66 research outputs found
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
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