Near-Integrability and Recurrence in FPU-Cells

2016 ◽  
Vol 26 (14) ◽  
pp. 1650230 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Hamiltonian Systems ◽  
Building Block ◽  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Chain Recurrence ◽  
Low Dimension ◽  
A Cell ◽  
Linearized System ◽  
A Chain ◽  
Three Degrees Of Freedom

In a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the [Formula: see text] resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in [Formula: see text] resonance.

The statistical distribution of the length of a rubber molecule

1948 ◽  
Vol 44 (3) ◽  
pp. 342-344 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Normal Distribution ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Statistical Distribution ◽  
Equal Length ◽  
Fixed Angle ◽  
A Chain ◽  
Image Position ◽  
Three Degrees Of Freedom

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the formwhere σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.

The Vibrations of the Engine with Neo-Hookean Suspension

2013 ◽  
Vol 430 ◽  
pp. 53-59 ◽  
Author(s):  
Nicolae Doru Stanescu ◽  
Dinel Popa
Keyword(s):  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Harmonic Force ◽  
Small Oscillations ◽  
Excited System ◽  
Beat Phenomenon ◽  
Three Degrees Of Freedom

Our paper realizes a study of the vibrations of an engine excited by a harmonic force and sustained by four identical neo-Hookean springs of negligible masses. The considered model is one with three degrees of freedom (one translation and two rotations) and we obtain for it the equations of motion. Using these equations, we determine for the unexcited system the equilibrium positions and their stability. We also study the small oscillations about the stable equilibrium positions and we find the fundamental eigenpulsations of the system. For the case of the excited system we perform a numerical study considering the situation when the pulsation of the excitation is far away from the eigenpulsations and the situation when the pulsation of the excitation is closed to one eigenpulsation, highlighting the beat phenomenon.

A class of C∞-integrable Hamiltonian systems

1989 ◽  
Vol 113 (3-4) ◽  
pp. 293-314 ◽  
Author(s):  
W. M. Oliva ◽  
M. S. A. C. Castilla
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
First Integrals ◽  
Integrable Hamiltonian Systems ◽  
Integral Of Motion ◽  
Toda System ◽  
Nonnegative Coefficients ◽  
Special Case ◽  
Three Degrees Of Freedom

SynopsisWe discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

Hamiltonian systems with three degrees of freedom, singular-point analysis, and chaotic behavior

1986 ◽  
Vol 33 (3) ◽  
pp. 2131-2133 ◽  
Author(s):  
W.-H. Steeb ◽  
J. A. Louw ◽  
P. G. L. Leach ◽  
F. M. Mahomed
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Point Analysis ◽  
Three Degrees Of Freedom
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