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## Orbit perturbations in Kepler's Problem

### Katarzyna Zdon

#### Abstract

The thesis concerns the area of Mechanics - the science about forces and movement of physical bodies. The thesis outlines the description of movements of planets in the Solar system with special emphasis on perturbations of orbits in the classical problem of Kepler. The approach is based on a mathematical description of the movement by ordinary differential equations. I use algebraic methods, but also present numerical simulations of selected problems using the R programming environment. In the first chapter, based on the available literature, I consider Kepler's problem with related ideas and concepts. I present definitions of the vector of position, velocity, acceleration and derivative of the vector. I introduce Newton's second law of motion and the resulting properties of motion in the gravitational field. In a formal way, I show thata physical body under the influence of gravity moves in a plane which contains the central body, with the segment joining them sweeping equal areas in equal times. I formulate the principle of conservation of angular momentum and show that the gravitational field is a conservative field, in which the total energy is constant. I consider the geometry of the component forces in the three-dimensional Cartesian system as well as the distribution of velocity and acceleration vectors in the radial and transversal direction. In the same chapter, I also show how to derive the equation for the trajectory of a body moving in a central force field (the classic Kepler problem) with a discussion of constants of integration and the relation to energy. In addition, I consider numerical integration of equations of motion using Euler's method. I also generalize Kepler's problem to the problem of the motion of two bodies, following the derivation used by Stefan Banach - a representative of the Lvov school of mathematics. In the next chapter, based on the literature, I introduce the problem of the circular restricted three-body problem. I present the general problem of motion equations in a system which rotates at constant speed. Next, I show the derivation of a certain first integral (Jacobi integral). The last chapter describes a specific, uncomplicated case of the three body motion, where we are dealing with the system referred to in the literature as a 'syzygy'. In that system we consider three bodies in a co-linear configuration, centrally symmetric about the central body. After stating the problem, I arrive at exact differential equations for the perturbed motion as well as the linearized equation with respect to the small mass of the orbiting bodies, checking the consistency of both approaches. This part of the thesis is the result of my own research.
Diploma type
Engineer's / Bachelor of Science
Diploma type
Bachelor thesis
Author
Katarzyna Zdon (FMIS) Katarzyna Zdon,, Faculty of Mathematics and Information Science (FMIS)
Title in Polish
Supervisor
Grzegorz Świątek (FMIS/DFE) Grzegorz Świątek,, Department of Functional Equations (FMIS/DFE)Faculty of Mathematics and Information Science (FMIS)
Certifying unit
Faculty of Mathematics and Information Science (FMIS)
Affiliation unit
Department of Functional Equations (FMIS/DFE)
Study subject / specialization
, Matematyka
Language
(pl) Polish
Status
Finished
Defense Date
21-09-2016
Issue date (year)
2016
Reviewers
Przemysław Górka (FMIS/DPDE) Przemysław Górka,, Department of Partial Differential Equations (FMIS/DPDE)Faculty of Mathematics and Information Science (FMIS) Grzegorz Świątek (FMIS/DFE) Grzegorz Świątek,, Department of Functional Equations (FMIS/DFE)Faculty of Mathematics and Information Science (FMIS)
Keywords in Polish
równania różniczkowe zwyczajne, prawa dynamiki Newtona, prawa Keplera, problem Keplera, zagadnienie dwóch ciał, linearyzacja, równanie na wariację, równania ruchu, ograniczone zagadnienie trzech ciał
Keywords in English
ordinary differential equations, Newton's laws, Kepler's laws, Kepler's problem, two-body problem, linearized, variation equation, motion equations, limited three-body problem
Abstract in Polish
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Identyfikator pracy APD: 1226

Uniform Resource Identifier
urn:pw-repo:WUTcf364083ad9e490d8a2350e5800363ef