Satellite Orbits and Relative Motion in Levi-Civita Coordinates

In this paper we consider satellite trajectories in central force field with quadratic drag using two formalisms. The first using polar coordinates in which the angular momentum plays a dominant role. The second is in Levi-Civita coordinates in which the energy plays a central role. We then unify these two formalisms by introducing polar coordinates in Levi-Civita space and derive a new equation for satellite orbits in which energy and and angular momentum are on equal footing {and thus characterize the orbit by its two invariants}. In the second part of the paper we derive in Levi-Civita coordinates a linearized equation for the relative motion of two satellites whose trajectories are in the same plane. We carry out also a numerical verification of these equations.Comment: 19 pages 3 figure

EMD Algorithm with Approximate Zero Crossings

The classical EMD algorithm has been used extensively in the literature to decompose signals that contain nonlinear waves. However when a signal contain two or more frequencies that are close to one another the decomposition might fail. In this paper we propose a new formulation of this algorithm which is based on the zero crossings of the signal and show that it performs well even when the classical algorithm fail. We address also the filtering properties and convergence rate of the new algorithm versus the classical EMD algorithm.Comment: 23 pages, 7 fig

Assessing Turbulence Strength via Lyaponuv Exponents

In this paper we study the link between `turbulence strength' in a flow and the leading Lyaponuv exponent that characterize it. To this end we use two approaches. The first, analytical, considers the truncated convection equations in 2-dimensions with three (Lorenz model) and six components and study their leading Lyaponuv exponent as a function of the Rayleigh number. For the second approach we analyze fifteen time series of measurements taken by a plane flying at constant height in the upper troposphere. For each of these time series we estimate the leading Lyaponuv exponent which we then correlate with the structure constant for the temperature.Comment: 15 pages 2 fig

A Generalized Cole-Hopf Transformation for Nonlinear ODES

We introduce a hybrid Cole-Hopf-Darboux transformation to relate solutions of nonlinear and linear second order differential equations and derive a sufficient condition for this correspondence. In particular we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the "class of special nonlinear equations" which correspond to the linear differential equations which define the special functions of mathematical physics.Comment: 15 page

Convective Equations and a Generalized Cole-Hopf Transformation

Differential equations with convective terms such as the Burger's equation appear in many applications and have been the subject of intense research. In this paper we use a generalized form of Cole-Hopf transformation to relate the solutions of some of these nonlinear equations to the solutions of linear equations. In particular we consider generalized forms of Burger's equation and second order nonlinear ordinary differential equations with convective terms which can represent steady state one-dimensional convection.Comment: 10 page

Axisymmetric, Rotating and Stratified Star

The paper considers Euler-Poisson equations which govern the steady state of a self gravitating, rotating, axi-symmetric stars under the additional assumption that it is composed of incompressible stratified fluid. The original system of six nonlinear equations is reduced to two equations, one for the mass density and the other for gravitational field. This reduction is carried out separately in cylindrical and spherical coordinates. As a "byproduct" we derive also expressions for the pressure. The resulting equations are then solved approximately and these analytic solutions are used then to determine the shape of the rotating star.Comment: preprint. ALL comments or feedback will be appreciate

A Modified EMD Algorithm and its Applications

The classical EMD algorithm has been used extensively in the literature to decompose signals that contain nonlinear waves. However when a signal contain two or more frequencies that are close to one another the decomposition might fail. In this paper we propose a new formulation of this algorithm which is based on the zero crossings of the signal and show that it performs well even when the classical algorithm fail. We address also the filtering properties and convergence rate of the new algorithm versus the classical EMD algorithm. These properties are compared then to those of the principal component algorithm (PCA). Finally we apply this algorithm to the detection of gravity waves in the atmosphere.Comment: To be presented at "The 2011 International Conference on Scientific Computing (CSC'11)" Las-Vegas Nevada July 17-21, 201

Fractional Darboux Transformations

In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of fractional Darboux transformations which can be recognized also as generalized Cole-Hopf transformations. We apply these transformations to find Schrodinger equations with isospectral potentials and to the linearization of some new classes of nonlinear partial differential equations.Comment: 15 pages,submitted N.C. B corrected some typos added two reference

Closed form Solutions to Some Nonlinear equations by a Generalized Cole-Hopf Transformation

In the first part of this paper we linearize and solve the Van der Pol and Lienard equations with some additional nonlinear terms by the application of a generalized form of Cole-Hopf transformation. We then show that the same transformation can be used to linearize Painleve III equation for certain combinations of its parameters. Finally we linearize new forms of Burger's and related convective equations with higher order nonlinearities.Comment: The paper is a merger of arXiv:1308.5965 and arXiv:1308.0858 with a new section in which I apply the generalized Cole-Hopf transformation to find analytical solutions to Painleve III equatio

A New Approach to Impulsive Rendezvous near Circular Orbit

A new approach is presented for the problem of optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a related characteristic-value function and this related optimization problem can be solved in closed form. The solution of this problem is shown to approach the solution of the original problem in the limit as the boundary conditions approach those of a circular orbit. Using a form of primer-vector theory the problem is formulated in a way that leads to relatively easy calculation of the optimal velocity increments. A certain vector that can easily be calculated from the boundary conditions determines the number of impulses required for solution of the optimization problem and also is useful in the computation of these velocity increments. Necessary and sufficient conditions for boundary conditions to require exactly three nonsingular non-degenerate impulses for solution of the related optimal rendezvous problem, and a means of calculating these velocity increments are presented. If necessary these velocity increments could be calculated from a hand calculator containing trigonometric functions. A simple example of a three-impulse rendezvous problem is solved and the resulting trajectory is depicted.Comment: An expanded version of a paper that appeared in Celest Mech Dyn Ast