI YU
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Decomposition Method ◽  
Normal Forms ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Multiple Parameters ◽  
Formal Basis ◽  
Parametric Normal Forms ◽  
Parametric Systems

We introduce a formal decomposition method for efficiently computing the parametric normal form of nonlinear dynamical systems with multiple parameters. Recently introduced notions of formal basis style and costyle are applied through formal decomposition method to obtain the simplest parametric normal form for degenerate nonlinear parametric center. The necessary formulas are derived and implemented using Maple to compute the simplest parametric normal form of degenerate and nondegenerate nonlinear centers. Our program computes the order of any planar parametric systems associated with this singularity.

The Application of Improved Recursive Formula for Computing Normal Forms of Four-Dimensional Nilpotent Dynamical Systems

2013 ◽  
Vol 437 ◽  
pp. 27-31
Author(s):  
Lei Guo ◽  
Qun Hong Li ◽  
Zi Gen Song
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Normal Form ◽  
Normal Forms ◽  
Recursive Formula ◽  
Computer Programs ◽  
Nonlinear Transformations ◽  
Identity Transformations

In this paper, an explicit recursive formula of normal forms under nonlinear near-identity transformations is introduced. By solving a series of algebra equations with the aid of Maple, not only the coefficients of k order normal form and the associated nonlinear transformations but also high (>k) order terms of the original equations can be obtained. An example about four-dimensional nilpotent dynamical system is given to show applicability of the recursive formula and the outline of symbolic computer programs is given to support application of the recursive formula.

Unique Normal Form for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (08) ◽  
pp. 1750131 ◽  
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Linear Part ◽  
Three Dimensional ◽  
First Integral ◽  
Second Order ◽  
Lie Brackets ◽  
The Given

In this paper, the unique normal form for a class of three-dimensional nilpotent vector fields with symmetry is investigated. The key technique used is a combination of multiple Lie brackets, linear grading function, new notations of block matrices and first integral of the linear part of the given vector field which avoids complicated calculations. The first and the second order normal forms are computed successively, and the uniqueness of the second order normal form is proved under certain conditions.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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.

scholarly journals Normal form approach in the motion planning of space robots: a case study

2021 ◽  
Author(s):  
Krzysztof Tchoń ◽  
Katarzyna Zadarnowska
Keyword(s):  
Normal Form ◽  
Motion Planning ◽  
Normal Forms ◽  
Inverse Dynamics ◽  
Planning Problem ◽  
Inverse Dynamics Problem ◽  
Velocity Level ◽  
Form Approach ◽  
Space Approach

AbstractWe examine applicability of normal forms of non-holonomic robotic systems to the problem of motion planning. A case study is analyzed of a planar, free-floating space robot consisting of a mobile base equipped with an on-board manipulator. It is assumed that during the robot’s motion its conserved angular momentum is zero. The motion planning problem is first solved at velocity level, and then torques at the joints are found as a solution of an inverse dynamics problem. A novelty of this paper lies in using the chained normal form of the robot’s dynamics and corresponding feedback transformations for motion planning at the velocity level. Two basic cases are studied, depending on the position of mounting point of the on-board manipulator. Comprehensive computational results are presented, and compared with the results provided by the Endogenous Configuration Space Approach. Advantages and limitations of applying normal forms for robot motion planning are discussed.

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