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Some theoretical and numerical aspects of the N-body problem

Gardarsson Myrdal, Kjartan Kari LU (2013) In Bachelor's Theses in Mathematical Sciences MATX01 20112
Mathematics (Faculty of Sciences)
Abstract
The N-body problem has been studied for many centuries and is still of interest in contemporary science. A lot of effort has gone into solving this problem but it's unlikely that a general solution will be found with the mathematical tools we have today. We review some of the progress that has been made over the centuries in solving it. We take a look at the first integrals, existence of solutions and where singularities can occur. We solve the two body problem and take a look at the special case of central configurations. We find all the possible three-body central configurations, which are known as Euler's and Lagrange's solutions. When analytic solutions are missing it is natural to use numerical methods. We implement and compare four... (More)
The N-body problem has been studied for many centuries and is still of interest in contemporary science. A lot of effort has gone into solving this problem but it's unlikely that a general solution will be found with the mathematical tools we have today. We review some of the progress that has been made over the centuries in solving it. We take a look at the first integrals, existence of solutions and where singularities can occur. We solve the two body problem and take a look at the special case of central configurations. We find all the possible three-body central configurations, which are known as Euler's and Lagrange's solutions. When analytic solutions are missing it is natural to use numerical methods. We implement and compare four numerical solvers for differential equations: Euler's method, Heun's method, the classical fourth-order Runge-Kutta scheme and Störmer-Verlet. Comparison of accuracy is made using the known solutions discussed in the previous parts of this thesis. (Less)
Popular Abstract
The famous physicist and mathematician Sir Isaac Newton formulated the universal gravitation theory about 300 years ago. This theory has allowed scientists to make calculations in order to predict movement of objects under the influence of gravitational force. In astronomy the movement of planets and asteroids can be formulated as a mathematical equation based on Newton's laws of gravitation. The equation describes how a force changes the speed and direction of the moving particles in space. In order to predict a future location one needs to solve this equation. This is referred to as the N-body problem. Still today the general problem is considered unsolved. In this thesis we will consider solutions to special cases of the problem such as... (More)
The famous physicist and mathematician Sir Isaac Newton formulated the universal gravitation theory about 300 years ago. This theory has allowed scientists to make calculations in order to predict movement of objects under the influence of gravitational force. In astronomy the movement of planets and asteroids can be formulated as a mathematical equation based on Newton's laws of gravitation. The equation describes how a force changes the speed and direction of the moving particles in space. In order to predict a future location one needs to solve this equation. This is referred to as the N-body problem. Still today the general problem is considered unsolved. In this thesis we will consider solutions to special cases of the problem such as when there are only two bodies. We will also take a look at numerical methods that can be used when analytical solutions don't exist. (Less)
Please use this url to cite or link to this publication:
author
Gardarsson Myrdal, Kjartan Kari LU
supervisor
organization
course
MATX01 20112
year
type
M2 - Bachelor Degree
subject
keywords
N-body problem, central configuration, Euler's solution, Lagrange's solution, numerical methods, Euler's method, Heun's method, the classical fourth-order Runge-Kutta scheme, Störmer-Verlet
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4025-2013
ISSN
1654-6229
other publication id
2013:K12
language
English
id
4780668
date added to LUP
2015-06-18 11:19:03
date last changed
2015-06-18 11:19:03
@misc{4780668,
  abstract     = {The N-body problem has been studied for many centuries and is still of interest in contemporary science. A lot of effort has gone into solving this problem but it's unlikely that a general solution will be found with the mathematical tools we have today. We review some of the progress that has been made over the centuries in solving it. We take a look at the first integrals, existence of solutions and where singularities can occur. We solve the two body problem and take a look at the special case of central configurations. We find all the possible three-body central configurations, which are known as Euler's and Lagrange's solutions. When analytic solutions are missing it is natural to use numerical methods. We implement and compare four numerical solvers for differential equations: Euler's method, Heun's method, the classical fourth-order Runge-Kutta scheme and Störmer-Verlet. Comparison of accuracy is made using the known solutions discussed in the previous parts of this thesis.},
  author       = {Gardarsson Myrdal, Kjartan Kari},
  issn         = {1654-6229},
  keyword      = {N-body problem,central configuration,Euler's solution,Lagrange's solution,numerical methods,Euler's method,Heun's method,the classical fourth-order Runge-Kutta scheme,Störmer-Verlet},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Theses in Mathematical Sciences},
  title        = {Some theoretical and numerical aspects of the N-body problem},
  year         = {2013},
}