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py_gmm G. Pellegrini [20] Python + Fortran A Python + Fortran 90 implementation of the Generalized Multiparticle Mie method, especially suited for plasmonics and near field computation. 2017 CELES A. Egel, L. Pattelli and G. Mazzamuto [21] MATLAB + CUDA Running on NVIDIA GPUs, with high performance for many spheres. 2020 QPMS M. Nečada [22] C ...
Here F N is the Newtonian force, m is the object's (gravitational) mass, a is its acceleration, μ (x) is an as-yet unspecified function (called the interpolating function), and a 0 is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires
In the spherical-coordinates example above, there are no cross-terms; the only nonzero metric tensor components are g rr = 1, g θθ = r 2 and g φφ = r 2 sin 2 θ. In his special theory of relativity, Albert Einstein showed that the distance ds between two spatial points is not constant, but depends on the motion of the observer.
the vector r is the position of one body relative to the other; r, v, and in the case of an elliptic orbit, the semi-major axis a, are defined accordingly (hence r is the distance) μ = Gm 1 + Gm 2 = μ 1 + μ 2, where m 1 and m 2 are the masses of the two bodies. Then: for circular orbits, rv 2 = r 3 ω 2 = 4π 2 r 3 /T 2 = μ
where F is the gravitational force acting between two objects, m 1 and m 2 are the masses of the objects, r is the distance between the centers of their masses, and G is the gravitational constant. The first test of Newton's law of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry ...
There are many useful features of the effective potential, such as . To find the radius of a circular orbit, simply minimize the effective potential with respect to , or equivalently set the net force to zero and then solve for : = After solving for , plug this back into to find the maximum value of the effective potential .
The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R 4 with coordinates (,,,) and the metric = + + + =. These coordinates actually cover all of R 4 . The flat space metric (or Minkowski metric ) is often denoted by the symbol η and is the metric used in special relativity .
For two pairwise interacting point particles, the gravitational potential energy is the work that an outside agent must do in order to quasi-statically bring the masses together (which is therefore, exactly opposite the work done by the gravitational field on the masses): = = where is the displacement vector of the mass, is gravitational force acting on it and denotes scalar product.