A new approach to the parameterization method for Lagrangian tori of hamiltonian systems

We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of O(e1/2)O(e1/2) , where ee is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.Peer ReviewedPostprint (author's final draft

A parameterization method for Lagrangian tori of exact symplectic maps of R2r

We are concerned with analytic exact symplectic maps of ${\mathbb R}^{2r}$ endowed with the standard symplectic form. We study the existence of a real analytic torus of dimension $r$, invariant by the map and carrying quasi-periodic motion with a prefixed Diophantine rotation vector. Therefore, this torus is a Lagrangian manifold. We address the problem by the parameterization method in KAM theory. The main aspect of our approach is that we do not look for the parameterization of the torus as a solution of the corresponding invariance equation. Instead, we consider a set of three equations that, all together, are equivalent to the invariance equation. These equations arise from the geometric and dynamical properties of the map and the torus. Suppose that an approximate solution of these equations is known and that a suitable nondegeneracy (twist) condition is satisfied. Then, this system of equations is solved by a quasi-Newton-like method, provided that the initial error is sufficiently small. By “quasi-Newton-like” we mean that the convergence is almost quadratic but that at each iteration we have to solve a nonlinear equation. Although it is straightforward to build a quasi-Newton method for the selected set of equations, proceeding in this way we improve the convergence condition. The selected definition of error reflects the level at which the error associated with each of these three equations contributes to the total error. The map is not required to be close to integrable or expressed in action-angle variables. Suppose the map is $\varepsilon$-close to an integrable one, and consider the portion of the phase space not filled up by Lagrangian invariant tori of the map. Then, the upper bound for the Lebesgue measure of this set that we may predict from the result is of ${\mathcal O}(\varepsilon^{1/2})$. In light of the classical KAM theory for exact symplectic maps, an upper bound of ${\mathcal O}(\varepsilon^{1/2})$ for this measure is the expected estimate. The result also has some implications for finitely differentiable mapsPeer ReviewedPostprint (author's final draft

Kolmogorov Theorem Revisited

Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian systems is revisited. In this paper we are mainly concerned with the lower bound on the constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be of O(ε1/4), where ε is the size of the perturbation. In this paper, by means of careful (but involved) estimates on Kolmogorov’smethod, we show that this lower bound can be weakened to be of O(ε1/2). This condition coincides with the optimal one of KAM Theorem. Moreover, we also obtain optimal estimates for the distance between the actions of the perturbed and unperturbed tori

L'aprenentatge basat en jocs millora els resultats de l'alumnat?

L'objectiu d'aquest TFM és analitzar si l'aprenentatge basat en jocs, com a innovació pedagògica, millora el rendiment acadèmic de l'alumnat i, en base als resultats, concloure si és recomanable (o no) utilitzar aquesta metodologia a l'aula. Per dur a terme aquest anàlisi, s'han dissenyat i executat dues programacions diferents de la UD de "Mecanismes" als tres grups de tercer d'ESO del centre de pràctiques: una amb activitats basades en jocs (programació J) i l'altra amb les mateixes activitats seguint un format més "tradicional" de classe magistral (programació T). Totes dues programacions (J i T) han compartit la impartició dels mateixos continguts teòrics i pràctics i, en el marc dels criteris d'avaluació, una prova individual idèntica. Els resultats d'aquesta prova individual, conjuntament amb l'observació directa, una enquesta a l'alumnat i un estudi al voltant de les activitats de repàs, han sigut la base de les conclusions d'aquest treball

Numerical computation of normal forms around some periodic orbits of the restricted three body problem

In this paper we introduce a general methodology for computing (numerically) the normal form around a periodic orbit of an autonomous analytic Hamiltonian system. The process follows two steps. First, we expand the Hamiltonian in suitable coordinates around the orbit and second, we perform a standard normal form scheme, based on the Lie series method. This scheme is carried out up to some finite order and, neglecting the remainder, we obtain an accurate description of the dynamics in a (small enough) neighbourhood of the orbit. In particular, we obtain the invariant tori that generalize the elliptic directions of the periodic orbit. On the other hand, bounding the remainder one obtains lower estimates for the diffusion time around the orbit. This procedure is applied to an elliptic periodic orbit of the spatial Restricted Three Body Problem. The selected orbit belongs to the Lyapunov family associated to the vertical oscillation of the equilibrium point $L_5$. The mass parameter $\mu$ has been chosen such that $L_5$ is unstable but the periodic orbit is still stable. This allows to show the existence of regions of effective stability near $L_5$ for values of $\mu$ bigger that the Routh critical value. The computations have been done using formal expansions with numerical coefficients

Constraining the Milky Way potential using the dynamical kinematic substructures

We present a method to constrain the potential of the non-axisymmetric components of the Galaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamical substructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodic or irregular orbits, which may be easily identified in orbital frequency space. We use the "observed" positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitational potentials. We then compute their characteristic frequencies, and study the structure present in the frequency maps. We find that the distribution of dynamical substructures in velocity- and frequency-space is best preserved when the integrations are performed in the "true" gravitational potential.Comment: 2 pages, 4 figures, to appear in the proceedings of "Assembling the Puzzle of the Milky Way", Le Grand Bornand (Apr. 17-22, 2011

On the persistence of lower dimensional invariant tori under quasiperiodic perturbations

In this work we consider time dependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the effect that this kind of perturbations has on lower dimensional invariant tori. Our results show that, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the perturbation to the ones they already have. The paper also contains estimates on the amount of surviving tori. The worst situation happens when the initial tori are normally elliptic. In this case, a torus (identified by the vector of intrinsic frequencies) can be continued with respect to a perturbative parameter $\epsilon\in[0,\epsilon_0]$, except for a set of $\epsilon$ of measure exponentially small with $\epsilon_0$. In case that $\epsilon$ is fixed (and sufficiently small), we prove the existence of invariant tori for every vector of frequencies close to the one of the initial torus, except for a set of frequencies of measure exponentially small with the distance to the unperturbed torus. As a particular case, if the perturbation is autonomous, these results also give the same kind of estimates on the measure of destroyed tori. Finally, these results are applied to some problems of celestial mechanics, in order to help in the description of the phase space of some concrete models

Effective Stability Around Periodic Orbits of the Spatial RTBP

In this work we study the dynamics around an elliptic periodic orbit of Hamiltonian systems. To this end we have developped an algorithm to compute a normal form (up to a finite order) around this orbit, that gives an accurate description of the dynamics close to it. If the remainder of this normal form can be bounded, it is not difficult to produce explicit bounds of the diffusion time of trajectories starting near the periodic orbit. In order to discuss the effectivity of the method, it will be explained at the same time that it is applied to a concrete example. The one used here has been the Spatial Restricted Three Body Problem (RTBP