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In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus (although to solve the electron/nucleus 2-body system correctly a quantum mechanical approach must be used). The two-body problem can be re-formulated as two one-body problems, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the ''n''-body problem for ''n'' ≥ 3) cannot be solved in terms of first integrals, except in special cases. == Reduction to two independent, one-body problems == Let x1 and x2 be the vector positions of the two bodies, and ''m''1 and ''m''2 be their masses. The goal is to determine the trajectories x1(''t'') and x2(''t'') for all times ''t'', given the initial positions x1(''t'' = 0) and x2(''t'' = 0) and the initial velocities v1(''t'' = 0) and v2(''t'' = 0). When applied to the two masses, Newton's second law states that : : where F12 is the force on mass 1 due to its interactions with mass 2, and F21 is the force on mass 2 due to its interactions with mass 1. The two dots on top of the x position vectors denote their second derivative, or their acceleration vectors. Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. ''Adding'' equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, ''subtracting'' equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x1(''t'') and x2(''t''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Two-body problem」の詳細全文を読む スポンサード リンク
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