Some theoretical and numerical aspects of the Nbody problem
(2013) In Bachelor's Theses in Mathematical Sciences MATX01 20112Mathematics (Faculty of Sciences)
 Abstract
 The Nbody 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 threebody 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 Nbody 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 threebody 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 fourthorder RungeKutta scheme and StörmerVerlet. 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 Nbody 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 Nbody 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:
http://lup.lub.lu.se/studentpapers/record/4780668
 author
 Gardarsson Myrdal, Kjartan Kari ^{LU}
 supervisor

 Erik Wahlén ^{LU}
 Joachim Hein ^{LU}
 organization
 course
 MATX01 20112
 year
 2013
 type
 M2  Bachelor Degree
 subject
 keywords
 Nbody problem, central configuration, Euler's solution, Lagrange's solution, numerical methods, Euler's method, Heun's method, the classical fourthorder RungeKutta scheme, StörmerVerlet
 publication/series
 Bachelor's Theses in Mathematical Sciences
 report number
 LUNFMA40252013
 ISSN
 16546229
 other publication id
 2013:K12
 language
 English
 id
 4780668
 date added to LUP
 20150618 11:19:03
 date last changed
 20150618 11:19:03
@misc{4780668, abstract = {The Nbody 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 threebody 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 fourthorder RungeKutta scheme and StörmerVerlet. Comparison of accuracy is made using the known solutions discussed in the previous parts of this thesis.}, author = {Gardarsson Myrdal, Kjartan Kari}, issn = {16546229}, keyword = {Nbody problem,central configuration,Euler's solution,Lagrange's solution,numerical methods,Euler's method,Heun's method,the classical fourthorder RungeKutta scheme,StörmerVerlet}, language = {eng}, note = {Student Paper}, series = {Bachelor's Theses in Mathematical Sciences}, title = {Some theoretical and numerical aspects of the Nbody problem}, year = {2013}, }