|
In the classical central-force problem of classical mechanics, some potential energy functions ''V''(''r'') produce motions or orbits that can be expressed in terms of well-known functions, such as the trigonometric functions and elliptic functions. This article describes these functions and the corresponding solutions for the orbits. ==General problem== The Binet equation for ''u''(φ) can be solved numerically for nearly any central force ''F''(1/''u''). However, only a handful of forces result in formulae for ''u'' in terms of known functions. The solution for φ can be expressed as an integral over ''u'' : A central-force problem is said to be "integrable" if this integration can be solved in terms of known functions. If the force is a power law, i.e., if ''F''(''r'') = α ''r''''n'', then ''u'' can be expressed in terms of circular functions and/or elliptic functions if ''n'' equals 1, -2, -3 (circular functions) and -7, -5, -4, 0, 3, 5, -3/2, -5/2, -1/3, -5/3 and -7/3 (elliptic functions).〔Whittaker, pp. 80–95.〕 If the force is the sum of a inverse quadratic law and a linear term, i.e., if ''F''(''r'') = α ''r''''-2'' + c r, the problem also is solved explicitly in terms of Weierstrass elliptic functions〔Izzo and Biscani〕 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「Exact solutions of classical central-force problems」の詳細全文を読む スポンサード リンク
|