Table of Contents
Table of Contents				2
Abstract				4
Introduction				5
1.0	Theory of the N-Body Problem			7
	1.1	The Background and History of the N-body Problem		7
	1.2	Newtonian Physics		8
		1.2.1	An Example of Newtonian Physics With Just Two Bodies	9
		1.2.2	An Example of Newtonian Physics With Four Bodies	10
		1.2.3	A Detailed Example of Newtonian Physics With Two Bodies	10
	1.3	Development of Methods to Solve the N-body Problem		17
	1.4	What Is the "Best" Method?		20
		1.4.1	The Efficiency of a Method	20
		1.4.2	The Accuracy of a Method	21
	1.5	Types of Deviations In Star Movement		23
		1.5.1	Gaining or Losing Energy	23
		1.5.2	Distortion	23
		1.5.3	Perihelion Shift	23
		1.5.4	Overstep Phenomenon	24
		1.5.5	Slingshot Effect	25
		1.5.6	Program Bugs	26
	1.6	Speed-up Techniques		27
	1.7	The Error Term		27
2.0	Types of N-body ODE Integration Methods			30
	2.1	One Step Methods		30
		2.1.1	Euler's Method (-m euler1)	30
		2.1.2	Taylor Series (not implemented)	31
		2.1.3	Runge-Kutta's Method (-m rk4)	31
	2.2	Multistep Methods		33
		2.2.1	Modified Taylor Series Method (-m taylor3)	34
		2.2.2	Adam-Bashford's Method (-m ab7 and -m ab4)	34
	2.3	Predictor-Corrector Methods		35
		2.3.1	Modified Euler's Method (not implemented)	36
		2.3.2	Adam-Moulton's Method (-m am7)	36
	2.4	Other formulas (Mid-point method) (none are implemented)		37
	2.5	The Gragg-Bulirsch-Stoer Method (-m gpemce8)		38
3.0	Variable Step Size N-Body ODE Integration Methods			40
	3.1	Method of Changing the Step Size		41
	3.2	Method of Changing Orders		41
	3.3	Method of Changing the ODE Integration Method		41
	3.4	Method of Internal Checking		42
4.0	Efficient Force Function Evaluation Methods			43
	4.1	Floating Point Rounding Errors and the Force Function		44
	4.2	Distant Stars Can Be Grouped Together		45
	4.3	Closely Packed Stars Can Be Moved at the Same Time		46
	4.4	Replace the Force Functions With a Potential Mesh		47
	4.5	Creating Tree Structures For Evaluating the Force Function		48
	4.6	Hybrid Force Evaluation Methods		49
5.0	Analysis of the ODE Integration Methods			51
6.0	Conclusion			55
References				56

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