arXiv subjects

Jaume Giné

Publications and source records attributed to Jaume Giné.

22 records · Page 2Linked to original sources

arXiv2005

On the origin of the gravitational quantization: The Titius--Bode Law

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post--Newtonian physics. As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account. Newtonian equations of motion, with post--Newtonian corrections, are often used to approximate the functional differential equations. In ``On the origin of quantum mechanics'', preprint, physics/0505181, May 2005, a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model was presented. The unique assumption was that the electrodynamic interaction has finite propagation speed. Are the finite propagation speeds also the origin of the gravitational quantization? In this work a simple gravitational model based on a functional differential equation gives an explanation of the modified Titius--Bode law.

physics.gen-ph↗

arXiv2005

A note on "Relaxation Oscillators with Exact Limit Cycles"

In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work of M.A. Abdelkader entitled ``Relaxation Oscillators with Exact Limit Cycles'', which appeared in J. Math. Anal. Appl. 218 (1998), 308--312. The result given in this note may be used to construct models of Liénard differential equations exhibiting a desired limit cycle.

math.DS↗

arXiv2005

The role of algebraic solutions in planar polynomial differential systems

We study a planar polynomial differential system, given by \dot{x}=P(x,y), \dot{y}=Q(x,y). We consider a function I(x,y)=\exp \{h_2(x) A_1(x,y) \diagup A_0(x,y) \} h_1(x) \prod_{i=1}^{\ell} (y-g_i(x))^{α_i}, where g_i(x) are algebraic functions, A_1(x,y)=\prod_{k=1}^r (y-a_k(x)), A_0(x,y)=\prod_{j=1}^s (y-\tilde{g}_j(x)) with a_k(x) and \tilde{g}_j(x) algebraic functions, A_0 and A_1 do not share any common factor, h_2(x) is a rational function, h(x) and h_1(x) are functions with a rational logarithmic derivative and α_i are complex numbers. We show that if I(x,y) is a first integral or an integrating factor, then I(x,y) is a Darboux function. In order to prove this result, we show that if g(x) is such that there exists an irreducible polynomial f(x,y) with f(x,g(x)) \equiv 0, then f(x,y)=0 is an invariant algebraic curve of the system. In relation with this fact, we give some characteristics related to particular solutions and functions of the form I(x,y) such as the structure of their cofactor. Moreover, we consider a function of the form Φ(x,y):= \exp \{h_2(x) A_1(x,y) / A_0 (x,y) \}. We show that if the derivative of Φ(x,y) with respect to the flow is well defined over A_0(x,y)=0 then Φ(x,y) gives rise to an exponential factor.

math.DS↗

arXiv2005

Integrability of planar polynomial differential systems through linear differential equations

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.

math.DS↗