The \u201cN-body problem\u201d is one that has plagued physicists since Newton\u2019s times. Newtonian\/Classical gravity is defined by\u00a0<\/span> We begin this analysis of gravitational interactions using the formula for gravitational force in conjunction with Newton\u2019s Second Law.<\/p>\n From this, we get This velocity rate of changes allow us to analyze the position of all of the celestial bodies in our system, since Method<\/strong><\/p>\n Equations of Motion<\/em><\/p>\n To model the n-body problem, I start by creating a set of input variables. This allows to me to create a very general program which should run whether we examine a system of 1 celestial body, 10, or 100. The program allows the input of number of bodies in the system, mass, initial position, initial speed, and initial direction of motion for each body.\u00a0<\/span><\/p>\n This is the code for collecting information from the user of the program. This excerpt of code applies to all objects other than the first.<\/p><\/div>\n Once the inputs have been obtained, we can proceed with the Euler-Cromer method to analyze the motions of however many bodies.\u00a0<\/span><\/p>\n The Euler-Cromer method uses a known rate of change for some desired variable, and adds the product of the rate of change with some step of the independent variable, to approximate some behavior. The distinction between the Euler-Cromer and Euler methods lies the fact that Euler-Cromer uses a newly calculated rate of change to calculate the new position. The Euler-Cromer method is effective for 2nd order differential equations, such as this case, where the Euler method fails. In the case of the n-body problem, this method takes the following form: However, the above form only accounts for an object interacting with one other object. To address this, I summed the acceleration due to each object in my code.<\/p>\n This excerpt of code includes the summation of the gravitational forces and their influence on each object’s velocity.<\/p><\/div>\n Plotting\/Animation<\/em><\/p>\n To understand how these interacting bodies behave, I examined their motion through graphing their paths and animating their motion. This plots demanded very little in terms of new plotting\/animation methods, but rather demanded a lot of finessing, not all of which paid off. The drawnow command in conjunction with “hold on” and “hold off” produced the animation plots, whereas the same command only with “hold on” provided the plotted paths.<\/p>\n This is the code which plots the path of each object as we cycle through each time step.<\/p><\/div>\n Here is the code which plots the animation of each object’s motion. The primary difference between this code excerpt than the previous one is the use of “hold off”.<\/p><\/div>\n Results<\/strong><\/p>\n It is hard to comprehensively discuss the results of this program because its input-based nature leaves infinite potential results. To examine how systems of n number of celestial bodies behave, I will discuss several cases as n increases.<\/p>\n Case 1:<\/em><\/p>\n This case is the most simple and best understood in case, in which two bodies interact gravitationally. The initial conditions of this case are as follows:<\/p>\n Graphing the path of each of these objects yields the following result:<\/p>\n Above is the path of each individual object in Case 1, on a plot in units AU.<\/p><\/div>\n This (messy) version of the previous plot is simply to provide clarity for each individual path of the objects.<\/p><\/div>\n Below is a clip of the animation for this case.<\/p>\n This is a 15 second clip of the animation of this case’s behavior. I could not include the entire animation because the video would not work, and GIFs are limited in length. If the animation is not playing automatically, click to view.<\/p><\/div>\n The results for Case 1 are precisely what one would expect from such a case. Object 1, the more massive object, experiences smaller acceleration due to the gravitational force from Object 2 than vice-versa, yielding a much smaller spiral of motion.<\/p>\n Case 2:<\/em><\/p>\n Maintaining the same conditions for the first two objects, we then consider adding a third into the system and see how it impacts the motion of the objects within the system. The initial conditions are as follows: The results of Case 2 are graphed below.<\/p>\n The paths of gravitationally interacting objects 1,2, and 3 are shown above, on a graph of units AU.<\/p><\/div>\n This graph is simply the plot above, with sketched color coding to clarify the paths of individual objects.<\/p><\/div>\n An animation of this case is shown below:<\/p>\n This GIF is 15 second clip of the animation of Case 2. If not playing automatically, click to view.<\/p><\/div>\n Case 2’s results are less predictable than the results of Case 1, because of the additional object included. The most massive object (Object 1) still changes position the least, with each object of a lower mass moving more significantly than the last. This is the first case whose results we could not have calculated analytically. We can understand the trajectory of an object influenced by more than one object if one or more of those objects are held still, but the fact that each and every object experiences a gravitational force due to every other object eliminates the possibility of an analytical solution.<\/p>\n Case 3:<\/em><\/p>\n Adding a fourth body to our system, we will further examine how motion evolves as n increases. The initial conditions of this case are listed below.\u00a0 This case yields the following results:<\/p>\n On a plot with axis units AU, objects 1,2,3, and 4’s paths are shown.<\/p><\/div>\n The same plot as above is shown here with sketched on color-coding the clarify the paths of each individual object.<\/p><\/div>\n The animation for this case is shown below.<\/p>\n <\/p>\n![\"\"](\"http:\/\/pages.vassar.edu\/magnes\/files\/2019\/05\/Screen-Shot-2019-05-13-at-12.06.10-AM.png\")
<\/a>The behavior of two objects, interacting through Newtonian gravity, is well understood, easy to analyze by hand, and easy to model. However, as we increase the number of objects in our system, the problem immediately becomes far more complicated. People have studied the n-body problem for centuries, searching for a solution. In the late 19th century, the king of Sweden offered a prize to anyone who could find a general solution to the n-body problem. This challenge was left unmet, and the award went to Poincar\u00e9 for an unrelated accomplishment. In more recent times, understanding the n-body problem has become far more feasible with technological advancements and the associated rise of computational physics. While there is no analytical solution to the n-body problem for 3 or more bodies, we can use computational methods to model and better understand gravitational interactions between celestial bodies and their resulting behavior.<\/p>\n<\/a>To analyze this behavior, we must examine the system component by component. This results in the rate of change for velocity<\/a><\/p>\n<\/a><\/p>\n<\/a><\/a><\/p>\n<\/a><\/a><\/a><\/a><\/p>\n<\/a><\/a><\/a><\/a><\/p>\n<\/a><\/a><\/a><\/a><\/p>\n<\/a><\/a>