Scattering from rough surfaces - Revision history

en>Logical1004 at 11:44, 12 November 2014

2014-11-12T11:44:30Z

← Older revision		Revision as of 13:44, 12 November 2014
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	~~{{About\|Bonnet's theorem in classical mechanics\|the Bonnet theorem in differential geometry\|Bonnet theorem}}~~		The author's title is Christy. Mississippi is exactly where her house is but her husband desires them to move. What me and my family members love is performing ballet but I've been taking on new issues lately. Credit authorising is how she makes a residing.<br><br>Feel free to visit my webpage telephone psychic ([http://kard.dk/?p=24252 Info])

	~~In [[classical mechanics]], '''Bonnet's theorem''' states that if '~~'~~n'' different [[force]] fields each produce the same geometric orbit (say, an ellipse of given dimensions) albeit with different [[speed]]~~s ~~''v''<sub>1</sub>, ''v''<sub>2</sub>,...,''v''<sub>''n''</sub> at a given point ''P'', then the same orbit will be followed if the speed at point ''P'' equals~~

	~~:<math>~~
	~~v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}~~
	~~</math>~~

	~~This theorem was first derived by [[Adrien-Marie Legendre]] in 1817,<ref>{{cite book \| last = Legendre \| first = A-M \| authorlink = Adrien-Marie Legendre \| year = 1817 \|~~ title ~~= Exercises de Calcul Intégral \| volume = 2 \| pages = 382–3 \|url = http://archive~~.~~org/stream/exercicescalculi02legerich#page/n403/mode/2up \| publisher = Courcier \| location = Paris}}</ref> but it~~ is ~~named after [[Pierre Ossian Bonnet]].~~

	~~==Derivation==~~

	~~The shape of an [[orbit]]~~ is ~~determined only by the [[centripetal force]]s at each point of the orbit, which are the forces acting perpendicular~~ to ~~the orbit~~. ~~By contrast, forces~~ '~~'along'' the orbit change only the speed, but not the direction, of the [[velocity]]~~.

	~~Let the instantaneous radius of curvature at~~ a ~~point ''P'' on the orbit be denoted as ''R''~~. ~~For the ''k''~~<~~sup~~>th<~~/sup~~> ~~force field that produces that orbit, the force normal~~ to ~~the orbit ''F''<sub>''k''</sub> must provide the~~ [~~[centripetal force]]~~

	:~~<math>~~
	~~F_{k} = \frac{m}{R} v_{k}^{2}~~
	</~~math>~~

	~~Adding all these forces together yields the equation~~

	~~:<math>~~
	~~\sum_{k=1}^{n} F_{k} = \frac{m}{R} \sum_{k=1}^{n} v_{k}^{2}~~
	</~~math>~~

	~~Hence, the combined force-field produces the same orbit if the speed at a point ''P'' is set equal to~~

	~~:<math>~~
	~~v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}~~
	</~~math>~~

	~~==References=~~=

	~~{{Reflist\|1}}~~

	~~[[Category:Classical mechanics~~]]


	~~{{classicalmechanics-stub}}~~

en>Monkbot: /* Description */Fix CS1 deprecated date parameter errors

2014-01-29T20:19:05Z

Description: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 22:19, 29 January 2014
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	~~Hi there. Allow me start by introducing~~ the ~~writer~~, ~~her title is Sophia Boon but she never truly liked~~ that ~~title. My spouse doesn~~'~~t like it the way I do but what I really like performing is caving but I don~~'~~t have~~ the ~~time recently. Office supervising is my occupation. Her family lives~~ [~~http:~~//~~ltreme~~.~~com~~/~~index.php?do~~=/~~profile~~-~~127790/info/ psychic readings online~~] ([http://~~chorokdeul~~.~~co.kr~~/~~index.php?document_srl~~=~~324263&mid~~=~~customer21 http:~~/~~/chorokdeul~~.~~co.kr/index.php?document_srl~~=~~324263&mid~~=~~customer21~~]~~) in Alaska~~ but ~~her husband desires them~~ to ~~transfer.~~<br><br>~~Also visit my site: psychic chat online (~~[~~http~~:/~~/checkmates.co.za/index.php?do~~=/~~profile~~-~~56347/info/ http~~:/~~/checkmates.co.za/index.php?do~~=~~/profile~~-~~56347/info/])~~		{{About\|Bonnet's theorem in classical mechanics\|the Bonnet theorem in differential geometry\|Bonnet theorem}}

			In [[classical mechanics]], '''Bonnet's theorem''' states that if ''n'' different [[force]] fields each produce the same geometric orbit (say, an ellipse of given dimensions) albeit with different [[speed]]s ''v''<sub>1</sub>, ''v''<sub>2</sub>,...,''v''<sub>''n''</sub> at a given point ''P'', then the same orbit will be followed if the speed at point ''P'' equals

			:<math>
			v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}
			</math>

			This theorem was first derived by [[Adrien-Marie Legendre]] in 1817,<ref>{{cite book \| last = Legendre \| first = A-M \| authorlink = Adrien-Marie Legendre \| year = 1817 \| title = Exercises de Calcul Intégral \| volume = 2 \| pages = 382–3 \|url = http://archive.org/stream/exercicescalculi02legerich#page/n403/mode/2up \| publisher = Courcier \| location = Paris}}</ref> but it is named after [[Pierre Ossian Bonnet]].

			==Derivation==

			The shape of an [[orbit]] is determined only by the [[centripetal force]]s at each point of the orbit, which are the forces acting perpendicular to the orbit. By contrast, forces ''along'' the orbit change only the speed, but not the direction, of the [[velocity]].

			Let the instantaneous radius of curvature at a point ''P'' on the orbit be denoted as ''R''. For the ''k''<sup>th</sup> force field that produces that orbit, the force normal to the orbit ''F''<sub>''k''</sub> must provide the [[centripetal force]]

			:<math>
			F_{k} = \frac{m}{R} v_{k}^{2}
			</math>

			Adding all these forces together yields the equation

			:<math>
			\sum_{k=1}^{n} F_{k} = \frac{m}{R} \sum_{k=1}^{n} v_{k}^{2}
			</math>

			Hence, the combined force-field produces the same orbit if the speed at a point ''P'' is set equal to

			:<math>
			v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}
			</math>

			==References==

			{{Reflist\|1}}

			[[Category:Classical mechanics]]


			{{classicalmechanics-stub}}

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2012-07-31T14:52:52Z

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Hi there. Allow me start by introducing the writer, her title is Sophia Boon but she never truly liked that title. My spouse doesn't like it the way I do but what I really like performing is caving but I don't have the time recently. Office supervising is my occupation. Her family lives [http://ltreme.com/index.php?do=/profile-127790/info/ psychic readings online] ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21]) in Alaska but her husband desires them to transfer.<br><br>Also visit my site: psychic chat online ([http://checkmates.co.za/index.php?do=/profile-56347/info/ http://checkmates.co.za/index.php?do=/profile-56347/info/])