On the averaging principle for one-frequency systems. An application to satellite motions

This paper is related to our previous works [1][2] on the error estimate of the averaging technique, for systems with one fast angular variable. In the cited references, a general method (of mixed analytical and numerical type) has been introduced to obtain precise, fully quantitative estimates on the averaging error. Here, this procedure is applied to the motion of a satellite in a polar orbit around an oblate planet, retaining only the J_2 term in the multipole expansion of the gravitational potential. To exemplify the method, the averaging errors are estimated for the data corresponding to two Earth satellites; for a very large number of orbits, computation of our estimators is much less expensive than the direct numerical solution of the equations of motion.Comment: LaTeX, 35 pages, 12 figures. The final version published in Nonlinear Dynamic

Separation of variables in multi-Hamiltonian systems: an application to the Lagrange top

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the reduction of the vector field and of the Poisson tensors. We show explicitly that, after the reduction on each one of the symplectic leaves, the vector field of the Lagrange top is separable in the sense of Hamilton-Jacobi.Comment: report to XVI NEEDS (Cadiz 2002): 15 pages, no figures, LaTeX. To appear in Theor. Math. Phy

On the constants in a Kato inequality for the Euler and Navier-Stokes equations

We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v . D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity) and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.Comment: LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412 by the same authors, not concerning the main result

On the Treves theorem for the AKNS equation

According to a theorem of Treves, the conserved functionals of the AKNS equation vanish on all pairs of formal Laurent series of a specified form, both of them with a pole of the first order. We propose a new and very simple proof for this statement, based on the theory of B\"acklund transformations; using the same method, we prove that the AKNS conserved functionals vanish on other pairs of Laurent series. The spirit is the same of our previous paper on the Treves theorem for the KdV, with some non trivial technical differences.Comment: LaTeX, 16 page

On approximate solutions of semilinear evolution equations

A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance of the latter from the approximate solution can be evaluated solving a one-dimensional "control" integral equation, where the unknown gives a bound on the previous distance as a function of time. For example, the control equation can be applied to the approximation methods based on the reduction of the evolution equation to finite-dimensional manifolds: among them, the Galerkin method is discussed in detail. To illustrate this framework, the nonlinear heat equation is considered. In this case the control equation is used to evaluate the error of the Galerkin approximation; depending on the initial datum, this approach either grants global existence of the solution or gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened version; the proof of Prop. 3.4. has been simplified

On the averaging principle for one-frequency systems. Seminorm estimates for the error

We extend some previous results of our work [1] on the error of the averaging method, in the one-frequency case. The new error estimates apply to any separating family of seminorms on the space of the actions; they generalize our previous estimates in terms of the Euclidean norm. For example, one can use the new approach to get separate error estimates for each action coordinate. An application to rigid body under damping is presented. In a companion paper [2], the same method will be applied to the motion of a satellite around an oblate planet.Comment: LaTeX, 23 pages, 4 figures. The final version published in Nonlinear Dynamic

On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian

It is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained trough a non canonical map whose form is directly suggested by the associated Nijenhuis tensor.Comment: 11 pages, AMS-LaTex 1.

Ground state and excitation dynamics in Ag doped helium clusters

We present a quantum Monte Carlo study of the structure and energetics of silver doped helium clusters AgHen for n up to 100. Our simulations show the first solvation shell of the Ag atom to include roughly 20 He atoms, and to possess a structured angular distribution. Moreover, the P-2(1/2)<--S-2(1/2) and P-2(3/2)<--S-2(1/2) electronic transitions of the embedded silver impurity have been studied as a function of the number of helium atoms. The computed spectra show a redshift for nless than or equal to15 and an increasing blueshift for larger clusters, a feature attributed to the effect of the second solvation shell of He atoms. For the largest cluster, the computed excitation spectrum is found in excellent agreement with the ones recorded in superfluid He clusters and bulk. No signature of the direct formation of the proposed AgHe2 exciplex is present in the computed spectrum of AgHe100. To explain the absence of the fluorescent D-2 line in the experiments, a relaxation mechanism between the P-2(3/2) and the P-2(1/2) states is proposed on the basis of the partial overlap of the excitation bands in the simulated spectra. (C) 2002 American Institute of Physics

Higher Order Potential Expansion for the Continuous Limits of the Toda Hierarchy

A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the lower ones and can be continued to be performed indefinitly. By introducing the higher order terms, the fewer equations in the Toda hierarchy are needed in the so-called recombination method to recover the KdV hierarchy. It is shown that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the KdV hierarchy in continuous limit.Comment: 20 pages, Latex, to be published in Journal of Physics