The powerful method of multipole expansion has found wide utility in classical electromagnetics and quantum-mechanics. In contrast, the gravitational mechanics traditionally have only seen peripheral mentions of the corresponding (mass-density) multipoles in specific applications. In this paper, we develop the general theory of the multipole formalism for the classical two-body gravitational dynamics, a most prevalent subject in planetary (and global Earth) dynamics. We treat two general, well-encountered configurations: (i) the interior “mantle-inner core gravitational (MICG)” type of configuration consisting of concentric bodies; and (ii) the exterior type of configuration of “planet + satellite” separate bodies in the general form of tidal interactions. We derive concise and exact formulas in terms of multipole expansions; by retaining the leading term(s) of relevance, typically up to the quadrupolar terms that include the planetary triaxiality, one can evaluate to the precision desired while bearing in mind that higher subtleties are readily available in the higher multipole terms. The two-body problems that are so formulated range from potential energy, satellite orbit, to tides, librations of sorts and precession/nutation, to planetary rotational normal modes and wobbles. We demonstrate that, via its various symmetry properties, the multipole formalism provides a theoretical unification framework for these gravitational phenomena that are conventionally treated topic-by-topic in textbook literature.

多极展开的强大方法已在经典电磁学和量子力学中得到广泛应用。相比之下，引力力学传统上只在特定应用中看到了相应（质量密度）多极子的外围提及。在本文中，我们为经典双体引力动力学发展了多极形式的一般理论，这是行星（和全球地球）动力学中最普遍的主题。我们处理两种常见的、常见的配置：（i）内部“地幔-内核引力（MICG）”类型的配置，由同心体组成；(ii)外部潮汐相互作用的一般形式中的“行星+卫星”分离体的配置类型。我们根据多极展开推导出简明准确的公式；通过保留相关的主要项，通常包括行星三轴性在内的四极项，人们可以评估所需的精度，同时牢记在更高的多极项中很容易获得更高的微妙之处。如此表述的二体问题范围从势能、卫星轨道到潮汐、各种振动和进动/章动，再到行星旋转正常模式和摆动。我们证明，通过其各种对称特性，