en>ChrisGualtieri: General Fixes using AWB

2013-10-25T23:25:25Z

General Fixes using AWB

← Older revision		Revision as of 01:25, 26 October 2013
Line 1:		Line 1:
	~~Hayley Rake~~ is ~~how I am named even though I will not truly like currently being termed like that~~. ~~One particular~~ of the ~~items I appreciate most~~ is ~~to accumulate greeting playing cards but~~ [~~http://search~~.~~about~~.~~com/?q~~=I+~~haven%27t I haven't~~] ~~built~~ a ~~dime with it. Interviewing~~ is what I do for a living but I prepare on altering it. Many years back we moved to Arkansas and I love just about every working day living listed here. I am not superior at webdesign but you could possibly want to verify my web page: ~~http~~://~~smartcamp.pl~~/wp-~~content~~/~~languages~~/~~cache~~/<br><br>~~Also visit my blog post~~ :: ~~nike air max~~; [http://~~smartcamp~~.pl/~~wp-content~~/~~languages/cache/ http~~:~~//smartcamp.pl/wp~~-~~content/languages/cache/~~],		The '''Einstein–Infeld–Hoffmann equations of motion''', jointly derived by [[Albert Einstein]], [[Leopold Infeld]] and [[Banesh Hoffmann]], are the [[differential equation\|differential]] [[equations of motion]] describing the approximate [[dynamics (physics)\|dynamics]] of a system of point-like masses due to their mutual gravitational interactions, including [[general relativity\|general relativistic]] effects. It uses a first-order [[post-Newtonian expansion]] and thus is valid in the limit where the velocities of the bodies are small compared to the speed of light and where the gravitational fields affecting them are correspondingly weak.

			Given a system of ''N'' bodies, labelled by indices ''A'' = 1, ..., ''N'', the barycentric acceleration vector of body ''A'' is given by:

			: <math>
			\begin{align}
			\vec{a}_A & = \sum_{B \not = A} \frac{G m_B \vec{n}_{BA}}{r_{AB}^2} \\
			& {} \quad{} + \frac{1}{c^2} \sum_{B \not = A}
			\frac{G m_B \vec{n}_{BA}}{r_{AB}^2}
			\left[ v_A^2+2v_B^2 - 4( \vec{v}_A \cdot \vec{v}_B) - \frac{3}{2} ( \vec{n}_{AB} \cdot \vec{v}_B)^2 \right. \\
			& {} \qquad {} \left. {} -
			4 \sum_{C \not = A} \frac{G m_C}{r_{AC}} -
			\sum_{C \not = B} \frac{G m_C}{r_{BC}} + \frac{1}{2}( (\vec{x}_B-\vec{x}_A) \cdot \vec{a}_B ) \right] \\
			& {}\quad{} + \frac{1}{c^2} \sum_{B \not = A} \frac{G m_B}{r_{AB}^2}\left[\vec{n}_{AB}\cdot(4\vec{v}_A-3\vec{v}_B)\right](\vec{v}_A-\vec{v}_B) \\
			& {} \quad {} + \frac{7}{2c^2} \sum_{B \not = A}{ \frac{G m_B \vec{a}_B }{r_{AB}}} + O (c^{-4})
			\end{align}
			</math>

			where:
			:<math>\vec{x}_A</math> is the barycentric position vector of body A
			:<math>\vec{v}_A=d\vec{x}_A/dt</math> is the barycentric velocity vector of body A
			:<math>\vec{a}_A=d^2\vec{x}_A/dt^2</math> is the barycentric acceleration vector of body A
			:<math>r_{AB}=\|\vec{x}_A-\vec{x}_B\|</math> is the coordinate distance between bodies A and B
			:<math>\vec{n}_{AB}=(\vec{x}_A-\vec{x}_B)/r_{AB}</math> is the unit vector pointing from body B to body A
			:<math>m_A</math> is the mass of body A.
			:<math>c</math> is the [[speed of light]]
			:<math>G</math> is the [[gravitational constant]]
			:and the [[big O notation]] is used to indicate that terms of order ''c''<sup>−4</sup> or beyond have been omitted.

			The coordinates used here are [[harmonic coordinate condition\|harmonic]]. The first term on the right hand side is the Newtonian gravitational acceleration at ''A''; in the limit as ''c'' → ∞, one recovers Newton's law of motion.

			The acceleration of a particular body depends on the accelerations of all the other bodies. Since the quantity on the left hand side also appears in the right hand side, this system of equations must be solved iteratively. In practice, using the Newtonian acceleration instead of the true acceleration provides sufficient accuracy.<ref>Standish, Williams. Orbital Ephemerides of the Sun, Moon, and Planets, Pg 4. http://iau-comm4.jpl.nasa.gov/XSChap8.pdf</ref>

			==References==
			{{reflist}}

			==Further reading==
			*{{cite journal \|last=Einstein \|first=A. \|last2=Infeld \|first2=L. \|last3=Hoffmann \|first3=B. \|year=1938 \|title=The Gravitational Equations and the Problem of Motion \|journal=[[Annals of Mathematics]] \|series=Second series \|volume=39 \|issue=1 \|pages=65–100 \|jstor=1968714 \|bibcode = 1938AnMat..39...65E }}
			*{{cite book \|last=Kovalevsky \|first=Jean \|last2=Seidelmann \|first2=P. Kenneth \|year=2004 \|title=Fundamentals of Astrometry \|location=New York \|publisher=Cambridge University Press \|page=173 \|isbn=0521642167 }}

			{{Einstein}}

			{{DEFAULTSORT:Einstein-Infeld-Hoffmann equations}}
			[[Category:Differential equations]]
			[[Category:General relativity]]
			[[Category:Albert Einstein]]

en>D.Lazard: wikilink for smooth

2012-01-04T13:55:16Z

wikilink for smooth

New page

Hayley Rake is how I am named even though I will not truly like currently being termed like that. One particular of the items I appreciate most is to accumulate greeting playing cards but [http://search.about.com/?q=I+haven%27t I haven't] built a dime with it. Interviewing is what I do for a living but I prepare on altering it. Many years back we moved to Arkansas and I love just about every working day living listed here. I am not superior at webdesign but you could possibly want to verify my web page: http://smartcamp.pl/wp-content/languages/cache/<br><br>Also visit my blog post :: nike air max; [http://smartcamp.pl/wp-content/languages/cache/ http://smartcamp.pl/wp-content/languages/cache/],

Motivic zeta function - Revision history

en>ChrisGualtieri: General Fixes using AWB

en>D.Lazard: wikilink for smooth