Periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

1992 ◽  
Vol 193 (2) ◽  
pp. 309-315 ◽  
Author(s):  
F. M. F. El-Sabaa
Keyword(s):  
Fixed Point ◽  
Rigid Body ◽  
Periodic Solutions ◽  
Kovalevskaya Case

Remarks on the Covering of the Possible Motion Area by Solutions in Rigid Body Systems

2019 ◽  
Vol 20 (3-4) ◽  
pp. 293-302
Author(s):  
Ivan Polekhin
Keyword(s):  
Fixed Point ◽  
Rigid Body ◽  
Integrable Case ◽  
Analytical Results ◽  
Routh Reduction ◽  
Gyroscopic Forces ◽  
Kovalevskaya Case

AbstractThe problem of motion of a rigid body with a fixed point is considered. We study qualitatively the solutions of the system after Routh reduction. For the Lagrange integrable case, we show that the trajectories of solutions starting at the boundary of a possible motion area can both cover and not cover the entire possible motion area. It distinguishes these systems from the systems without gyroscopic forces, where the trajectories always cover the possible motion area. We also present some numerical and analytical results on the same matter for the Kovalevskaya case.

scholarly journals The periodic rotary motions of a rigid body in a new domain of angular velocity

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Fixed Point ◽  
Angular Velocity ◽  
Rigid Body ◽  
Periodic Solutions ◽  
Nonlinear Differential Equations ◽  
First Integral ◽  
High Energy ◽  
The Body ◽  
New Technique ◽  
Large Parameter

AbstractIn the previous works, the limiting case for the motion of a rigid body about a fixed point in a Newtonian force field, which comes from a gravity center lies on Z-axis, is solved. The authors apply the small parameter technique which is achieved giving the body a sufficiently large angular velocity component ro about the fixed z-axis of the body. The periodic solutions of motion are obtained in neighborhood ro tends to $$\infty$$ ∞ . In our work, we aim to find periodic solutions to the problem of motion in the neighborhood of r0 tends to $$0$$ 0 . So, we give a new assumption that: ro is sufficiently small. Under this assumption, we must achieve a large parameter and search for another technique for solving this problem. This technique is named; a large parameter technique instead of the small one well known previously. We see the advantage of the new technique which appears in saving high energy used to begin the motion and give the solution of the problem in another domain. The obtained solutions by the new technique depend on ro. We consider that the center of mass of this body does not necessarily coincide with the fixed point O. We reduce the six nonlinear differential equations of the body and their three first integrals to a quasilinear autonomous system of two degrees of freedom and one first integral. We solve the rational case when the frequencies of the generating system are rational except $$(\,\omega = \,1,\,2,1/2,3,1/3, \ldots )$$ ( ω = 1 , 2 , 1 / 2 , 3 , 1 / 3 , … ) under the condition $$\gamma^{\prime\prime}_{0} = \cos \theta_{o} \approx 0$$ γ 0 ″ = cos θ o ≈ 0 . We use the fourth-order Runge–Kutta method to find the periodic solutions in the closed interval of the time t and to compare the analytical method with the numerical one.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

scholarly journals Existence of Positive Periodic Solutions for a Class ofn-Species Competition Systems with Impulses

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Peilian Guo ◽  
Yansheng Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Species Competition ◽  
Positive Periodic Solutions ◽  
The Fixed Point Theorem

By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class ofn-species competition systems with impulses. Meanwhile, we point out that the conclusion of (Yan, 2009) is incorrect.

scholarly journals Multiplicity of positive periodic solutions to second order differential equations

2006 ◽  
Vol 73 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Jifeng Chu ◽  
Xiaoning Lin ◽  
Daqing Jiang ◽  
Donal O'Regan ◽  
R. P. Agarwal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Second Order Differential Equations

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

scholarly journals Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Meiqiang Feng
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Type Equation ◽  
Index Theorem ◽  
Rayleigh Equation ◽  
Deviating Arguments ◽  
Nontrivial Periodic Solutions ◽  
Existence Of Periodic Solutions

The Rayleigh equation with two deviating argumentsx′′(t)+f(x'(t))+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=e(t)is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

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