# 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*

**.**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

*the included computer programs given in*

**RUN****The**

**n****-body Problem**

**M**

**onograph**

**.**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 (**Galil****e****o**'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, P**art 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 **M****onograph**, **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

*from its Assembly (in the*

**Report****Appendix**

**T**

**o**book), Session of 1856, "

*," Doc. No. 6, by Wm. C.*

**Annual Report of the Quarter-Master and Adjutant-General****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

*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.*

**proxy** Added a five-part free ebook, **A Principia Mathematica**, 698 pages, which contains almost all the mathematical articles from the 1960s through 1984 **Encyclopaedia Britannica**s (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

**.**

**Astronomy**

**Astronomy**