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Difference between revisions of "Principle of least reaction"

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A corollary of Gauss' principle (cf. [[Gauss principle|Gauss principle]]), obtained from the latter by using the equations representing Newton's second law for points of a constrained system (see [[#References|[1]]]). According to the principle of least reaction, for the real motion of a system the quantity
 
A corollary of Gauss' principle (cf. [[Gauss principle|Gauss principle]]), obtained from the latter by using the equations representing Newton's second law for points of a constrained system (see [[#References|[1]]]). According to the principle of least reaction, for the real motion of a system the quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074800/p0748001.png" /></td> </tr></table>
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$$\sum_\nu\frac{R_\nu^2}{2m_\nu}$$
  
is minimal with respect to all motions conceivable in Gauss' sense. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074800/p0748002.png" /> are the reactions of the constraints and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074800/p0748003.png" /> the masses of the points in the system.
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is minimal with respect to all motions conceivable in Gauss' sense. Here $R_\nu$ are the reactions of the constraints and $m_\nu$ the masses of the points in the system.
  
 
====References====
 
====References====

Latest revision as of 20:17, 11 April 2014

A corollary of Gauss' principle (cf. Gauss principle), obtained from the latter by using the equations representing Newton's second law for points of a constrained system (see [1]). According to the principle of least reaction, for the real motion of a system the quantity

$$\sum_\nu\frac{R_\nu^2}{2m_\nu}$$

is minimal with respect to all motions conceivable in Gauss' sense. Here $R_\nu$ are the reactions of the constraints and $m_\nu$ the masses of the points in the system.

References

[1] N.G. Chetaev, "Stability of motion" , Moscow (1965) (In Russian)


Comments

References

[a1] R.B. Lindsay, H. Margenau, "Foundations of physics" , Dover, reprint (1957)
How to Cite This Entry:
Principle of least reaction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principle_of_least_reaction&oldid=31557
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article