Nonintegrability of an Infinite-Degree-of-Freedom Model for Unforced and Undamped, Straight Beams

2003 ◽  
Vol 70 (5) ◽  
pp. 732-738
Author(s):  
K. Yagasaki
Keyword(s):  
Differential Equation ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Degree Of Freedom ◽  
Differential Galois Theory ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Infinite Degree ◽  
Simulation Results

We study a mathematical model for unforced and undamped, initially straight beams. This system is governed by an integro-partial differential equation, and its energy is conserved: It is an infinite-degree-of-freedom Hamiltonian system. We can derive “exact” finite-degree-of-freedom mode truncations for it. Using the differential Galois theory for Hamiltonian systems, we prove that any two or more modal truncations for the model are nonintegrable in the following sense: The Hamiltonian systems do not have the same number of “meromorphic” first complex integrals which are independent and in involution, as the number of their degrees of freedom, when they are regarded as Hamiltonian systems with complex time and coordinates. This also means the nonintegrability of the infinite-degree-of-freedom model for the beams. We present numerical simulation results and observe that chaotic motions occur as in typical nonintegrable Hamiltonian systems.

scholarly journals DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY

2009 ◽  
Vol 06 (08) ◽  
pp. 1357-1390 ◽  
Author(s):  
ANDRZEJ J. MACIEJEWSKI ◽  
MARIA PRZYBYLSKA
Keyword(s):  
Galois Group ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Homogeneous Potentials ◽  
Particular Solution

This paper is an overview of our works that are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realize the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

scholarly journals Second proof of the irreducibility of the first differential equation of painlevé

1990 ◽  
Vol 117 ◽  
pp. 125-171 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Galois Theory ◽  
Rigorous Proof ◽  
Mathematical Foundation ◽  
Differential Galois Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

A Novel 3-DOF Parallel Robot and its Kinematic Analysis

2014 ◽  
Vol 607 ◽  
pp. 759-763
Author(s):  
Xiao Bo Liu ◽  
Xiao Dong Yuan ◽  
Xiao Feng Wei ◽  
Wei Ni
Keyword(s):  
Motion Control ◽  
Kinematic Analysis ◽  
Degrees Of Freedom ◽  
Parallel Robot ◽  
Degree Of Freedom ◽  
The Novel ◽  
Forward Displacement ◽  
Simulation Based ◽  
Simulation Results

This paper deals with the design and analysis of a novel and simple two-translation and one-rotation (3 degrees of freedom, 3-dof) mechanism for alignment. Firstly, degree of freedom of the parallel robot is solved based on the theory of screw. Secondly considering the demand of motion control, we have conducted the analysis on the 3-dof parallel robot, which includes inverse displacement, forward displacement, and simulation based on SolidWorks Motion. The simulation results indicate that the novel 3-dof robot is suitable for performing the required operations.

scholarly journals Birational automorphism groups and differential equations

1990 ◽  
Vol 119 ◽  
pp. 1-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Rigorous Proof ◽  
Differential Galois Theory ◽  
General Attitude ◽  
Transcendental Functions ◽  
Birational Automorphism

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

Integrable and chaotic motions of four vortices. I. The case of identical vortices

1982 ◽  
Vol 380 (1779) ◽  
pp. 359-387 ◽  
Keyword(s):  
Degrees Of Freedom ◽  
Heteroclinic Orbit ◽  
Elliptic Functions ◽  
Explicit Construction ◽  
Resonant Interaction ◽  
Two Dimensions ◽  
Degree Of Freedom ◽  
Chaotic Motions ◽  
Interaction Terms ◽  
Coupled Nonlinear Oscillators

It is shown that the three-vortex problem in two-dimensional hydrodynamics is integrable, whereas the motion of four identical vortices is not. A sequence of canonical transformations is obtained that reduces the N -degree-of-freedom Hamiltonian, which describes the interaction of N identical vortices, to one with N — 2 degrees of freedom. For N = 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions. For N = 4 the reduction procedure leads to an effective Hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators. Resonant interaction terms in this Hamiltonian suggest non-integrable behaviour and this is verified by numerical experiments. Explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible. The relevance of the results obtained to fundamental problems in hydrodynamics, such as the question of integrability of Euler’s equation in two dimensions, is discussed. The paper also contains a general exposition of the Hamiltonian and Poisson-bracket formalism for point vortices.

Chaotic Dynamics of a Quasi-Periodically Forced Beam

1992 ◽  
Vol 59 (1) ◽  
pp. 161-167 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Single Mode ◽  
Homoclinic Orbits ◽  
Equation Of Motion ◽  
Degree Of Freedom ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Higher Modes ◽  
Straight Beam

A straight beam with fixed ends, forced with two frequencies is considered. By using Galerkin’s method, the equation of motion of the beam is reduced to a finite degree-of-freedom system. The resulting equation is transformed into a multi-frequency system and the averaging method is applied. It is shown, by using Melnikov’s method, that there exist transverse homoclinic orbits in the averaged system associated with the first-mode equation. This implies that chaotic motions may occur in the single-mode equation. Furthermore, the effect of higher modes and the implications of this result for the full beam motions are described.

scholarly journals Degrees of freedom and Hamiltonian formalism for f(T) gravity

2020 ◽  
Vol 35 (02n03) ◽  
pp. 2040022
Author(s):  
María José Guzmán ◽  
Rafael Ferraro
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Essential Feature ◽  
Hamiltonian Formalism ◽  
Local Symmetry ◽  
Degree Of Freedom ◽  
Constrained Hamiltonian Systems ◽  
Dirac Formalism

The existence of an extra degree of freedom (d.o.f.) in [Formula: see text] gravity has been recently proved by means of the Dirac formalism for constrained Hamiltonian systems. We will show a toy model displaying the essential feature of [Formula: see text] gravity, which is the pseudo-invariance of [Formula: see text] under a local symmetry, to understand the nature of the extra d.o.f.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

2005 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Periodic Motion ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a 2-DOF nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the energy incremental energy approach. The Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The chaotic and quasiperiodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian systems. The resonant-periodic motions are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

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