Periodic time dependent hamiltonian systems and applications

Zusammenfassung

A dynamical system is one that evolves with time. This de nition is so di use that seems to be completely useless, however, gives a good insight of the vast range of applicability of this eld of Mathematics has. It is hard to track back in the history of science to nd the origins of this discipline. The works by Fibonacci, in the twelfth century, concerning the population growth rate of rabbits can be already considered to belong to the above mentioned eld. Newton's legacy changed the prism through the humankind watched the universe and established the starting shot of several areas of knowledge including the study of di erential equations. Newton's second law relates the acceleration, the second derivative of the position of a body with the net force acting upon it. The formulation of the law of universal gravitation settled the many body problem, the fundamental question around the eld of celestial mechanics has grown. Newton itself solved the two body problem, providing an analytical proof of Kepler's laws. In the subsequent years a number of authors, among of them Euler and Lagrange, exhausted Newton's powerful ideas but none of them was able to nd a closed solution of the many body problem. By the end of the nineteenth century, Poincare changed again the point of view: The french mathematician realized that the many body problem could not be solved in the sense his predecessors expected, however, many other fundamental questions could be addressed by studying the solutions of not quantitatively but by means of their geometrical and topological properties. The ideas that bloomed in Poincare's mind are nowadays a source of inspiration for modern scientist facing problems located along all the spectrum of human knowledge. Hamiltonian systems are central to the study of dynamical systems. These are governed by a conservation law. If the conserved quantity, the so called Hamiltonian function, does not depend on time, it is constant when it is evaluated along the trajectories of the system. As a matter of fact, a Hamiltonian smooth function H = H(t; Q; P), where t denotes the time and (Q; P) 2 R2n, determines the evolution of the system through the associated rst xix order di erential equation _P = 􀀀 @ @Q H(t; Q; P); _Q = @ @P H(t; Q; P): Here, the integer n is named the number of degrees of freedom of the Hamiltonian systems, the coordinates Q are the positions and P are refereed as the momenta. When the Hamiltonian function does not depend on the time variable t, the system is said to be autonomous, otherwise, it is called nonautonomous. The phase space, has dimension twice the degrees of freedom of the system and displays a very speci c geometrical property: It has a symplectic structure. That is, a solution of a Hamiltonian system t 7! '(t) 2 R2n can be written as '_ = JrH('); where J =  0 I 􀀀I 0  ; and I is the identity matrix of dimension n. This geometrical structure induces properties on the solutions of the systems, in particular, the ow preserves the Lebesgue measure of regions of phase space and, therefore, Hamiltonian systems do not have attractors. Let us focus, for the moment being, in autonomous Hamiltonians. When these have n conserved quantities, I1; I2; : : : ; In whose gradients are linearly independent at each point of the phase space and are in involution; Xn j=1  @Ik @Qj @Il @Pj 􀀀 @Ik @Pj @Il @Qj  = 0 if k 6= l; for each k; l 2 f1; : : : ; ng, are said to be integrable. The phase space of such systems is foliated by n-dimensional invariant sets. When these sets are a compact connected manifold, they are di eomorphic to the torus of dimension n. Many problems in physics (and other sciences) can be modeled by means of Hamiltonian systems, moreover, a large number of those models are perturbations of integrable Hamiltonians. It is natural to address the question of whether the invariant tori of integrable systems persist under perturbation. This problem was originally undertaken by Kolmogorov [Kol54] and, later, completed by Arnold [Arn63a, Arn63b] and Moser [Mos62]. The set of results concerning this issue is nowadays known as KAM theory 1. The solution of the problem 1KAM theory is still a hot topic today and it has been generalized to systems that are not Hamiltonians. was that most, in a measure theory sense, of the invariant tori persist. The size of the perturbation needed to destroy each torus depends (among other hypotheses) on arithmetical properties of its vector of basic frequencies. The space let by the tori after their destruction contains with chaotic motion, trajectories which, even though they are deterministic, seem to obey some kind of randomness. This thesis is concerned with nonautonomous Hamiltonian systems whose time dependence is periodic and, by using a suitable rename, one can think of the temporal variable as an angular one. This angular variables rises by a half the number of degrees of freedom, henceforth, periodic time dependent Hamiltonian systems are said to be of n and a half degrees of freedom. A standard tool to study periodically time dependent Hamiltonian systems is the so-called stroboscopic map i.e. the map obtained by evaluating the ow of the system at the period of the Hamiltonian function. The stroboscopic map of a periodically time dependent Hamiltonians is a di eomorphism P of the phase space that preserve the symplectic structure DPT JDP = J: Symplectic maps are the discrete counterpart to Hamiltonian systems. Most of the properties of Hamiltonians systems can also be derived for maps, in particular, the measure of regions of the phase space is preserved by iteration under a symplectic map. When n = 1, the set of symplectic maps coincides with the set of Area Preserving Maps (APM), those whose di erential matrix has determinant equal to one in the whole of the phase space. To study symplectic maps has certain advantages with respect to Hamiltonian di erential equations. For instance, the limits of KAM theory are better understood in the case of APM: the destruction of KAM tori occurs by the presence of unstable and stable invariant manifolds related to high period periodic points. The chaotic motion, as well, is induced by the tangles created by the intersection of these manifolds. When KAM tori are destroyed, their remainings are a cantor invariant set, the cantori. The studies on the fate of KAM tori under suciently strong perturbations are called Aubry-Mather theory and it is only complete for the case of APM. Notice that one degree of freedom Hamiltonian systems are integrable and their phase space is foliated by invariant tori. The simplest systems in which the dynamics does not consists essentially on invariant tori are one and a half degrees of freedom Hamiltonian systems which, by means of the stroboscopic map, can be regarded as APM. It is common that periodically time dependent Hamiltonian systems are a periodic perturbation of autonomous systems, it is the case, in fact, of all the systems appearing in this dissertation. The phase space of periodically time dependent perturbations of autonomous systems has an inherited structure. Each quasi-periodic invariant structure gains, generically, the frequency of the perturbation: to perturb periodically in time is to periodically shake the phase space, see [JV97b].