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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gravitational potential ： ウィキペディア英語版 Gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be done by the force of gravity if an object were moved from its location in space to a fixed reference location. It is analogous to the electric potential with mass playing the role of charge. The reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance. In mathematics the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory. ==Potential energy== The gravitational potential (''V'') is the gravitational potential energy (''U'') per unit mass: :$U\; =\; m\; V,$ where ''m'' is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity. If the body has a mass of 1 unit, then the potential energy to be assigned to that body is equal to the gravitational potential. So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity. In some situations, the equations can be simplified by assuming a field that is nearly independent of position. For instance, in daily life, in the region close to the surface of the Earth, the gravitational acceleration can be considered constant. In that case, the difference in potential energy from one height to another is to a good approximation linearly related to the difference in height: :$\backslash Delta\; U\; =\; mg\; \backslash Delta\; h.$ ==Mathematical form== The potential ''V'' of a unit mass ''m'' at a distance ''x'' from a point mass of mass ''M'' can be defined as the work ''W'' done by the gravitational field ''F'' bringing the unit mass in from infinity to that point: :$V(x)\; =\; \backslash frac\; =\; \backslash frac\; \backslash int\backslash limits\_^\; F\; \backslash \; dx\; =\; \backslash frac\; \backslash int\backslash limits\_^\; \backslash frac\; dx\; =\; -\backslash frac,$ where ''G'' is the gravitational constant. The potential has units of energy per unit mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as ''x'' tends to infinity, it approaches zero. The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is :$\backslash mathbf\; =\; -\backslash frac\; \backslash mathbf\; =\; -\backslash frac\; \backslash hat\}$ is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law: :$|\backslash mathbf|\; =\; \backslash frac.$ The potential associated with a mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x1, ..., x''n'' and have masses ''m''1, ..., ''m''''n'', then the potential of the distribution at the point x is :$V(\backslash mathbf)\; =\; \backslash sum\_^n\; -\backslash frac|\}.$ If the mass distribution is given as a mass measure ''dm'' on three-dimensional Euclidean space R3, then the potential is the convolution of −G/|r| with ''dm''. In good cases this equals the integral :$V(\backslash mathbf)\; =\; -\backslash int\_\; \backslash frac|\}\backslash ,dm(\backslash mathbf),$ where |x − r| is the distance between the points x and r. If there is a function ''ρ''(r) representing the density of the distribution at r, so that '', where ''dv''(r) is the Euclidean volume element, then the gravitational potential is the volume integral :$V(\backslash mathbf)\; =\; -\backslash int\_\; \backslash frac|\}\backslash ,\backslash rho(\backslash mathbf)dv(\backslash mathbf).$ If ''V'' is a potential function coming from a continuous mass distribution ''ρ''(r), then ''ρ'' can be recovered using the Laplace operator, Δ: :$\backslash rho(\backslash mathbf)\; =\; \backslash frac\backslash Delta\; V(\backslash mathbf).$ This holds pointwise whenever ''ρ'' is continuous and is zero outside of a bounded set. In general, the mass measure ''dm'' can be recovered in the same way if the Laplace operator is taken in the sense of distributions. As a consequence, the gravitational potential satisfies Poisson's equation. See also Green's function for the three-variable Laplace equation and Newtonian potential. 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「Gravitational potential」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.