# n-Body Problem Closed-Form Solution

## Preface

The n-Body Problem Closed-form Solution Monograph below (is a freebie ebook), now the 4th Edition, is the principal pdf file presented below. Its thrust is the derivation of a closed-form, algebraic, 3D generalization of NEWTON's Law of Universal Gravitation -- which has been transformed (coded) into an easy to use computer algorithms (FORTRAN and BASIC versions) for computing the static gravitational interacting forces for any number of isolated bodies knowing their masses and instantaneous positions. This solution is the classical (static) n-body problem. That static solution can be made time dependent if pass known planetary forces, positions and velocities are used in small, piecewise steps to develop 3D planetary trajectories which are then used to predict, accurately, future planet positions. The problem in the pass was not knowing the coupled, interactive planetary forces. (The latter problem has to be iterated for all nine planets, plus moons, etc., many times looping through the basic algorithm at least nine times for each step, is left to the reader to solve -- it takes up here too much computer space.)

This webpage has been under construction since 2013...and admittedly is somewhat disorganized. You can view and download any pdf file to your personal PC by clicking that file and downloading it. By 8/27/2017, there were more than 10000+ hits on this webpage from people from all over the world -- and counting -- about an average of 7 hit a day. There probably are many, tens-of-thousands more, but not recordable hits because they have turned on their “do not track” buttons. If you have questions, etc., you can contact the author at mathrudiness@gmail.com.

Again: all pdf files are free to download for private use -- which are described in more detail below. Also, free compilers and interpreters for miscellaneous programming languages are available at https://www.thefreecountry.com/compilers/miscellaneous.shtml. These computer tools can be used to RUN the included computer programs given in The n-body Problem Monograph. To that point, there is an example problem which gives INPUT and OUTPUT files illustrating how the two versions, BASIC and FORTRAN, programs work.

One suggestion in this principal Monograph is the electromagnetic energy flux issuing from the atomic nuclei is a cause of gravity. The flux, however weak, doesn't stop at the valence electron, but travels on effecting other atom's electrons, is the cause of gravity. This premise of a mechanism for gravity -- is assumed to be the result of van der WAALS like forces (see pdf file). That is, all the Earth's atom's nuclei radiate electromagnetic energy – and this energy is the energy which holds their electrons in orbit, including the outer-most valance electrons. These nuclei forces don't stop at the valance electrons, but continue weakly, spreading on outward like van der WAALS forces, attracting other electrons, which in turn gives rise to gravity. Hence, it is assumed Earth's mechanism of gravity is via the resulting manifestation of all Earth's nuclei's attractive, diffused electromagnetic flux, extended by-passing forces, which gives rise to gravity.

As a result, Earth's smeared gravitational field pulls on all the electrons attached to all the nuclei of every body equally, regardless of mass, weight, shape, etc.; so (Galileo's Law) that all bodies are pulled down to Earth equally at the same rate if starting from the same distance. Experimental evidence via ROTHEN's experiments, coupled with the observations of van der WAALS' forces, does provide more than a sufficient reason for the above conjecture; i.e., for the mechanism of gravity. See: the n-Body Problem Closed-form Solution for further discussion below. It's algorithm may be used to determine those gravitational interaction forces.

Astronomy, Parts 1, 2, 3, and 4 is a single monograph (divided up into four Parts), mainly serving as technical background, with some historical background given too, to augment the n-body Problem ebook. It has been completely rewritten and now identified as the 4nd Edition . Astronomy, Part 1 and 2, consists of a collection of essays about the Antikythera astronomy machine, Greek astronomy, Aristotle's Physics, Islamic astronomy, Copernicus, Galileo, Bacon versus Descartes, NEWTON, celestial mechanics, and more. Specifically, in the Second section of Part 1, it explains how the Sun's wobble creates, by gravitational forces, a planetoid's translation acceleration. And further, it hypothesizes in this same section that the Sun's powerful Eddy currents are responsible for effecting the Sun's electromagnetic nodal rings, the gravitational force field (potential) wells encircling the Sun, the planets and planetoid are embedded (trapped) in, which, when combined with the Sun's forces of these nodal ring gravitational wells, wherein the planets and planetoids natural gravitational forces and their centrifugal forces, cause orbital stability to be maintained for all the orbiting bodies orbiting the Sun. This phenomenological scenario is described step-by-step in much more detail in the Second section of Part 1. More astronomical history, followed by more celestial mechanics, is presented in Part 2.

See Part 3 and 4 Prefaces of Astronomy for their contents. These Parts contains considerably more technical papers then Parts 1 and 2. For example, Part 4 contains WOODARD's paper on calculating orbits; and gives the essential pages from the book, Astronomical Paper..., Vol. XII, Coordinates of the Five Outer Planets, 1653-2060 (and the book's equations for the coordinates too).

Added to the principal Monograph, n-Body Problem Closed-Form Solution, 2nd Edition, was the "Planet Forces" section, which refers to LIBESKIND's work. HERGET's Computation of Orbits out-of-print textbook was added on 1/21/2019 in two Parts: Part I, Chapters 1 through 5; and Part II (a separate document), Chapters 6 through 8; plus some more references (contained in the Main References section).

Added 10/06/2021 below, and updated (i.e., completed) on 3/23/2022, the uploaded book, Report from Iron Mountain on the Possibility and Desirability of Peace , by Leonard C. LEWIS, is a political satire. Later will be added: Turkey: A Past and A Future, by A. J. TOYNBEE, George H. Doran Co., 1917 (which is long out-of-print; and those existing copies are very rare and few indeed), is TOYNBEE's account about the 1915 genocide of the 1.5 million Turkish Armenians (this account is not in the Blue Book). Also included below is an official State of California Report from its Assembly (in the Appendix To book), Session of 1856, "Annual Report of the Quarter-Master and Adjutant-General," Doc. No. 6, by Wm. C. KIBBE: he list the names of California's State sponsored and paid, self-appointed local Militia Officers effecting the Northern California geocide during the 1850s (and later); and list all of the Federal Government's proxy war materials (via ARMY materials) supplied to California in support of the State of California's genocide efforts. The latter is essentially the proof of the Federal and State sponsored genocide of the Northern California Native Americans.

Added a five-part free ebook, A Principia Mathematica, 698 pages, which contains almost all the mathematical articles from the 1960s through 1984 Encyclopaedia Britannicas (encyclopedias now mostly out-of-print). The latter ebook articles were written mainly by 20-century mathematical experts. To this section too was added articles on logarithm's, tensors, manifolds, and other relevant articles.

Added a page on "Shear Flow" and more technical stuff: the intent is to add hard to find and out-of-print stress-structural articles, and more, for my grandchildren.

n-body_Problem_4thEd.pdf

## Astronomy

Ancient_Astronomy_Pt_1.pdf
Ancient_Astronomy_Pt_2.pdf
Ancient_Astronomy_Pt_3.pdf
Ancient_Astronomy_Pt_4.pdf
Van_Der_WAALS_Forces.pdf
Planetary_Motion_in_Rectrangular_Coord.pdf
HERGET_Master - a.pdf
HERGET_Master - b.pdf
Vagabond _Moons.pdf